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Use chemistry problems as a tool for mastering chemistry concepts. Some of these examples show using formulas while others include lists of examples.

Acids, Bases, and pH Chemistry Problems

Learn about acids and bases. See how to calculate pH, pOH, K a , K b , pK a , and pK b .

• Practice calculating pH.
• Get example pH, pK a , pK b , K a , and K b calculations.
• Get examples of amphoterism.

Atomic Structure Problems

Learn about atomic mass, the Bohr model, and the part of the atom.

• Practice identifying atomic number, mass number, and atomic mass.
• Get examples showing ways to find atomic mass.
• Use Avogadro’s number and find the mass of a single atom .
• Review the Bohr model of the atom.
• Find the number of valence electrons of an element’s atom.

Chemical Bonds

Learn how to use electronegativity to determine whether atoms form ionic or covalent bonds. See chemistry problems drawing Lewis structures.

• Identify ionic and covalent bonds.
• Learn about ionic compounds and get examples.
• Practice identifying ionic compounds.
• Get examples of binary compounds.
• Learn about covalent compounds and their properties.
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• Practice drawing Lewis structures.
• Practice calculating bond energy.

Chemical Equations

Practice writing and balancing chemical equations.

• Learn the steps of balancing equations.
• Practice balancing chemical equations (practice quiz).
• Get examples finding theoretical yield.
• Practice calculating percent yield.
• Learn to recognize decomposition reactions.
• Practice recognizing synthesis reactions.
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• Find the mole ratio between chemical species in an equation.

Concentration and Solutions

Learn how to calculate concentration and explore chemistry problems that affect chemical concentration, including freezing point depression, boiling point elevation, and vapor pressure elevation.

• Get example concentration calculations in several units.
• Practice calculating normality (N).
• Practice calculating molality (m).
• Explore example molarity (M) calculations.
• Get examples of colligative properties of solutions.
• See the definition and examples of saturated solutions.
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• Get examples of miscible and immiscible liquids.

Error Calculations

Learn about the types of error and see worked chemistry example problems.

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• Practice absolute and relative error calculations.
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• See how to find standard deviation.
• Calculate mean, median, and mode.
• Review the difference between accuracy and precision.

Equilibrium Chemistry Problems

Learn about Le Chatelier’s principle, reaction rates, and equilibrium.

• Solve activation energy chemistry problems.
• Review factors that affect reaction rate.
• Practice calculating the van’t Hoff factor.

Practice chemistry problems using the gas laws, including Raoult’s law, Graham’s law, Boyle’s law, Charles’ law, and Dalton’s law of partial pressures.

• Calculate vapor pressure.
• Solve Avogadro’s law problems.
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• Solve combined gas law problems.
• Solve Gay-Lussac’s law problems.

Some chemistry problems ask you identify examples of states of matter and types of mixtures. While there are any chemical formulas to know, it’s still nice to have lists of examples.

• Practice density calculations.
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• Get the definition and examples of solids.
• Get the definition and examples of gases.
• See the definition and examples of liquids.
• Learn what melting point is and get a list of values for different substances.
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• See how to calculate specific volume of a gas.
• Get examples of physical properties of matter.
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• Review the states of matter.

Molecular Structure Chemistry Problems

See chemistry problems writing chemical formulas. See examples of monatomic and diatomic elements.

• Practice empirical and molecular formula problems.
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• Get examples of the monatomic elements.
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• Calculate the number of atoms and molecules in a drop of water.

Nomenclature

Practice chemistry problems naming ionic compounds, hydrocarbons, and covalent compounds.

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Nuclear Chemistry

These chemistry problems involve isotopes, nuclear symbols, half-life, radioactive decay, fission, fusion.

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Periodic Table

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• Know the trends in the periodic table.
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Physical Chemistry

Explore thermochemistry and physical chemistry, including enthalpy, entropy, heat of fusion, and heat of vaporization.

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• See example activation energy calculations.

Spectroscopy and Quantum Chemistry Problems

See chemistry problems involving the interaction between light and matter.

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Stoichiometry Chemistry Problems

Practice chemistry problems balancing formulas for mass and charge. Learn about reactants and products.

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• Calculate percent yield.
• Learn how to assign oxidation numbers.
• Get the definition and examples of reactants in chemistry.
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Physics Problems with Solutions

Displacement and distance: problems with solutions.

Problems with detailed solutions on displacement and distance of moving objects.

Problem 3: An object moves from point A to B to C to D and finally to A along the circle shown in the figure below. a) Find the distance covered by the moving object. b) Find the magnitude and direction of the displacement of the object.

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Solved Speed, Velocity, and Acceleration Problems

Simple problems on speed, velocity, and acceleration with descriptive answers are presented for the AP Physics 1 exam and college students. In each solution, you can find a brief tutorial. 

Speed and velocity Problems: 

Problem (1): What is the speed of a rocket that travels $8000\,{\rm m}$ in $13\,{\rm s}$?

Solution : Speed is defined in physics  as the total distance divided by the elapsed time,  so the rocket's speed is \[\text{speed}=\frac{8000}{13}=615.38\,{\rm m/s}\]

Problem (2): How long will it take if you travel $400\,{\rm km}$ with an average speed of $100\,{\rm m/s}$?

Solution : Average speed is the ratio of the total distance to the total time. Thus, the elapsed time is \begin{align*} t&=\frac{\text{total distance}}{\text{average speed}}\\ \\ &=\frac{400\times 10^{3}\,{\rm m}}{100\,{\rm m/s}}\\ \\ &=4000\,{\rm s}\end{align*} To convert it to hours, it must be divided by $3600\,{\rm s}$ which gives $t=1.11\,{\rm h}$.

Problem (3): A person walks $100\,{\rm m}$ in $5$ minutes, then $200\,{\rm m}$ in $7$ minutes, and finally $50\,{\rm m}$ in $4$ minutes. Find its average speed. 

Solution : First find its total distance traveled ($D$) by summing all distances in each section, which gets $D=100+200+50=350\,{\rm m}$. Now, by definition of average speed, divide it by the total time elapsed $T=5+7+4=16$ minutes.

But keep in mind that since the distance is in SI units, so the time traveled must also be in SI units, which is $\rm s$. Therefore, we have\begin{align*}\text{average speed}&=\frac{\text{total distance} }{\text{total time} }\\ \\ &=\frac{350\,{\rm m}}{16\times 60\,{\rm s}}\\ \\&=0.36\,{\rm m/s}\end{align*}

Problem (4): A person walks $750\,{\rm m}$ due north, then $250\,{\rm m}$ due east. If the entire walk takes $12$ minutes, find the person's average velocity. 

Solution : Average velocity , $\bar{v}=\frac{\Delta x}{\Delta t}$, is displacement divided by the elapsed time. Displacement is also a vector that obeys the addition vector rules. Thus, in this velocity problem, add each displacement to get the total displacement . 

In the first part, displacement is $\Delta x_1=750\,\hat{j}$ (due north) and in the second part $\Delta x_2=250\,\hat{i}$ (due east). The total displacement vector is $\Delta x=\Delta x_1+\Delta x_2=750\,\hat{i}+250\,\hat{j}$ with magnitude of  \begin{align*}|\Delta x|&=\sqrt{(750)^{2}+(250)^{2}}\\ \\&=790.5\,{\rm m}\end{align*} In addition, the total elapsed time is $t=12\times 60$ seconds. Therefore, the magnitude of the average velocity is \[\bar{v}=\frac{790.5}{12\times 60}=1.09\,{\rm m/s}\]

Problem (5): An object moves along a straight line. First, it travels at a velocity of $12\,{\rm m/s}$ for $5\,{\rm s}$ and then continues in the same direction with $20\,{\rm m/s}$ for $3\,{\rm s}$. What is its average speed?

Solution: Average velocity is displacement divided by elapsed time, i.e., $\bar{v}\equiv \frac{\Delta x_{tot}}{\Delta t_{tot}}$.

Here, the object goes through two stages with two different displacements, so add them to find the total displacement. Thus,\[\bar{v}=\frac{x_1 + x_2}{t_1 +t_2}\] Again, to find the displacement, we use the same equation as the average velocity formula, i.e., $x=vt$. Thus, displacements are obtained as $x_1=v_1\,t_1=12\times 5=60\,{\rm m}$ and $x_2=v_2\,t_2=20\times 3=60\,{\rm m}$. Therefore, we have \begin{align*} \bar{v}&=\frac{x_1+x_2}{t_1+t_2}\\ \\&=\frac{60+60}{5+3}\\ \\&=\boxed{15\,{\rm m/s}}\end{align*}

Problem (6): A plane flies the distance between two cities in $1$ hour and $30$ minutes with a velocity of $900\,{\rm km/h}$. Another plane covers that distance at $600\,{\rm km/h}$. What is the flight time of the second plane?

Solution: first find the distance between two cities using the average velocity formula $\bar{v}=\frac{\Delta x}{\Delta t}$ as below \begin{align*} x&=vt\\&=900\times 1.5\\&=1350\,{\rm km}\end{align*} where we wrote one hour and a half minutes as $1.5\,\rm h$. Now use again the same kinematic equation above to find the time required for another plane \begin{align*} t&=\frac xv\\ \\ &=\frac{1350\,\rm km}{600\,\rm km/h}\\ \\&=2.25\,{\rm h}\end{align*} Thus, the time for the second plane is $2$ hours and $0.25$ of an hour, which converts to minutes as $2$ hours and ($0.25\times 60=15$) minutes.

Problem (7): To reach a park located south of his jogging path, Henry runs along a 15-kilometer route. If he completes the journey in 1.5 hours, determine his speed and velocity.

Solution:  Henry travels his route to the park without changing direction along a straight line. Therefore, the total distance traveled in one direction equals the displacement, i.e, \[\text{distance traveled}=\Delta x=15\,\rm km\]Velocity is displacement divided by the time of travel \begin{align*} \text{velocity}&=\frac{\text{displacement}}{\text{time of travel}} \\\\ &=\frac{15\,\rm km}{1.5\,\rm h} \\\\ &=\boxed{10\,\rm km/h}\end{align*} and by definition, its average speed is \begin{align*} \text{speed}&=\frac{\text{distance covered}}{\text{time interval}}\\\\&=\frac{15\,\rm km}{1.5\,\rm h}\\\\&=\boxed{10\,\rm km/h}\end{align*} Thus, Henry's velocity is $10\,\rm km/h$ to the south, and its speed is $10\,\rm km/h$. As you can see, speed is simply a positive number, with units but velocity specifies the direction in which the object is moving. 

Problem (8): In 15 seconds, a football player covers the distance from his team's goal line to the opposing team's goal line and back to the midway point of the field having 100-yard-length. Find, (a) his average speed, and (b) the magnitude of the average velocity.

Solution:  The total length of the football field is $100$ yards or in meters, $L=91.44\,\rm m$. Going from one goal's line to the other and back to the midpoint of the field takes $15\,\rm s$ and covers a distance of $D=100+50=150\,\rm yd$. 

Distance divided by the time of travel gets the average speed, \[\text{speed}=\frac{150\times 0.91}{15}=9.1\,\rm m/s\] To find the average velocity, we must find the displacement of the player between the initial and final points. 

The initial point is her own goal line and her final position is the midpoint of the field, so she has displaced a distance of $\Delta x=50\,\rm yd$ or $\Delta x=50\times 0.91=45.5\,\rm m$. Therefore, her velocity is calculated as follows \begin{align*} \text{velocity}&=\frac{\text{displacement}}{\text{time elapsed}} \\\\ &=\frac{45.5\,\rm m}{15\,\rm s} \\\\&=\boxed{3.03\quad \rm m/s}\end{align*} Contrary to the previous problem, here the motion is not in one direction, hence, the displacement is not equal to the distance traveled. Accordingly, the average speed is not equal to the magnitude of the average velocity.

Problem (9): You begin at a pillar and run towards the east (the positive $x$ direction) for $250\,\rm m$ at an average speed of $5\,\rm m/s$. After that, you run towards the west for $300\,\rm m$ at an average speed of $4\,\rm m/s$ until you reach a post. Calculate (a) your average speed from pillar to post, and (b) your average velocity from pillar to post. 

Solution : First, you traveled a distance of $L_1=250\,\rm m$ toward east (or $+x$ direction) at $5\,\rm m/s$. Time of travel in this route is obtained as follows \begin{align*} t_1&=\frac{L_1}{v_1}\\\\ &=\frac{250}{5}\\\\&=50\,\rm s\end{align*} Likewise, traveling a distance of $L_2=300\,\rm m$ at $v_2=4\,\rm m/s$ takes \[t_2=\frac{300}{4}=75\,\rm s\]  (a) Average speed is defined as the distance traveled (or path length) divided by the total time of travel \begin{align*} v&=\frac{\text{path length}}{\text{time of travel}} \\\\ &=\frac{L_1+L_2}{t_1+t_2}\\\\&=\frac{250+300}{50+75} \\\\&=4.4\,\rm m/s\end{align*} Therefore, you travel between these two pillars in $125\,\rm s$ and with an average speed of $4.4\,\rm m/s$. 

(b) Average velocity requires finding the displacement between those two points. In the first case, you move $250\,\rm m$ toward $+x$ direction, i.e., $L_1=+250\,\rm m$. Similarly, on the way back, you move $300\,\rm m$ toward the west ($-x$ direction) or $L_2=-300\,\rm m$. Adding these two gives us the total displacement between the initial point and the final point, \begin{align*} L&=L_1+L_2 \\\\&=(+250)+(-300) \\\\ &=-50\,\rm m\end{align*} The minus sign indicates that you are generally displaced toward the west. 

Finally, the average velocity is obtained as follows: \begin{align*} \text{average velocity}&=\frac{\text{displacement}}{\text{time of travel}} \\\\ &=\frac{-50}{125} \\\\&=-0.4\,\rm m/s\end{align*} A negative average velocity indicating motion to the left along the $x$-axis. 

This speed problem better makes it clear to us the difference between average speed and average speed. Unlike average speed, which is always a positive number, the average velocity in a straight line can be either positive or negative. 

Problem (10): What is the average speed for the round trip of a car moving uphill at 40 km/h and then back downhill at 60 km/h? 

Solution : Assuming the length of the hill to be $L$, the total distance traveled during this round trip is $2L$ since $L_{up}=L_{down}=L$. However, the time taken for going uphill and downhill was not provided. We can write them in terms of the hill's length $L$ as $t=\frac L v$. 

Applying the definition of average speed gives us \begin{align*} v&=\frac{\text{distance traveled}}{\text{total time}} \\\\ &=\frac{L_{up}+L_{down}}{t_{up}+t_{down}} \\\\ &=\cfrac{2L}{\cfrac{L}{v_{up}}+\cfrac{L}{v_{down}}} \end{align*} By reorganizing this expression, we obtain a formula that is useful for solving similar problems in the AP Physics 1 exams. \[\text{average speed}=\frac{2v_{up} \times v_{down}}{v_{up}+v_{down}}\] Substituting the numerical values into this, yields \begin{align*} v&=\frac{2(40\times 60)}{40+60} \\\\ &=\boxed{48\,\rm m/s}\end{align*} What if we were asked for the average velocity instead? During this round trip, the car returns to its original position, and thus its displacement, which defines the average velocity, is zero. Therefore, \[\text{average velocity}=0\,\rm m/s\]

Acceleration Problems

Problem (9): A car moves from rest to a speed of $45\,\rm m/s$ in a time interval of $15\,\rm s$. At what rate does the car accelerate? 

Solution : The car is initially at rest, $v_1=0$, and finally reaches $v_2=45\,\rm m/s$ in a time interval $\Delta t=15\,\rm s$. Average acceleration is the change in velocity, $\Delta v=v_2-v_1$, divided by the elapsed time $\Delta t$, so \[\bar{a}=\frac{45-0}{15}=\boxed{3\,\rm m/s^2} \] 

Problem (10): A car moving at a velocity of $15\,{\rm m/s}$, uniformly slows down. It comes to a complete stop in $10\,{\rm s}$. What is its acceleration?

Solution:  Let the car's uniform velocity be $v_1$ and its final velocity $v_2=0$.   Average acceleration is the difference in velocities divided by the time taken, so we have: \begin{align*}\bar{a}&=\frac{\Delta v}{\Delta t}\\\\&=\frac{v_2-v_1}{\Delta t}\\\\&=\frac{0-15}{10}\\\\ &=\boxed{-1.5\,{\rm m/s^2}}\end{align*}The minus sign indicates the direction of the acceleration vector, which is toward the $-x$ direction.

Problem (11): A car moves from rest to a speed of $72\,{\rm km/h}$ in $4\,{\rm s}$. Find the acceleration of the car.

Solution: Known: $v_1=0$, $v_2=72\,{\rm km/h}$, $\Delta t=4\,{\rm s}$.  Average acceleration is defined as the difference in velocities divided by the time interval between those points \begin{align*}\bar{a}&=\frac{v_2-v_1}{t_2-t_1}\\\\&=\frac{20-0}{4}\\\\&=5\,{\rm m/s^2}\end{align*} In above, we converted $\rm km/h$ to the SI unit of velocity ($\rm m/s$) as \[1\,\frac{km}{h}=\frac {1000\,m}{3600\,s}=\frac{10}{36}\, \rm m/s\] so we get \[72\,\rm km/h=72\times \frac{10}{36}=20\,\rm m/s\] 

Problem (12): A race car accelerates from an initial velocity of $v_i=10\,{\rm m/s}$ to a final velocity of $v_f = 30\,{\rm m/s}$ in a time interval of $2\,{\rm s}$. Determine its average acceleration.

Solution:  A change in the velocity of an object $\Delta v$ over a time interval $\Delta t$ is defined as an average acceleration. Known: $v_i=10\,{\rm m/s}$, $v_f = 30\,{\rm m/s}$, $\Delta t=2\,{\rm s}$. Applying definition of average acceleration, we get \begin{align*}\bar{a}&=\frac{v_f-v_i}{\Delta t}\\&=\frac{30-10}{2}\\&=10\,{\rm m/s^2}\end{align*}

Problem (13): A motorcycle starts its trip along a straight line with a velocity of $10\,{\rm m/s}$ and ends with $20\,{\rm m/s}$ in the opposite direction in a time interval of $2\,{\rm s}$. What is the average acceleration of the car?

Solution:  Known: $v_i=10\,{\rm m/s}$, $v_f=-20\,{\rm m/s}$, $\Delta t=2\,{\rm s}$, $\bar{a}=?$. Using average acceleration definition we have \begin{align*}\bar{a}&=\frac{v_f-v_i}{\Delta t}\\\\&=\frac{(-20)-10}{2}\\\\ &=\boxed{-15\,{\rm m/s^2}}\end{align*}Recall that in the definition above, velocities are vector quantities. The final velocity is in the opposite direction from the initial velocity so a negative must be included.

Problem (14): A ball is thrown vertically up into the air by a boy. After $4$ seconds, it reaches the highest point of its path. How fast does the ball leave the boy's hand?

Solution : At the highest point, the ball has zero speed, $v_2=0$. It takes the ball $4\,\rm s$ to reach that point. In this problem, our unknown is the initial speed of the ball, $v_1=?$. Here, the ball accelerates at a constant rate of $g=-9.8\,\rm m/s^2$ in the presence of gravity.

When the ball is tossed upward, the only external force that acts on it is the gravity force. 

Using the average acceleration formula $\bar{a}=\frac{\Delta v}{\Delta t}$ and substituting the numerical values into this, we will have \begin{gather*} \bar{a}=\frac{\Delta v}{\Delta t} \\\\ -9.8=\frac{0-v_1}{4} \\\\ \Rightarrow \boxed{v_1=39.2\,\rm m/s} \end{gather*} Note that $\Delta v=v_2-v_1$. 

Problem (15): A child drops crumpled paper from a window. The paper hit the ground in $3\,\rm s$. What is the velocity of the crumpled paper just before it strikes the ground? 

Solution : The crumpled paper is initially in the child's hand, so $v_1=0$. Let its speed just before striking be $v_2$. In this case, we have an object accelerating down in the presence of gravitational force at a constant rate of $g=-9.8\,\rm m/s^2$. Using the definition of average acceleration, we can find $v_2$ as below \begin{gather*} \bar{a}=\frac{\Delta v}{\Delta t} \\\\ -9.8=\frac{v_2-0}{3} \\\\ \Rightarrow v_2=3\times (-9.8)=\boxed{-29.4\,\rm m/s} \end{gather*} The negative shows us that the velocity must be downward, as expected!

Problem (16): A car travels along the $x$-axis for $4\,{\rm s}$ at an average velocity of $10\,{\rm m/s}$ and $2\,{\rm s}$ with an average velocity of $30\,{\rm m/s}$ and finally $4\,{\rm s}$ with an average velocity $25\,{\rm m/s}$. What is its average velocity across the whole path?

Solution: There are three different parts with different average velocities. Assume each trip is done in one dimension without changing direction. Thus, displacements associated with each segment are the same as the distance traveled in that direction and is calculated as below: \begin{align*}\Delta x_1&=v_1\,\Delta t_1\\&=10\times 4=40\,{\rm m}\\ \\ \Delta x_2&=v_2\,\Delta t_2\\&=30\times 2=60\,{\rm m}\\ \\ \Delta x_3&=v_3\,\Delta t_3\\&=25\times 4=100\,{\rm m}\end{align*}Now use the definition of average velocity, $\bar{v}=\frac{\Delta x_{tot}}{\Delta t_{tot}}$, to find it over the whole path\begin{align*}\bar{v}&=\frac{\Delta x_{tot}}{\Delta t_{tot}}\\ \\&=\frac{\Delta x_1+\Delta x_2+\Delta x_3}{\Delta t_1+\Delta t_2+\Delta t_3}\\ \\&=\frac{40+60+100}{4+2+4}\\ \\ &=\boxed{20\,{\rm m/s}}\end{align*}

Problem (17): An object moving along a straight-line path. It travels with an average velocity $2\,{\rm m/s}$ for $20\,{\rm s}$ and $12\,{\rm m/s}$ for $t$ seconds. If the total average velocity across the whole path is $10\,{\rm m/s}$, then find the unknown time $t$.

Solution: In this velocity problem, the whole path $\Delta x$ is divided into two parts $\Delta x_1$ and $\Delta x_2$ with different average velocities and times elapsed, so the total average velocity across the whole path is obtained as \begin{align*}\bar{v}&=\frac{\Delta x}{\Delta t}\\\\&=\frac{\Delta x_1+\Delta x_2}{\Delta t_1+\Delta t_2}\\\\&=\frac{\bar{v}_1\,t_1+\bar{v}_2\,t_2}{t_1+t_2}\\\\10&=\frac{2\times 20+12\times t}{20+t}\\\Rightarrow t&=80\,{\rm s}\end{align*}

Note : whenever a moving object, covers distances $x_1,x_2,x_3,\cdots$ in $t_1,t_2,t_3,\cdots$ with constant or average velocities $v_1,v_2,v_3,\cdots$ along a straight-line without changing its direction, then its total average velocity across the whole path is obtained by one of the following formulas

• Distances and times are known:\[\bar{v}=\frac{x_1+x_2+x_3+\cdots}{t_1+t_2+t_3+\cdots}\]
• Velocities and times are known: \[\bar{v}=\frac{v_1\,t_1+v_2\,t_2+v_3\,t_3+\cdots}{t_1+t_2+t_3+\cdots}\]
• Distances and velocities are known:\[\bar{v}=\frac{x_1+x_2+x_3+\cdots}{\frac{x_1}{v_1}+\frac{x_2}{v_2}+\frac{x_3}{v_3}+\cdots}\]

Problem (18): A car travels one-fourth of its path with a constant velocity of $10\,{\rm m/s}$, and the remaining with a constant velocity of $v_2$. If the total average velocity across the whole path is $16\,{\rm m/s}$, then find the $v_2$?

Solution: This is the third case of the preceding note. Let the length of the path be $L$ so \begin{align*}\bar{v}&=\frac{x_1+x_2}{\frac{x_1}{v_1}+\frac{x_2}{v_2}}\\\\16&=\frac{\frac 14\,L+\frac 34\,L}{\frac{\frac 14\,L}{10}+\frac{\frac 34\,L}{v_2}}\\\\\Rightarrow v_2&=20\,{\rm m/s}\end{align*}

Problem (19): An object moves along a straight-line path. It travels for $t_1$ seconds with an average velocity $50\,{\rm m/s}$ and $t_2$ seconds with a constant velocity of $25\,{\rm m/s}$. If the total average velocity across the whole path is $30\,{\rm m/s}$, then find the ratio $\frac{t_2}{t_1}$?

Solution: the velocities and times are known, so we have \begin{align*}\bar{v}&=\frac{v_1\,t_1+v_2\,t_2}{t_1+t_2}\\\\30&=\frac{50\,t_1+25\,t_2}{t_1+t_2}\\\\ \Rightarrow \frac{t_2}{t_1}&=4\end{align*} 

Read more related articles:  

Kinematics Equations: Problems and Solutions

Position vs. Time Graphs

Velocity vs. Time Graphs

In the following section, some sample AP Physics 1 problems on acceleration are provided.

Problem (20): An object moves with constant acceleration along a straight line. If its velocity at instant of $t_1 = 3\,{\rm s}$ is $10\,{\rm m/s}$ and at the moment of $t_2 = 8\,{\rm s}$ is $20\,{\rm m/s}$, then what is its initial speed?

Solution: Let the initial speed at time $t=0$ be $v_0$. Now apply average acceleration definition in the time intervals $[t_0,t_1]$ and $[t_0,t_2]$ and equate them.\begin{align*}\text{average acceleration}\ \bar{a}&=\frac{\Delta v}{\Delta t}\\\\\frac{v_1 - v_0}{t_1-t_0}&=\frac{v_2-v_0}{t_2-t_0}\\\\ \frac{10-v_0}{3-0}&=\frac{20-v_0}{8-0}\\\\ \Rightarrow v_0 &=4\,{\rm m/s}\end{align*} In the above, $v_1$ and $v_2$ are the velocities at moments $t_1$ and $t_2$, respectively. 

Problem (21): For $10\,{\rm s}$, the velocity of a car that travels with a constant acceleration, changes from $10\,{\rm m/s}$ to $30\,{\rm m/s}$. How far does the car travel?

Solution: Known: $\Delta t=10\,{\rm s}$, $v_1=10\,{\rm m/s}$ and $v_2=30\,{\rm m/s}$. 

Method (I) Without computing the acceleration: Recall that in the case of constant acceleration, we have the following kinematic equations for average velocity and displacement:\begin{align*}\text{average velocity}:\,\bar{v}&=\frac{v_1+v_2}{2}\\\text{displacement}:\,\Delta x&=\frac{v_1+v_2}{2}\times \Delta t\\\end{align*}where $v_1$ and $v_2$ are the velocities in a given time interval. Now we have \begin{align*} \Delta x&=\frac{v_1+v_2}{2}\\&=\frac{10+30}{2}\times 10\\&=200\,{\rm m}\end{align*}

Method (II) with computing acceleration: Using the definition of average acceleration, first determine it as below \begin{align*}\bar{a}&=\frac{\Delta v}{\Delta t}\\\\&=\frac{30-10}{10}\\\\&=2\,{\rm m/s^2}\end{align*} Since the velocities at the initial and final points of the problem are given so use the below time-independent kinematic equation to find the required displacement \begin{align*} v_2^{2}-v_1^{2}&=2\,a\Delta x\\\\ (30)^{2}-(10)^{2}&=2(2)\,\Delta x\\\\ \Rightarrow \Delta x&=\boxed{200\,{\rm m}}\end{align*}

Problem (22): A car travels along a straight line with uniform acceleration. If its velocity at the instant of $t_1=2\,{\rm s}$ is $36\,{\rm km/s}$ and at the moment $t_2=6\,{\rm s}$ is $72\,{\rm km/h}$, then find its initial velocity (at $t_0=0$)?

Solution: Use the equality of definition of average acceleration $a=\frac{v_f-v_i}{t_f-t_i}$ in the time intervals $[t_0,t_1]$ and $[t_0,t_2]$ to find the initial velocity as below \begin{align*}\frac{v_2-v_0}{t_2-t_0}&=\frac{v_1-v_0}{t_1-t_0}\\\\ \frac{20-v_0}{6-0}&=\frac{10-v_0}{2-0}\\\\ \Rightarrow v_0&=\boxed{5\,{\rm m/s}}\end{align*}

All these kinematic problems on speed, velocity, and acceleration are easily solved by choosing an appropriate kinematic equation. Keep in mind that these motion problems in one dimension are of the uniform or constant acceleration type. Projectiles are also another type of motion in two dimensions with constant acceleration.

Author:   Dr. Ali Nemati

Date Published: 9/6/2020

Updated: Jun 28,  2023

© 2015 All rights reserved. by Physexams.com

7 Science Fair Projects that Solve Problems

• August 5, 2023

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Science fair projects that solve problems are a great way for students to test their interest and aptitude for a career in STEM (science-technology-engineering-math). But they shouldn’t choose just any old topic. To make the most of the opportunity, try to focus on projects with real-world applications. This will give them hands-on experience directly related to a good-paying job field, like  engineering .

With planning and hard work, the right science fair project might bump up a student’s chances for a scholarship or a trip to one of the science competitions sponsored by the Society for Science .

Do your students need help sketching the experimental set-up for a science fair presentation? Check out these resources:

• No-Prep Worksheets – How to Draw like an Engineer and Isometric Drawing
• 3D Isometric Drawing and Design for Middle School
• My Engineering Draw & Write Journal for Kids : 48 Fun Drawing and Writing Prompts to Learn about the Engineering Design Process.

Don’t get me wrong — creating foaming volcanoes or diagramming the human circulatory system are fun and classic ideas for a science fair project. But unless your student plans to go to med school or major in geology, these typical projects won’t do much to advance his or her future career. Far more practical engineering jobs will be available in the 21st century.

In this post you’ll find seven problem-solving science fair projects gleaned from the Education.com website. They provide simple, but realistic, introductions to real-world careers in electronics, robotics & automation, and construction engineering.

For more help with choosing a science fair topic, setting up your experiment, collecting and analyzing the data, and presenting your results, visit NASA’s video page on How to do a Science Fair Project .

Solving problems in Smart Technology

Consider the hottest topic in industry today – Smart Manufacturing, or Industry 4.0, sometimes called the Industrial Internet of Things (IIOT). Industry 4.0 is just one facet of the global push towards Smart Cities, Smart Homes, and Smart Agriculture.

All these concepts center on wireless connectivity between machines using cellular networks. So, for Smart Homes, this means your utilities, fridge, lights, security, HVAC, and other systems would be connected through an app on your smartphone. From there you can track and control these systems to keep your home safe and comfortable, while reducing water and energy use.

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For Industry 4.0, companies are connecting the machines used in their manufacturing and power generation plants at different locations around the world. On top of that, they are creating “digital twins” of each machine, which are 3D animated computer models of the machines.

The idea is to collect real-time data from each machine and then use that data, along with artificial intelligence (AI), machine vision, and even virtual reality simulations, to:

• Design new products
• Predict when a machine will need maintenance BEFORE something goes wrong
• Optimize the output of the machines and harmonize them to work together

Solving problems in Robotics

Another major topic in industry is  robotics and automation . Automation means that machines are programmed to perform tasks without human help. Some robots are standalone, “service” robots, like the Roomba. Others, like robotic arms in factories and warehouses, pick and place items to be processed.

The more human-friendly “collaborative” robots can improve human capacity and are safe to work around. Put together, these technologies allow some manufacturing plants to run “lights out,” without any human input for days.

Robots are boosting agriculture, both in planting and harvesting fields and in packaging food. With Smart Agriculture technology, farmers collect data in their fields with mobile apps applying artificial intelligence (AI) software to reduce fertilizer needs and optimize water use.

Help students sketch their experimental set-up for science fair presentations with these resources: No-Prep Worksheets – How to Draw like an Engineer and Isometric Drawing 3D Isometric Drawing and Design for Middle School My Engineering Draw & Write Journal for Kids : 48 Fun Drawing and Writing Prompts to Learn about the Engineering Design Process.

Solving Engineering Problems

Most science fair projects on the internet seem to focus on the basic sciences, like biology and chemistry. But in light of the skills gap we are now experiencing between the available job force and manufacturing industry requirements, I believe engineering-focused science fair projects that solve problems in Industry 4.0, robotics, automation, and construction may be better choices for building up tomorrow’s workforce.

Here are 7 science fair project ideas that focus on solving problems:

1. cell phone dead zones science fair project.

https://www.education.com/science-fair/article/cell-phone-dead-zones/

Students learn how wireless networks work, find dead zones where wireless signals are lost, and determine ways to reduce these zones – important preparation for students who hope to work on Smart Homes, Smart Factories, Smart Cities, or Smart Agriculture.

2. App development science fair project

https://www.education.com/science-fair/article/iphone-application-design/

An app on a phone or tablet can be an interactive game, a navigational device, a business software package, or just about anything else you can imagine. This project allows you to get a head start in the growing app design field by designing your own app for popular smartphones.

3. Smoke detector science fair project

https://www.education.com/science-fair/article/smoke-detectors-working/?source=related_materials&order=2

Sensors of all kinds solve problems for smart technologies and robotics engineering. Sensors can detect motion, gases, light, heat, and other changes in the environment to allow robots to avoid collisions or Smart Homes to detect a fire, for example. This project compares the effectiveness of two types of sensors in a smoke detector.

4. Faraday’s experiment science fair project

https://www.education.com/science-fair/article/faraday-experiment-current-generated-magnet/

Electric currents create their own magnetic fields, and the movement of magnets induces , or creates, current in a wire. Motors and generators use magnetic movement to create current and send electricity to do useful work to power machines. In this lab, you will recreate Michael Faraday’s famous experiment by building a solenoid  (a coil of wire) and experiment with moving magnets to produce current.

5 & 6. EMFs science fair projects

https://www.education.com/science-fair/article/smart-card-electromagnetic-fields/

https://www.education.com/science-fair/article/EMF-affect-us/

Radio Frequency Identification (RFID) is an electronic technology used in credit cards, ID Cards, and theft prevention systems, as well as in manufacturing, warehousing and shipping products. The first project measures the electromagnetic fields (EMFs) given off by various RFID transmitters, which may have harmful effects on people. The second project looks directly at how EMFs can affect us physically.

7. Rust prevention science fair project

https://www.education.com/science-fair/article/bust-that-rust/

Metals rust, and that can be a big problem when it comes to bridges, buildings, cars, and any object exposed to air and water. This project examines the process of oxidation (not just rust) that ultimately breaks down every physical object and looks at ways to prevent that from happening.

For more problem-solving science fair project ideas, follow the STEM-Inspirations Science Fair Projects board on Pinterest.

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15 Common Problem-Solving Interview Questions

In an interview for a big tech company, I was asked if I’d ever resolved a fight — and the exact way I went about handling it. I felt blindsided, and I stammered my way through an excuse of an answer.

It’s a familiar scenario to fellow technical job seekers — and one that risks leaving a sour taste in our mouths. As candidate experience becomes an increasingly critical component of the hiring process, recruiters need to ensure the problem-solving interview questions they prepare don’t dissuade talent in the first place. 

Interview questions designed to gauge a candidate’s problem-solving skills are more often than not challenging and vague. Assessing a multifaceted skill like problem solving is tricky — a good problem solver owns the full solution and result, researches well, solves creatively and takes action proactively. 

It’s hard to establish an effective way to measure such a skill. But it’s not impossible.

We recommend taking an informed and prepared approach to testing candidates’ problem-solving skills . With that in mind, here’s a list of a few common problem-solving interview questions, the science behind them — and how you can go about administering your own problem-solving questions with the unique challenges of your organization in mind.

Key Takeaways for Effective Problem-Solving Interview Questions

• Problem solving lies at the heart of programming. 
• Testing a candidate’s problem-solving skills goes beyond the IDE. Problem-solving interview questions should test both technical skills and soft skills.
• STAR, SOAR and PREP are methods a candidate can use to answer some non-technical problem-solving interview questions.
• Generic problem-solving interview questions go a long way in gauging a candidate’s fit. But you can go one step further by customizing them according to your company’s service, product, vision, and culture. 

Technical Problem-Solving Interview Question Examples

Evaluating a candidates’ problem-solving skills while using coding challenges might seem intimidating. The secret is that coding challenges test many things at the same time — like the candidate’s knowledge of data structures and algorithms, clean code practices, and proficiency in specific programming languages, to name a few examples.

Problem solving itself might at first seem like it’s taking a back seat. But technical problem solving lies at the heart of programming, and most coding questions are designed to test a candidate’s problem-solving abilities.

Here are a few examples of technical problem-solving questions:

1. Mini-Max Sum  

This well-known challenge, which asks the interviewee to find the maximum and minimum sum among an array of given numbers, is based on a basic but important programming concept called sorting, as well as integer overflow. It tests the candidate’s observational skills, and the answer should elicit a logical, ad-hoc solution.

2. Organizing Containers of Balls  

This problem tests the candidate’s knowledge of a variety of programming concepts, like 2D arrays, sorting and iteration. Organizing colored balls in containers based on various conditions is a common question asked in competitive examinations and job interviews, because it’s an effective way to test multiple facets of a candidate’s problem-solving skills.

3. Build a Palindrome

This is a tough problem to crack, and the candidate’s knowledge of concepts like strings and dynamic programming plays a significant role in solving this challenge. This problem-solving example tests the candidate’s ability to think on their feet as well as their ability to write clean, optimized code.

4. Subarray Division

Based on a technique used for searching pairs in a sorted array ( called the “two pointers” technique ), this problem can be solved in just a few lines and judges the candidate’s ability to optimize (as well as basic mathematical skills).

5. The Grid Search 

This is a problem of moderate difficulty and tests the candidate’s knowledge of strings and searching algorithms, the latter of which is regularly tested in developer interviews across all levels.

Common Non-Technical Problem-Solving Interview Questions 

Testing a candidate’s problem-solving skills goes beyond the IDE . Everyday situations can help illustrate competency, so here are a few questions that focus on past experiences and hypothetical situations to help interviewers gauge problem-solving skills.

1. Given the problem of selecting a new tool to invest in, where and how would you begin this task? 

Key Insight : This question offers insight into the candidate’s research skills. Ideally, they would begin by identifying the problem, interviewing stakeholders, gathering insights from the team, and researching what tools exist to best solve for the team’s challenges and goals. 

2. Have you ever recognized a potential problem and addressed it before it occurred? 

Key Insight: Prevention is often better than cure. The ability to recognize a problem before it occurs takes intuition and an understanding of business needs. 

3. A teammate on a time-sensitive project confesses that he’s made a mistake, and it’s putting your team at risk of missing key deadlines. How would you respond?

Key Insight: Sometimes, all the preparation in the world still won’t stop a mishap. Thinking on your feet and managing stress are skills that this question attempts to unearth. Like any other skill, they can be cultivated through practice.

4. Tell me about a time you used a unique problem-solving approach. 

Key Insight: Creativity can manifest in many ways, including original or novel ways to tackle a problem. Methods like the 10X approach and reverse brainstorming are a couple of unique approaches to problem solving. 

5. Have you ever broken rules for the “greater good?” If yes, can you walk me through the situation?

Key Insight: “Ask for forgiveness, not for permission.” It’s unconventional, but in some situations, it may be the mindset needed to drive a solution to a problem.

6. Tell me about a weakness you overcame at work, and the approach you took. 

Key Insight: According to Compass Partnership , “self-awareness allows us to understand how and why we respond in certain situations, giving us the opportunity to take charge of these responses.” It’s easy to get overwhelmed when faced with a problem. Candidates showing high levels of self-awareness are positioned to handle it well.

7. Have you ever owned up to a mistake at work? Can you tell me about it?

Key Insight: Everybody makes mistakes. But owning up to them can be tough, especially at a workplace. Not only does it take courage, but it also requires honesty and a willingness to improve, all signs of 1) a reliable employee and 2) an effective problem solver.

8. How would you approach working with an upset customer?

Key Insight: With the rise of empathy-driven development and more companies choosing to bridge the gap between users and engineers, today’s tech teams speak directly with customers more frequently than ever before. This question brings to light the candidate’s interpersonal skills in a client-facing environment.

9. Have you ever had to solve a problem on your own, but needed to ask for additional help? How did you go about it? 

Key Insight: Knowing when you need assistance to complete a task or address a situation is an important quality to have while problem solving. This questions helps the interviewer get a sense of the candidate’s ability to navigate those waters. 

10. Let’s say you disagree with your colleague on how to move forward with a project. How would you go about resolving the disagreement?

Key Insight: Conflict resolution is an extremely handy skill for any employee to have; an ideal answer to this question might contain a brief explanation of the conflict or situation, the role played by the candidate and the steps taken by them to arrive at a positive resolution or outcome. 

Strategies for Answering Problem-Solving Questions

If you’re a job seeker, chances are you’ll encounter this style of question in your various interview experiences. While problem-solving interview questions may appear simple, they can be easy to fumble — leaving the interviewer without a clear solution or outcome. 

It’s important to approach such questions in a structured manner. Here are a few tried-and-true methods to employ in your next problem-solving interview.

1. Shine in Interviews With the STAR Method

S ituation, T ask, A ction, and R esult is a great method that can be employed to answer a problem-solving or behavioral interview question. Here’s a breakdown of these steps:

• Situation : A good way to address almost any interview question is to lay out and define the situation and circumstances. 
• Task : Define the problem or goal that needs to be addressed. Coding questions are often multifaceted, so this step is particularly important when answering technical problem-solving questions.
• Action : How did you go about solving the problem? Try to be as specific as possible, and state your plan in steps if you can.
• Result : Wrap it up by stating the outcome achieved. 

2. Rise above difficult questions using the SOAR method

A very similar approach to the STAR method, SOAR stands for S ituation, O bstacle, A ction, and R esults .

• Situation: Explain the state of affairs. It’s important to steer clear of stating any personal opinions in this step; focus on the facts.
• Obstacle: State the challenge or problem you faced.
• Action: Detail carefully how you went about overcoming this obstacle.
• Result: What was the end result? Apart from overcoming the obstacle, did you achieve anything else? What did you learn in the process? 

3. Do It the PREP Way

Traditionally used as a method to make effective presentations, the P oint, R eason, E xample, P oint method can also be used to answer problem-solving interview questions.  

• Point : State the solution in plain terms. 
• Reasons: Follow up the solution by detailing your case — and include any data or insights that support your solution. 
• Example: In addition to objective data and insights, drive your answer home by contextualizing the solution in a real-world example.
• Point : Reiterate the solution to make it come full circle.

How to Customize Problem-Solving Interview Questions 

Generic problem-solving interview questions go a long way in gauging a candidate’s skill level, but recruiters can go one step further by customizing these problem-solving questions according to their company’s service, product, vision, or culture. 

Here are some tips to do so:

• Break down the job’s responsibilities into smaller tasks. Job descriptions may contain ambiguous responsibilities like “manage team projects effectively.” To formulate an effective problem-solving question, envision what this task might look like in a real-world context and develop a question around it.  
• Tailor questions to the role at hand. Apart from making for an effective problem-solving question, it gives the candidate the impression you’re an informed technical recruiter. For example, an engineer will likely have attended many scrums. So, a good question to ask is: “Suppose you notice your scrums are turning unproductive. How would you go about addressing this?” 
• Consider the tools and technologies the candidate will use on the job. For example, if Jira is the primary project management tool, a good problem-solving interview question might be: “Can you tell me about a time you simplified a complex workflow — and the tools you used to do so?”
• If you don’t know where to start, your company’s core values can often provide direction. If one of the core values is “ownership,” for example, consider asking a question like: “Can you walk us through a project you owned from start to finish?” 
• Sometimes, developing custom content can be difficult even with all these tips considered. Our platform has a vast selection of problem-solving examples that are designed to help recruiters ask the right questions to help nail their next technical interview.

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Discover Frequently Asked Physics Questions and Their Answers

• Two charges of # -7 C # and # 4 C # are at points # (4, 7 ,-8) # and # ( -1 , 3, -8 )#, respectively. Assuming that both coordinates are in meters, what is the force between the two points?
• Two charges of # -1 C # and # 4 C # are at points # (-2 ,6,-8 ) # and # ( 2,-4,1 )#, respectively. Assuming that both coordinates are in meters, what is the force between the two points?
• A charge of #4 C# is passing through points A and B on a circuit. If the charge's electric potential changes from #35 J# to #9 J#, what is the voltage between points A and B?
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• A charge of #24 C# passes through a circuit every #6 s#. If the circuit can generate #12 W# of power, what is the circuit's resistance?
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• An electric toy car with a mass of #5 kg# is powered by a motor with a voltage of #12 V# and a current supply of #1 A#. How long will it take for the toy car to accelerate from rest to #3 m/s#?
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• Two charges of # 6 C # and # -7 C# are positioned on a line at points # -7 # and # 8 #, respectively. What is the net force on a charge of # 6 C# at # 2 #?
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Top 20 Problem Solving Interview Questions (Example Answers Included)

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By Mike Simpson

When candidates prepare for interviews, they usually focus on highlighting their leadership, communication, teamwork, and similar crucial soft skills . However, not everyone gets ready for problem-solving interview questions. And that can be a big mistake.

Problem-solving is relevant to nearly any job on the planet. Yes, it’s more prevalent in certain industries, but it’s helpful almost everywhere.

Regardless of the role you want to land, you may be asked to provide problem-solving examples or describe how you would deal with specific situations. That’s why being ready to showcase your problem-solving skills is so vital.

If you aren’t sure who to tackle problem-solving questions, don’t worry, we have your back. Come with us as we explore this exciting part of the interview process, as well as some problem-solving interview questions and example answers.

What Is Problem-Solving?

When you’re trying to land a position, there’s a good chance you’ll face some problem-solving interview questions. But what exactly is problem-solving? And why is it so important to hiring managers?

Well, the good folks at Merriam-Webster define problem-solving as “the process or act of finding a solution to a problem.” While that may seem like common sense, there’s a critical part to that definition that should catch your eye.

What part is that? The word “process.”

In the end, problem-solving is an activity. It’s your ability to take appropriate steps to find answers, determine how to proceed, or otherwise overcome the challenge.

Being great at it usually means having a range of helpful problem-solving skills and traits. Research, diligence, patience, attention-to-detail , collaboration… they can all play a role. So can analytical thinking , creativity, and open-mindedness.

But why do hiring managers worry about your problem-solving skills? Well, mainly, because every job comes with its fair share of problems.

While problem-solving is relevant to scientific, technical, legal, medical, and a whole slew of other careers. It helps you overcome challenges and deal with the unexpected. It plays a role in troubleshooting and innovation. That’s why it matters to hiring managers.

How to Answer Problem-Solving Interview Questions

Okay, before we get to our examples, let’s take a quick second to talk about strategy. Knowing how to answer problem-solving interview questions is crucial. Why? Because the hiring manager might ask you something that you don’t anticipate.

Problem-solving interview questions are all about seeing how you think. As a result, they can be a bit… unconventional.

These aren’t your run-of-the-mill job interview questions . Instead, they are tricky behavioral interview questions . After all, the goal is to find out how you approach problem-solving, so most are going to feature scenarios, brainteasers, or something similar.

So, having a great strategy means knowing how to deal with behavioral questions. Luckily, there are a couple of tools that can help.

First, when it comes to the classic approach to behavioral interview questions, look no further than the STAR Method . With the STAR method, you learn how to turn your answers into captivating stories. This makes your responses tons more engaging, ensuring you keep the hiring manager’s attention from beginning to end.

Now, should you stop with the STAR Method? Of course not. If you want to take your answers to the next level, spend some time with the Tailoring Method , too.

With the Tailoring Method, it’s all about relevance. So, if you get a chance to choose an example that demonstrates your problem-solving skills, this is really the way to go.

We also wanted to let you know that we created an amazing free cheat sheet that will give you word-for-word answers for some of the toughest interview questions you are going to face in your upcoming interview. After all, hiring managers will often ask you more generalized interview questions!

Click below to get your free PDF now:

Get Our Job Interview Questions & Answers Cheat Sheet!

FREE BONUS PDF CHEAT SHEET: Get our " Job Interview Questions & Answers PDF Cheat Sheet " that gives you " word-word sample answers to the most common job interview questions you'll face at your next interview .

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Top 3 Problem-Solving-Based Interview Questions

Alright, here is what you’ve been waiting for: the problem-solving questions and sample answers.

While many questions in this category are job-specific, these tend to apply to nearly any job. That means there’s a good chance you’ll come across them at some point in your career, making them a great starting point when you’re practicing for an interview.

So, let’s dive in, shall we? Here’s a look at the top three problem-solving interview questions and example responses.

1. Can you tell me about a time when you had to solve a challenging problem?

In the land of problem-solving questions, this one might be your best-case scenario. It lets you choose your own problem-solving examples to highlight, putting you in complete control.

When you choose an example, go with one that is relevant to what you’ll face in the role. The closer the match, the better the answer is in the eyes of the hiring manager.

EXAMPLE ANSWER:

“While working as a mobile telecom support specialist for a large organization, we had to transition our MDM service from one vendor to another within 45 days. This personally physically handling 500 devices within the agency. Devices had to be gathered from the headquarters and satellite offices, which were located all across the state, something that was challenging even without the tight deadline. I approached the situation by identifying the location assignment of all personnel within the organization, enabling me to estimate transit times for receiving the devices. Next, I timed out how many devices I could personally update in a day. Together, this allowed me to create a general timeline. After that, I coordinated with each location, both expressing the urgency of adhering to deadlines and scheduling bulk shipping options. While there were occasional bouts of resistance, I worked with location leaders to calm concerns and facilitate action. While performing all of the updates was daunting, my approach to organizing the event made it a success. Ultimately, the entire transition was finished five days before the deadline, exceeding the expectations of many.”

2. Describe a time where you made a mistake. What did you do to fix it?

While this might not look like it’s based on problem-solving on the surface, it actually is. When you make a mistake, it creates a challenge, one you have to work your way through. At a minimum, it’s an opportunity to highlight problem-solving skills, even if you don’t address the topic directly.

When you choose an example, you want to go with a situation where the end was positive. However, the issue still has to be significant, causing something negative to happen in the moment that you, ideally, overcame.

“When I first began in a supervisory role, I had trouble setting down my individual contributor hat. I tried to keep up with my past duties while also taking on the responsibilities of my new role. As a result, I began rushing and introduced an error into the code of the software my team was updating. The error led to a memory leak. We became aware of the issue when the performance was hindered, though we didn’t immediately know the cause. I dove back into the code, reviewing recent changes, and, ultimately, determined the issue was a mistake on my end. When I made that discovery, I took several steps. First, I let my team know that the error was mine and let them know its nature. Second, I worked with my team to correct the issue, resolving the memory leak. Finally, I took this as a lesson about delegation. I began assigning work to my team more effectively, a move that allowed me to excel as a manager and help them thrive as contributors. It was a crucial learning moment, one that I have valued every day since.”

3. If you identify a potential risk in a project, what steps do you take to prevent it?

Yes, this is also a problem-solving question. The difference is, with this one, it’s not about fixing an issue; it’s about stopping it from happening. Still, you use problem-solving skills along the way, so it falls in this question category.

If you can, use an example of a moment when you mitigated risk in the past. If you haven’t had that opportunity, approach it theoretically, discussing the steps you would take to prevent an issue from developing.

“If I identify a potential risk in a project, my first step is to assess the various factors that could lead to a poor outcome. Prevention requires analysis. Ensuring I fully understand what can trigger the undesired event creates the right foundation, allowing me to figure out how to reduce the likelihood of those events occurring. Once I have the right level of understanding, I come up with a mitigation plan. Exactly what this includes varies depending on the nature of the issue, though it usually involves various steps and checks designed to monitor the project as it progresses to spot paths that may make the problem more likely to happen. I find this approach effective as it combines knowledge and ongoing vigilance. That way, if the project begins to head into risky territory, I can correct its trajectory.”

17 More Problem-Solving-Based Interview Questions

In the world of problem-solving questions, some apply to a wide range of jobs, while others are more niche. For example, customer service reps and IT helpdesk professionals both encounter challenges, but not usually the same kind.

As a result, some of the questions in this list may be more relevant to certain careers than others. However, they all give you insights into what this kind of question looks like, making them worth reviewing.

Here are 17 more problem-solving interview questions you might face off against during your job search:

• How would you describe your problem-solving skills?
• Can you tell me about a time when you had to use creativity to deal with an obstacle?
• Describe a time when you discovered an unmet customer need while assisting a customer and found a way to meet it.
• If you were faced with an upset customer, how would you diffuse the situation?
• Tell me about a time when you had to troubleshoot a complex issue.
• Imagine you were overseeing a project and needed a particular item. You have two choices of vendors: one that can deliver on time but would be over budget, and one that’s under budget but would deliver one week later than you need it. How do you figure out which approach to use?
• Your manager wants to upgrade a tool you regularly use for your job and wants your recommendation. How do you formulate one?
• A supplier has said that an item you need for a project isn’t going to be delivered as scheduled, something that would cause your project to fall behind schedule. What do you do to try and keep the timeline on target?
• Can you share an example of a moment where you encountered a unique problem you and your colleagues had never seen before? How did you figure out what to do?
• Imagine you were scheduled to give a presentation with a colleague, and your colleague called in sick right before it was set to begin. What would you do?
• If you are given two urgent tasks from different members of the leadership team, both with the same tight deadline, how do you choose which to tackle first?
• Tell me about a time you and a colleague didn’t see eye-to-eye. How did you decide what to do?
• Describe your troubleshooting process.
• Tell me about a time where there was a problem that you weren’t able to solve. What happened?
• In your opening, what skills or traits make a person an exceptional problem-solver?
• When you face a problem that requires action, do you usually jump in or take a moment to carefully assess the situation?
• When you encounter a new problem you’ve never seen before, what is the first step that you take?

Putting It All Together

At this point, you should have a solid idea of how to approach problem-solving interview questions. Use the tips above to your advantage. That way, you can thrive during your next interview.

FREE : Job Interview Questions & Answers PDF Cheat Sheet!

Download our " Job Interview Questions & Answers PDF Cheat Sheet " that gives you word-for-word sample answers to some of the most common interview questions including:

• What Is Your Greatest Weakness?
• What Is Your Greatest Strength?
• Tell Me About Yourself
• Why Should We Hire You?

Click Here To Get The Job Interview Questions & Answers Cheat Sheet

Co-Founder and CEO of TheInterviewGuys.com. Mike is a job interview and career expert and the head writer at TheInterviewGuys.com.

His advice and insights have been shared and featured by publications such as Forbes , Entrepreneur , CNBC and more as well as educational institutions such as the University of Michigan , Penn State , Northeastern and others.

Learn more about The Interview Guys on our About Us page .

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Co-Founder and CEO of TheInterviewGuys.com. Mike is a job interview and career expert and the head writer at TheInterviewGuys.com. His advice and insights have been shared and featured by publications such as Forbes , Entrepreneur , CNBC and more as well as educational institutions such as the University of Michigan , Penn State , Northeastern and others. Learn more about The Interview Guys on our About Us page .

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Home » Riddles » 80 Best Science Riddles (with Answers)

80 Best Science Riddles (with Answers)

Science riddles can take riddling to a new, more intellectual level. Many of the riddles in this article rely upon some scientific knowledge, but we also have those that can be answered by just about anyone. Our article begins with some easy Science Riddles for Kids. Then we have a selection of Science Riddles for Students, arranged in an ever-increasing level of difficulty. Our Science Riddles for Adults may prove challenging for some of you who struggled with science in school. Finally, we have some Scientific Puzzles, guaranteed to give your mind a good workout. See how well your brain can deal with our awesome collection of Science Riddles!

Table of Contents

Science Riddles for Kids

Science riddles for kids can help create a passion for science and can add to the fun of learning scientific concepts. They also provide an opportunity for lateral learning and problem solving; kids develop new, creative ways to think. Our collection of science riddles for kids can be used by teachers and for fun icebreakers anywhere kids gather.

• You will find me in Mercury, Earth, Mars and Jupiter, but not in Venus or Neptune. What am I? Show answer The letter R
• What are the three R’s that keep our planet clean? Show answer The three R’s are Reduce, Reuse and Recycle.
• Which weighs more, a ton of concrete or a ton of feathers? Show answer They both weigh the same (a ton).
• I can rush, be still, be hot, be cold, and be hard. I can slip through almost anything. What am I? Show answer Water
• What can go up and come down without moving? Show answer The temperature
• What has a foot on each side and one in the middle? Show answer A yardstick
• What grows only upwards and can never come down? Show answer Our height
• I am excellent to taste, but horrible to smell. What am I? Show answer A tongue
• I touch your face. I am in your words. I am a lack of space and beloved by birds. What am I? Show answer Air
• What breaks but never falls? Show answer Dawn
• What falls but never breaks? Show answer Dusk
• What has a mouth but cannot chew? Show answer A river
• What goes around and around the wood but never goes into the wood? Show answer The bark of a tree
• What kind of rocks are on the bottom of the Mississippi River? Show answer Wet rocks
• What is the moon worth? Show answer $1, because it has 4 quarters
• What do the numbers 11, 69, and 88 all have in common? Show answer They read the same right side up and upside down.
• What is the center of gravity? Show answer The letter V
• Give it food and it will live; give it water and it will die. What is it? Show answer Fire
• How can you tell a tree is a dogwood tree? Show answer By its bark
• What is a tornado’s favorite game to play? Show answer Twister
• What runs faster, cold or hot? Show answer Hot, because you can catch a cold.
• Why did the scientist take out his doorbell? Show answer He wanted to win the no-bell prize.
• What was the first animal to go into space? Show answer The cow that jumped over the moon
• How did Ben Franklin feel after discovering electricity? Show answer Shocked
• What did one earthquake say to the other? Show answer “It’s not my fault!”
• What kind of tree can fit into your hand? Show answer A palm tree

Science Riddles for Students

Riddles help make the study of science fun for students. We have some that rely on relatively simple scientific information and some that require high school or college science proficiency. We begin our list with come easy riddles and end with some sure to test your brain power. Enjoy!

• When the son of the water returns to the parent, it dies. What is it? Show answer Ice
• What can be measured, but has no length, width, or height? Show answer The temperature
• What is full of holes but still holds water? Show answer A sponge.
• What period of time weighs the least? Show answer A light-year
• I am the hottest body in the solar system, and I keep everyone warm. Who am I? Show answer The Sun
• I am a unit of chemical elements, but I can never be trusted. What am I? Show answer An atom, they make up everything.
• What can eat a lot of iron without getting sick? Show answer Rust
• You are in a room with 3 monkeys. One has a banana, one has a stick, one has nothing. Which primate in the room is the smartest? Show answer You are, because you are a primate, too.
• Why was the science teacher angry? Show answer He was a mad scientist.
• How do we know that Saturn was married more than once? Show answer Because she has a lot of rings!
• How did the astronaut serve dinner in outer space? Show answer On flying saucers
• What is neither water nor land, and is always soaking wet? Show answer Wetlands
• You can’t see me, but I can see you. To be more specific, I see through. What am I? Show answer An X-ray
• What did the limestone say to the geologist? Show answer “Don’t take me for granite!”
• Did you hear the one about a chemist who was reading a book about helium? Show answer He couldn’t put it down
• I am a god, a planet, and I can measure heat. What am I? Show answer Mercury
• What do you do with a dead chemist’s corpse? Show answer You just Barium.
• What kind of chemical element hates to be a follower? Show answer Lead
• I am a gas that is helpful to plants, but I cannot be bought at a gas station. What am I? Show answer Carbon dioxide
• Number one is hydrogen. Twenty-five is manganese. Sodium is eleven. What list contains all of these? Show answer The periodic table
• What do chemists call a benzene ring with iron atoms replacing the carbon atoms? Show answer A ferrous wheel
• Cesium and iodine love to watch television together. What is their favorite show? Show answer CSI. CS is the periodic symbol for Cesium and I is the symbol for Iodine.
• I can be good for you, I can be bad, I can be found all over inside and outside your body. I am microscopic, single-celled and have no nucleus. Show answer Bacteria

Science Riddles for Adults

Most adults know some basic scientific information, but how much do you remember from your high school and college science classes? Our collection of science riddles for adults will test your memory and provide a few challenges.

• Of all the glands, I am known as the master, because I boss around the other endocrine glands and tell them what to do. But even though I am a bit bossy, you need me for many things, such as secreting hormones required for sexual development and promoting bone and muscle growth. Show answer Pituitary gland
• Your mom and dad each gave you 23 of these threadlike strands and they helped to make you who you are. Show answer Chromosomes
• What two periodic elements, when combined, heal? Show answer Helium and Aluminum (HE + AL)
• When two light nuclei become one, I liberate quite a bit of nuclear energy? What am I? Show answer Nuclear fusion
• What is the most uninteresting of all the periodic elements? Show answer Boron
• I was once an old massive star and soon I will be a brightly colored gas cloud, but for now I am a massive explosion. What am I called? Show answer Supernova
• What are the only two periodic elements to have the state of a liquid? Show answer Mercury and Bromine
• I make up about three-fourths of all the universe and almost nothing is known about me. What am I? Show answer Dark energy
• I am a hypothetical tunnel, a short cut if you will, of space-time which connects far away regions. I am a what? Show answer A wormhole
• What did the scientist say when he found 2 atoms of helium? Show answer “He, he!”
• I was once called an embryo, but I have now gone through more than eight weeks of development, so my name is now changed to what? Show answer A fetus
• I am a black hole’s equivalent of a one-way street. What am I? Show answer The event horizon (a boundary in which nothing can ever leave, only get pulled in)
• What is black when you buy it, red when you use it, and gray when you throw it away? Show answer Charcoal
• What is the loneliest of all physics concepts? Show answer The singularity
• What four periodic elements, when combined, make up something that terrifies criminals? Show answer Carbon, Oxygen, Phosphorous, and Sulfur (C + O + P + S)
• What element is derived from a Norse god? Show answer Thorium
• I am the toe that every scientist would love to have as their own, yet there has never been a single person who has had a toe that everyone agreed is the best. What kind of toe am I? Show answer Theory of everything (also called a TOE in the scientific community)

Scientific Puzzles

A riddle is a type of puzzle, and we have found some scientific puzzles that will really challenge your thinking. We begin this section with some longer story-type scientific puzzlers, and end with a collection of short, fun riddles designed to work your brain cells.

Shake It Up!

There is an earthquake which is one point higher on the Richter scale than another earthquake which is ten times powerful. Now how much powerful do you think the earthquake will be if it was just 1/2 a point higher on the Richter scale?

A Matter of Growth

When Henry was six years old, he hammered a nail into his favorite tree to mark his height. Five years later at age eleven, Henry returned to see how much higher the nail was. If the tree grew by ten inches each year, how much higher would the nail be?

The Bouncy Ball

Martha throws a ball from a 90-foot building. The ball is quite bouncy and when it hits the ground, it bounces back half way up. It keeps bouncing back to half way up. How many bounces will the ball take before it comes to a permanent halt?

Deadly Laundry

Andrew was going to bleach his socks because they had gotten muddy the day before. As he was pouring the bleach into the washing machine, he spilled some on the floor. He got some cleaning fluid and mopped it up with a rag. Minutes later Andrew was dead. Why?

A Matter of Where

Suppose we drop: A 50 kg metal ball A 40 kg ball of silk From a height of 100 meters. Which will reach the ground first?

Some Short, Fun, but Tricky Science Puzzlers

• H, Be, F, S, Mn, Kr, In, Gd, and Tl? What’s the next in the sequence? Show answer Fm. These are the chemical elements whose atomic numbers are perfect squares.
• What number do Nickel and Neon make when combined? Show answer Nine (NI + NE)
• You cannot see me, I cannot be touched, you cannot feel me, but I can cook your lunch. Show answer A microwave particle
• I make up about three-fourths of all the universe and almost nothing is known about me. What am I ? Show answer Dark energy
• What scientist is the least interesting to listen to? Show answer Niels Bohr
• I know you may like to eat sugar, but when you eat too much of it and your blood-sugar levels rise, then my good friend Pancreas releases me into the bloodstream to restore blood sugar levels to their norm. What am I? Show answer Insulin
• Without me as a part of your brain, you would lose your posture and balance, and be unable to coordinate muscle movements. I always hang out near my good friend the brainstem. What am I? Show answer Cerebellum
• I come as a gas and I get in line with neon. I am also known as the home of a very super person. What am I? Show answer Krypton

There you have it; a collection of scientific riddles designed to please everyone, from young kids to college graduates. See how many you can answer, then share them with your family and friends. Riddle on and have fun!

Susan majored in English with a double minor in Humanities and Business at Arizona State University and earned a Master’s degree in Educational Administration from Liberty University. She taught grades four through twelve in both public and private schools. Subjects included English, U.S. and world history and geography, math, earth and physical science, Bible, information technologies, and creative writing.

Susan has been freelance writing for over ten years, during which time she has written and edited books, newspaper articles, biographies, book reviews, guidelines, neighborhood descriptions for realtors, Power Point presentations, resumes, and numerous other projects.

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54+ Science Trivia Questions & Answers (Weird, Hard, Interesting)

Home » Articles » 54+ Science Trivia Questions & Answers (Weird, Hard, Interesting)

by Kyle Boureston | June 2nd, 2023

Kyle is the founder of Mantelligence, a relationship & dating coach, and a conversation & communication expert. His work has been featured on Marriage.com, Reader's Digest, Vice, Ask Men, and Refinery29. He ... Read Full Bio

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via: Pexels / ThisIsEngineering

Science is incredible, and you can learn even more about this wonderful subject by answering science trivia questions .

When you answer these science trivia questions, you can learn more about science and the entire world around you - and that’s always a satisfying feeling.

Who knows - maybe you’ll use your newfound scientific knowledge to help get to know someone when you’re playing ice breaker games .

Perhaps you’ll discover a new scientific discipline that really captures your imagination and interests.

Whatever the case may be, these science trivia  are guaranteed to be incredibly interesting.

4 best science trivia questions

via: Unsplash / ThisisEngineering RAEng

If you really want to have fun with your friends as you learn more about this amazing topic, it’s best to choose only the best science trivia questions.

1. The Astronomical Unit (AU) is a unit of measurement based on the average distance between what two bodies?

View answer.

The Earth and the Sun

More Information

When you start dealing with vast distances across space, it’s often easier to view time and space as one interconnected unit, which is where we get the concept of a lightyear .

2. What general name is also given to natural satellites?

The Earth’s moon is relatively unique because of its large size in comparison to the planet it orbits (Earth). It is also unique in that it always shows its same side to us. 

3. Humans and chimpanzees share roughly how much DNA?

Interestingly enough, humans also share 60% of their DNA with a banana, and over 90% of their DNA with a housecat, so sometimes the genetic similarities can be misleading. 

4. What is the heaviest organ in the human body?

The liver may be the heaviest organ in the human body, but it’s not the largest. That honor goes instead to our skin, which covers more surface area than any other organ. 

3 funny science trivia questions

via: Unsplash / Chris Coudron

Science doesn’t have to feel like a cold, sterile subject. With these funny trivia questions , you can maintain your sense of humor while being a scientist.

5. What is known as the “master gland” of the human body?

Pituitary gland

The pituitary gland is responsible for hormone production in the human body. These hormones are important for blood pressure and many other functions. 

6. How much taller is the Eiffel Tower during the summer?

If you’re wondering why the Eiffel Tower gets taller during summer, the answer is quite simple: metal expands when heated. This is why it’s easier to open a jar when you run hot water over the lid first. 

7. What is the only planet that spins clockwise?

Venus is also the closest planet to Earth, and many have proposed that it might be a good candidate for a future space colony. However, the planet is incredibly inhospitable and has tremendous amounts of atmospheric pressure. 

3 fun science trivia questions

via: Pexels / Tobias Bjørkli

These science-based and fun trivia questions are great if you’re doing research and you want to remain inspired and motivated.

8. The Horsehead Nebula can be found in what constellation?

Some researchers believe that the pyramids of Giza are actually aligned with the stars that make up Orion’s belt. 

9. The world’s fastest-growing plant is a species of what?

Bamboo grows so quickly that in some cases, it has been known to grow over one foot in just 24 hours. 

10. How often does Halley’s Comet appear in the sky?

Every 75-76 years

Comets may be beautiful, but they also pose a threat to our planet. During the Tunguska event of 1908, a meteoroid impacted the Earth with as much as 30 megatons of explosive force. 

3 science trivia for kids

via: Pexels / Tatiana Syrikova

Making science seem interesting for kids can be easy when you choose this science trivia for kids . They’ll have loads of fun when answering these questions.

11. What melted rock eventually becomes lava?

Lava comes from volcanoes, and these explosive mountains have erupted throughout human history - even destroying the entire city of Pompeii during the ancient times. 

12. On what continent would you not find bees?

Interestingly enough, bees are considered “vegetarian” wasps because they have evolved to feed from plants alone. 

13. How many planets in our solar system have moons?

Many of the moons throughout our solar system may be potential spots for future space colonies, especially the water-rich moon of Titan. 

3 science trivia questions for adults

via: Unsplash / Niketh Vellanki

If you’re an adult, you may be more interested in science questions that are a little more mature. Try these trivia questions for adults .

14. In science, how long is an eon?

1 billion years

The Earth is approximately 4.54 billion years old, or 4.54 billion eons old. Interestingly, the first signs of life appear not long after the Earth was formed. 

15. In what year did the Apollo 7 human spaceflight take place?

The Apollo 7 mission may have been a success, but the Apollo 13 mission was not. In fact, the astronauts aboard very nearly never made it back to Earth. 

16. Big bang machine is also known as?

Large Hadron

The Large Hadron collider is one of the most interesting developments in the modern scientific world, mostly because it allows researchers to find the so-called “God particle.”

3 easy science trivia questions

via: Unsplash / Jongsun Lee

It’s always a good idea to add in a few easy trivia questions when creating your science trivia - just to give people a bit of relief from time to time.

17. When the moon appears to block the sun, as seen from Earth, what is the event called?

Solar eclipse

The fact that the moon is exactly large enough to completely block out the sun from our point of view is one of science’s biggest mysteries. 

18. The brain is divided into how many lobes?

The frontal lobe is dedicated to problem solving, emotional expression, language, and sexual behavior. The occipital lobe is primarily dedicated to vision. 

19. What is the name for trees that never lose their leaves?

Evergreen trees are actually much older than their deciduous counterparts on the evolutionary timeline. In fact, some mammals evolved before flowering plants. 

3 hard science trivia questions

via: Unsplash / Mohammad Asadi

If you’re up for a challenge, it’s time to try out these hard trivia questions . These scientific questions will challenge even the most accomplished scientists.

20. How long does it take for light from the Sun to reach Earth?

8 minutes and 20 seconds

There have been many theories when it comes to faster-than-light space travel, although most scientists believe that it is physically impossible. 

21. What is the end cause of every human death?

Cerebral hypoxia

Cerebral hypoxia is the lack of oxygen to the brain. When people die, their brains essentially starve, and that may be the cause of near-death-experiences and hallucinations. 

22. What is the only rock that floats?

Pumice is actually created from lava, and it forms from the froth at the top of the laval flow that then cools very rapidly. 

3 random science trivia questions and answers

via: Unsplash / Brad Barmore

Sometimes, the most spontaneous trivia questions are the most thought-provoking. Try out these random trivia questions and answers.

23. The oldest living tree is 4,843 years old and can be found where?

Also known as Methuselah, this bristlecone pine has been around since before the first building blocks were laid on the foundation of Ancient Rome. 

24. Who first proposed the concept of contact lenses?

Leonardo da Vinci

Leonardo da Vinci was a truly innovative mind. He also created the concept of the first flying machine, various siege engines, and much more. 

25. How many constellations are in the night sky?

Due to increased satellites in the Earth’s orbit, stars are becoming more and more difficult to see because of light pollution . 

4 science bar trivia questions

via: Pexels / Pixabay

Science bar trivia questions are perfect for an informal night of interesting games and fun. These questions are bound to activate your brainwaves.

26. How many electrons does a hydrogen atom have?

Hydrogen was of course used to create the first thermonuclear explosives, which are considered an improvement on the first-generation nuclear weapons. 

27. What is the name of the world’s largest reef?

Great Barrier Reef

One of the coolest things about the Great Barrier Reef is the fact that it's actually alive. That being said, human impact is threatening to destroy this natural wonder. 

28. What metal is the best conductor of electricity?

Silver may be used extensively for jewelry, but it also has very important uses for things like circuitry. Gold is also quite useful, especially in terms of space travel. 

30. What is the scientific name for the job or role an organism plays in its habitat?

While the word “niche” has a very specific meaning in terms of biology, it can also mean the purpose or target market of a particular business. 

4 animal science trivia questions

via: Unsplash / Yogendra Singh

Animals are some of the most fascinating additions to our planet, and you can learn more about them with these fascinating animal trivia questions .

31. Which large mammal’s tail is so strong it can stand on it and lift its hind legs off the ground?

The kangaroo

The kangaroo may be a mammal, but it’s also part of the marsupial family. This means that it gives birth to immature young that then spend a period of time developing further in a marsupial pouch. 

32. What is the scientific name for the trunk of an elephant?

The word “proboscis” also means “nose,” which means you have a proboscis too - just like an elephant. However, elephant probosci are unique for very obvious reasons. 

33. What are the male honey bees called that are the only members of the colony allowed to mate with the queen?

Drones might seem like they have a nice life, as they don’t really have to do any work. However, every so often all of the drones get kicked out of the hive, never to return. 

34. What animal is the closest living relative to the T-Rex

Birds in general are actually quite closely related to the extinct dinosaurs. In fact, some researchers believe that dinosaurs were actually feathered creatures for the most part. 

4 science movie trivia questions

via: Unsplash / Pim Chu

Some of the best movies in the world were based on very real science - after all, that’s where we get the term “science fiction” from. Check out these science movie trivia questions .

35. A xenomorph took on a shaved-head Ripley in this sci-fi horror?

Many people believe Alien 3 to be one of the weakest films of the Alien franchise, but others feel that it is somewhat misunderstood and underrated. 

36. How long did it take the evil aliens to defeat the combined armies of the human race in the 2000 sci-fi film "Battlefield Earth"?

Battlefield Earth has earned a reputation as one of the worst films ever made, although it did show John Travolta in a very non-standard role as an alien villain. 

37. Actress Scarlett Johansson starred in this 2017 film based on what popular Japanese comic?

Ghost in the Shell

The original Ghost in the Shell anime film explored concepts connected to the merging of humanity with machines, to the point where the only thing that remains is the brain or perhaps just the consciousness. 

38. 'Star Trek' marked the film debut of this actor, a future Marvel superhero.

Chris Hemsworth

Prior to his short cameo in Star Trek, Hemsworth was mostly known within Australia for his roles in various soap operas. Of course, he would go on to star in a range of other films after this debut. 

4 science in the 80s trivia questions and answers

via: Unsplash / Lians Jadan

Just like any other decade, there was plenty of scientific innovation and discovery during the 80s. How well do you remember it? Find out with these 80s trivia questions and answers .

39. In 1980, Mount St. Helens erupted in what US state killing 57 people?

Washington state

Mount St. Helens is still an active volcano today, and it’s actually considered one of the most dangerous in the United States. 

40. What power plant near Kiev, Ukraine, released radioactive fallout across much of Europe and renders thousands of acres of land near the accident site uninhabitable for thousands of years?

Chernobyl nuclear power plant

The official cause of the Chernobyl disaster has been stated as a flawed reactor design. Combined with inadequately trained staff, the power plant was always a dangerous prospect. 

41. In 1986 an American space shuttle disintegrated after launch, killing all aboard, what was the name of the shuttle?

Space Shuttle Challenger

The Space Shuttle Challenger disaster was caused by the explosion of an external fuel tank. It was even more tragic than usual because the shuttle contained the first woman in space, Sally Ride. 

42. Scientists first identified the AIDS virus in 1981, what does AIDS stand for?

Acquired Immunodeficiency Syndrome

Since its discovery in the 80s, AIDS/HIV has claimed many lives. However, much progress has been made during recent years, and today those diagnosed with AIDS can live long and normal lives with the right medications. 

3 history of science trivia questions

via: Unsplash / Andrew Neel

The history of science is very interesting, and it tells a story of human accomplishment and development. If you’d like to learn more about the history of science, check out these history trivia questions .

43. Who invented the Morse code?

Samuel Finley Breese Morse and Alfred Vail

It’s easy to look back on Morse code today and think of it as a primitive form of communication.

44. What islands were extensively studied by Charles Darwin?

Galapagos Islands

The thing that really formed the basis of Darwin’s theory of evolution was the various sub-species of finches, which all developed different traits after being isolated on different islands. 

45. Which dwarf planet was discovered by Clyde Tombaugh while working at the Lowell Observatory in Flagstaff in 1930?

Today, there is still quite a bit of debate as to whether or not Pluto is actually a legitimate planet. That being said, the general consensus is that Pluto is not actually a planet. 

46. Which civilization is regarded as the first to have a functional theory of the planets?

The ancient Sumerians had an incredible understanding of the planets, and we can see by their carvings that they understood, in basic terms, how the planets behaved in the night sky. 

3 marvel science trivia questions

via: Pexels / Daniel Xavier

Marvel is famous for basing its comics in science, and some of the most interesting heroes are scientists who made incredible discoveries that changed their lives forever. Check out these Marvel trivia questions .

47. What type of doctor is Stephen Strange?

Neurosurgeon

Stephen Strange was a talented neurosurgeon - at least until he injured his hands in a terrible car accident. This caused him to travel the world in search for ways to heal himself. 

48. What is the alien race Loki sends to invade Earth in The Avengers?

The Chitauri

In case you weren’t aware, Loki and all of the other Asgardians are based on real mythological figures from Norse legends. 

49. Thor's method of travel can be supported by a real theory called?

The Einstein Rosen Bridge

The Einstein Rosen Bridge is also known by its more common name, the wormhole. If a wormhole could be controlled, perhaps people could achieve faster than light travel. 

3 Star Wars science trivia questions

via: Unsplash / Agnieszka Kowalczyk

Do you love Star Wars? If so, you’ll really enjoy answering these incredible Star Wars trivia questions.

50. The echo base is hidden in Hoth, a frozen world. What types of precipitation will likely fall on Hoth?

The rebel base on Hoth was attacked by a viscous detachment of Imperial ground forces, including massive Imperial walkers. 

51. Tatoonie copies carry heavy cargo and are adapted to the desert. What Earth animal would be similar?

The camel is famous for its three humps and its ability to survive for long periods of time without drinking any water. 

52. Han and Lando can breathe outside Cloud City. What element must be in that atmosphere?

Lando and Han go way back. In fact, Lando once owned the Millennium Falcon before he lost it to Han as part of a bet. 

2 sports science trivia questions

via: Unsplash / Ruben Leija

The sporting world definitely revolves around scientific principles and you can learn a lot about sports when you answer these sports trivia questions in order to look at everything from a scientific point of view.

53. True or false? Lower tension on a tennis racquet produces more control and less power.

Most racket manufacturers recommend that you use 50 to 70 pounds of tension when adjusting your racket, but it totally depends on your personal preference as a player. 

54. What was the fastest recorded tennis serve?

55 mph (250 kph)

Sometimes, a fast serve can win entire tennis matches. However, most of the time it’s a combination of serving technique and other skills that win players matches. 

How To Pick The Best Science trivia Questions

via: Unsplash / Jud Mackrill

Picking the best science trivia questions isn’t always easy, but you can make this process more manageable by following this simple guide. Follow these steps, and you can choose science trivia questions that everyone will love.

1. Choose questions based on a wide range of disciplines

via: Pexels / Yogendra Singh

Don’t just focus on biology or chemistry. Instead, make sure that your science trivia questions cover a wide range of topics so that people are always interested and surprised when learning about new things.

2. Avoid questions that are too hard (or too easy)

via: Pexels / Wallace Chuck

If you pick too many hard questions, people are just going to get frustrated. After all, science is a tricky subject, and it’s easy to get overwhelmed. On the other hand, you should avoid too many easy questions as well, as this can make things boring.

3. Choose relevant questions

via: Pexels / Martin Péchy

There are some branches of science that are very relevant in today’s world. For example, you might want to pick questions based on climate science or animal conservation. Relevant questions are almost always more interesting.

Downloadable List of Science Trivia Questions

Here is a downloadable and printable list of science trivia questions (right-click the image and select Save Image As...) :

via: Mantelligence

More Awesome Trivia Questions

Science may be pretty interesting, but it’s not the only subject that can capture your imagination and passions. If you’ve finished these trivia questions and you’re still looking for more, check out these additional options:

• Who doesn’t love Harry Potter? Find out how big of a fan you are with these awesome Harry Potter trivia questions.
• Think you know the bible inside and out? Well, you haven’t truly been tested until you’ve tried these Bible trivia questions and answers .
• Everyone loves music, but not everyone can call themselves an expert. Find out if you can earn this title by answering these music trivia questions and answers .
• Do you love Thanksgiving? If so, these Thanksgiving trivia questions are bound to get you excited.
• If you just can’t wait until Christmas why not try out these Christmas trivia questions while you tick off the days on your calendar?

If you’ve enjoyed answering these science trivia questions , it’s time to take the next step. Find out which topics are most interesting to you and delve even deeper into this amazing subject.

In doing so, you can come up with new science trivia questions that are even more interesting than the last. One of the benefits of answering trivia questions is that you can spark a new interest in science for not only you, but also your friends and classmates.

Science can be a lot of fun - especially if you’re using these science trivia  during ice breaker games . Whatever the case may be, science is one of the most wonderful parts of human existence, and it makes sense to learn all that you can about it over the course of your life.

About The Author

Kyle Boureston

Kyle is the founder of Mantelligence, a relationship & dating coach, and a conversation & communication expert. His work has been featured on Marriage.com, Reader's Digest, Vice, Ask Men, and Refinery29. He lives in Austin, TX with his loving wife and his energetic Border Collie.

One of the greatest discoveries a man makes, one of his great surprises, is to find he can do what he was afraid he couldn’t do.

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Mantelligence founder launching World's First Highly-Effective Mosquito Trap, Powered by Essential Oils.

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• Define a problem and its specifications 📝
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Table of Contents

• What is Data Science?

Data Science combines statistics, maths, specialised programs, artificial intelligence, machine learning etc. Data Science is simply the application of specific principles and analytic techniques to extract information from data used in strategic planning, decision making, etc. Simply, data science means analysing data for actionable insights, incorporating top data science skills such as data visualization, Processing large data sets, Statistical analysis and computing to unlock the value in data.

10 Most Asked Data Science Interview Questions 

• Differentiate between Data Analytics and Data Science
• What are the differences between supervised and unsupervised learning?
• Explain the steps in making a decision tree.
• Differentiate between univariate, bivariate, and multivariate analysis.
• How should you maintain a deployed model?
• What is a Confusion Matrix?
• How is logistic regression done?
• What is the significance of p-value?
• Mention some techniques used for sampling.

Basic and Advanced Data Science Interview Questions

Here's a list of the most popular data science interview questions on the technical concept which you can expect to face, and how to frame your answers.

1. What are the differences between supervised and unsupervised learning?

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2. How is logistic regression done?

Logistic regression measures the relationship between the dependent variable (our label of what we want to predict) and one or more independent variables (our features) by estimating probability using its underlying logistic function (sigmoid).

The image shown below depicts how logistic regression works:

The formula and graph for the sigmoid function are as shown:

3. Explain the steps in making a decision tree.

• Take the entire data set as input
• Calculate entropy of the target variable, as well as the predictor attributes
• Calculate your information gain of all attributes (we gain information on sorting different objects from each other)
• Choose the attribute with the highest information gain as the root node 
• Repeat the same procedure on every branch until the decision node of each branch is finalized

For example, let's say you want to build a decision tree to decide whether you should accept or decline a job offer. The decision tree for this case is as shown:

It is clear from the decision tree that an offer is accepted if:

• Salary is greater than $50,000
• The commute is less than an hour 
• Incentives are offered 

4. How do you build a random forest model?

A random forest is built up of a number of decision trees. If you split the data into different packages and make a decision tree in each of the different groups of data, the random forest brings all those trees together.

Steps to build a random forest model:

• Randomly select 'k' features from a total of 'm' features where k << m
• Among the 'k' features, calculate the node D using the best split point
• Split the node into daughter nodes using the best split
• Repeat steps two and three until leaf nodes are finalized 
• Build forest by repeating steps one to four for 'n' times to create 'n' number of trees 

5. How can you avoid overfitting your model?

Overfitting refers to a model that is only set for a very small amount of data and ignores the bigger picture. There are three main methods to avoid overfitting :

• Keep the model simple—take fewer variables into account, thereby removing some of the noise in the training data
• Use cross-validation techniques, such as k folds cross-validation 
• Use regularization techniques, such as LASSO, that penalize certain model parameters if they're likely to cause overfitting

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6. Differentiate between univariate, bivariate, and multivariate analysis.

Univariate data contains only one variable. The purpose of the univariate analysis is to describe the data and find patterns that exist within it. 

Example: height of students 

The patterns can be studied by drawing conclusions using mean, median, mode, dispersion or range, minimum, maximum, etc.

Bivariate data involves two different variables. The analysis of this type of data deals with causes and relationships and the analysis is done to determine the relationship between the two variables.

Example: temperature and ice cream sales in the summer season

Here, the relationship is visible from the table that temperature and sales are directly proportional to each other. The hotter the temperature, the better the sales.

Multivariate

Multivariate data involves three or more variables, it is categorized under multivariate. It is similar to a bivariate but contains more than one dependent variable.

Example: data for house price prediction 

The patterns can be studied by drawing conclusions using mean, median, and mode, dispersion or range, minimum, maximum, etc. You can start describing the data and using it to guess what the price of the house will be.

7. What are the feature selection methods used to select the right variables?

There are two main methods for feature selection, i.e, filter, and wrapper methods.

Filter Methods

This involves: 

• Linear discrimination analysis

The best analogy for selecting features is "bad data in, bad answer out." When we're limiting or selecting the features, it's all about cleaning up the data coming in. 

Wrapper Methods

• Forward Selection: We test one feature at a time and keep adding them until we get a good fit
• Backward Selection: We test all the features and start removing them to see what works better
• Recursive Feature Elimination: Recursively looks through all the different features and how they pair together

Wrapper methods are very labor-intensive, and high-end computers are needed if a lot of data analysis is performed with the wrapper method. 

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8. In your choice of language, write a program that prints the numbers ranging from one to 50.

But for multiples of three, print "Fizz" instead of the number, and for the multiples of five, print "Buzz." For numbers which are multiples of both three and five, print "FizzBuzz" 

The code is shown below:

Note that the range mentioned is 51, which means zero to 50. However, the range asked in the question is one to 50. Therefore, in the above code, you can include the range as (1,51).

The output of the above code is as shown:

9. You are given a data set consisting of variables with more than 30 percent missing values. How will you deal with them?

The following are ways to handle missing data values:

If the data set is large, we can just simply remove the rows with missing data values. It is the quickest way; we use the rest of the data to predict the values.

For smaller data sets, we can substitute missing values with the mean or average of the rest of the data using the pandas' data frame in python. There are different ways to do so, such as df.mean(), df.fillna(mean).

10. For the given points, how will you calculate the Euclidean distance in Python?

plot1 = [1,3]

plot2 = [2,5]

The Euclidean distance can be calculated as follows:

euclidean_distance = sqrt( (plot1[0]-plot2[0])**2 + (plot1[1]-plot2[1])**2 )

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11. What are dimensionality reduction and its benefits?

The Dimensionality reduction refers to the process of converting a data set with vast dimensions into data with fewer dimensions (fields) to convey similar information concisely. 

This reduction helps in compressing data and reducing storage space. It also reduces computation time as fewer dimensions lead to less computing. It removes redundant features; for example, there's no point in storing a value in two different units (meters and inches). 

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12. How will you calculate eigenvalues and eigenvectors of the following 3x3 matrix?

The characteristic equation is as shown:

Expanding determinant:

(-2 – λ) [(1-λ) (5-λ)-2x2] + 4[(-2) x (5-λ) -4x2] + 2[(-2) x 2-4(1-λ)] =0

- λ3 + 4λ2 + 27λ – 90 = 0,

λ3 - 4 λ2 -27 λ + 90 = 0

Here we have an algebraic equation built from the eigenvectors.

By hit and trial:

33 – 4 x 32 - 27 x 3 +90 = 0

Hence, (λ - 3) is a factor:

λ3 - 4 λ2 - 27 λ +90 = (λ – 3) (λ2 – λ – 30)

Eigenvalues are 3,-5,6:

(λ – 3) (λ2 – λ – 30) = (λ – 3) (λ+5) (λ-6),

Calculate eigenvector for λ = 3

-5 - 4Y + 2Z =0,

-2 - 2Y + 2Z =0

Subtracting the two equations: 

3 + 2Y = 0,

Subtracting back into second equation:

Y = -(3/2) 

Similarly, we can calculate the eigenvectors for -5 and 6.

13. How should you maintain a deployed model?

The steps to maintain a deployed model are:

Constant monitoring of all models is needed to determine their performance accuracy. When you change something, you want to figure out how your changes are going to affect things. This needs to be monitored to ensure it's doing what it's supposed to do.

Evaluation metrics of the current model are calculated to determine if a new algorithm is needed. 

The new models are compared to each other to determine which model performs the best. 

The best-performing model is re-built on the current state of data.

14. What are recommender systems?

A recommender system predicts what a user would rate a specific product based on their preferences. It can be split into two different areas:

Collaborative Filtering

As an example, Last.fm recommends tracks that other users with similar interests play often. This is also commonly seen on Amazon after making a purchase; customers may notice the following message accompanied by product recommendations: "Users who bought this also bought…"

Content-based Filtering

As an example: Pandora uses the properties of a song to recommend music with similar properties. Here, we look at content, instead of looking at who else is listening to music.

15. How do you find RMSE and MSE in a linear regression model?

RMSE and MSE are two of the most common measures of accuracy for a linear regression model. 

RMSE indicates the Root Mean Square Error. 

MSE indicates the Mean Square Error.

16. How can you select k for k-means? 

We use the elbow method to select k for k-means clustering . The idea of the elbow method is to run k-means clustering on the data set where 'k' is the number of clusters.

Within the sum of squares (WSS), it is defined as the sum of the squared distance between each member of the cluster and its centroid. 

17. What is the significance of p-value?

p-value typically ≤ 0.05

This indicates strong evidence against the null hypothesis; so you reject the null hypothesis.

p-value typically > 0.05

This indicates weak evidence against the null hypothesis, so you accept the null hypothesis. 

p-value at cutoff 0.05 

This is considered to be marginal, meaning it could go either way.

18. How can outlier values be treated?

You can drop outliers only if it is a garbage value. 

Example: height of an adult = abc ft. This cannot be true, as the height cannot be a string value. In this case, outliers can be removed.

If the outliers have extreme values, they can be removed. For example, if all the data points are clustered between zero to 10, but one point lies at 100, then we can remove this point.

If you cannot drop outliers, you can try the following:

• Try a different model. Data detected as outliers by linear models can be fit by nonlinear models. Therefore, be sure you are choosing the correct model.
• Try normalizing the data. This way, the extreme data points are pulled to a similar range.
• You can use algorithms that are less affected by outliers; an example would be random forests . 

19. How can time-series data be declared as stationery?

It is stationary when the variance and mean of the series are constant with time. 

Here is a visual example: 

In the first graph, the variance is constant with time. Here, X is the time factor and Y is the variable. The value of Y goes through the same points all the time; in other words, it is stationary.

In the second graph, the waves get bigger, which means it is non-stationary and the variance is changing with time.

Find Our Data Science Course in Top Cities India United States Other Countries 20. How can you calculate accuracy using a confusion matrix? Consider this confusion matrix : You can see the values for total data, actual values, and predicted values. The formula for accuracy is: Accuracy = (True Positive + True Negative) / Total Observations = (262 + 347) / 650 = 609 / 650 As a result, we get an accuracy of 93 percent. 21. Write the equation and calculate the precision and recall rate. Consider the same confusion matrix used in the previous question. Precision = (True positive) / (True Positive + False Positive) = 262 / 277 Recall Rate = (True Positive) / (Total Positive + False Negative) = 262 / 288 22. 'People who bought this also bought…' recommendations seen on Amazon are a result of which algorithm? The recommendation engine is accomplished with collaborative filtering. Collaborative filtering explains the behavior of other users and their purchase history in terms of ratings, selection, etc.  The engine makes predictions on what might interest a person based on the preferences of other users. In this algorithm, item features are unknown. For example, a sales page shows that a certain number of people buy a new phone and also buy tempered glass at the same time. Next time, when a person buys a phone, he or she may see a recommendation to buy tempered glass as well. 23. Write a basic SQL query that lists all orders with customer information. Usually, we have order tables and customer tables that contain the following columns: Order Table  customerId  OrderNumber TotalAmount Customer Table  The SQL query is: SELECT OrderNumber, TotalAmount, FirstName, LastName, City, Country JOIN Customer ON Order.CustomerId = Customer.Id 24. You are given a dataset on cancer detection. You have built a classification model and achieved an accuracy of 96 percent. Why shouldn't you be happy with your model performance? What can you do about it? Cancer detection results in imbalanced data. In an imbalanced dataset, accuracy should not be based as a measure of performance. It is important to focus on the remaining four percent, which represents the patients who were wrongly diagnosed. Early diagnosis is crucial when it comes to cancer detection, and can greatly improve a patient's prognosis. Hence, to evaluate model performance, we should use Sensitivity (True Positive Rate), Specificity (True Negative Rate), F measure to determine the class wise performance of the classifier. 25. Which of the following machine learning algorithms can be used for inputting missing values of both categorical and continuous variables? K-means clustering Linear regression  K-NN (k-nearest neighbor) Decision trees  The K nearest neighbor algorithm can be used because it can compute the nearest neighbor and if it doesn't have a value, it just computes the nearest neighbor based on all the other features.  When you're dealing with K-means clustering or linear regression , you need to do that in your pre-processing, otherwise, they'll crash. Decision trees also have the same problem, although there is some variance. Looking forward to becoming a Data Scientist? Check out the   and get certified today. 26. Below are the eight actual values of the target variable in the train file. What is the entropy of the target variable? [0, 0, 0, 1, 1, 1, 1, 1]  Choose the correct answer. -(5/8 log(5/8) + 3/8 log(3/8)) 5/8 log(5/8) + 3/8 log(3/8) 3/8 log(5/8) + 5/8 log(3/8) 5/8 log(3/8) – 3/8 log(5/8) The target variable, in this case, is 1.  The formula for calculating the entropy is: Putting p=5 and n=8, we get  Entropy = A = -(5/8 log(5/8) + 3/8 log(3/8)) 27. We want to predict the probability of death from heart disease based on three risk factors: age, gender, and blood cholesterol level. What is the most appropriate algorithm for this case? Choose the correct option: Logistic Regression  Linear Regression K-means clustering  Apriori algorithm The most appropriate algorithm for this case is A, logistic regression.  28. After studying the behavior of a population, you have identified four specific individual types that are valuable to your study. You would like to find all users who are most similar to each individual type. Which algorithm is most appropriate for this study? Linear regression Association rules Decision trees As we are looking for grouping people together specifically by four different similarities, it indicates the value of k. Therefore, K-means clustering (answer A) is the most appropriate algorithm for this study. 29. You have run the association rules algorithm on your dataset, and the two rules {banana, apple} => {grape} and {apple, orange} => {grape} have been found to be relevant. What else must be true? Choose the right answer: {banana, apple, grape, orange} must be a frequent itemset {banana, apple} => {orange} must be a relevant rule {grape} => {banana, apple} must be a relevant rule {grape, apple} must be a frequent itemset The answer is A: {grape, apple} must be a frequent itemset 30. Your organization has a website where visitors randomly receive one of two coupons. It is also possible that visitors to the website will not receive a coupon. You have been asked to determine if offering a coupon to website visitors has any impact on their purchase decisions. Which analysis method should you use? One-way ANOVA  Association rules  Student's t-test  The answer is A: One-way ANOVA 31. What do you understand about true positive rate and false-positive rate? The True Positive Rate (TPR) defines the probability that an actual positive will turn out to be positive.  The True Positive Rate (TPR) is calculated by taking the ratio of the [True Positives (TP)] and [True Positive (TP) & False Negatives (FN) ].  The formula for the same is stated below - TPR=TP/TP+FN The False Positive Rate (FPR) defines the probability that an actual negative result will be shown as a positive one i.e the probability that a model will generate a false alarm.  The False Positive Rate (FPR) is calculated by taking the ratio of the [False Positives (FP)] and [True Positives (TP) & False Positives(FP)]. FPR=FP/TN+FP 32. What is the ROC curve? The graph between the True Positive Rate on the y-axis and the False Positive Rate on the x-axis is called the ROC curve and is used in binary classification. The False Positive Rate (FPR) is calculated by taking the ratio between False Positives and the total number of negative samples, and the True Positive Rate (TPR) is calculated by taking the ratio between True Positives and the total number of positive samples. In order to construct the ROC curve, the TPR and FPR values are plotted on multiple threshold values. The area range under the ROC curve has a range between 0 and 1. A completely random model, which is represented by a straight line, has a 0.5 ROC. The amount of deviation a ROC has from this straight line denotes the efficiency of the model. The image above denotes a ROC curve example. 33. What is a Confusion Matrix? The Confusion Matrix is the summary of prediction results of a particular problem. It is a table that is used to describe the performance of the model. The Confusion Matrix is an n*n matrix that evaluates the performance of the classification model. 34. What do you understand about the true-positive rate and false-positive rate? TRUE-POSITIVE RATE: The true-positive rate gives the proportion of correct predictions of the positive class. It is also used to measure the percentage of actual positives that are accurately verified. FALSE-POSITIVE RATE: The false-positive rate gives the proportion of incorrect predictions of the positive class. A false positive determines something is true when that is initially false. 35. How is Data Science different from traditional application programming? The primary and vital difference between Data Science and traditional application programming is that in traditional programming, one has to create rules to translate the input to output. In Data Science, the rules are automatically produced from the data. 36. What is the difference between the long format data and wide format data? LONG FORMAT DATA: It contains values that repeat in the first column. In this format, each row is a one-time point per subject. WIDE FORMAT DATA: In the Wide Format Data, the data’s repeated responses will be in a single row, and each response can be recorded in separate columns. Long format Table: NAME ATTRIBUTE VALUE RAMA HEIGHT  182 SITA HEIGHT 160 Wide format Table: NAME HEIGHT RAMA 182 SITA 160 37. Mention some techniques used for sampling.  Sampling is the selection of individual members or a subset of the population to estimate the characters of the whole population. There are two types of Sampling, namely Probability and Non-Probability Sampling. 38. Why is Python used for Data Cleaning in DS? Data Scientists and technical analysts must convert a huge amount of data into effective ones. Data Cleaning includes removing malwared records, outliners, inconsistent values, redundant formatting etc. Matplotlib, Pandas etc are the most used Python Data Cleaners. 39. What are the popular libraries used in Data Science? The popular libraries used in Data Science are  Tensor Flow 40. What is variance in Data Science? Variance is the value that depicts the individual figures in a set of data which distributes themselves about the mean and describes the difference of each value from the mean value. Data Scientists use variance to understand the distribution of a data set. 41. What is pruning in a decision tree algorithm? In Data Science and Machine Learning, Pruning is a technique which is related to decision trees. Pruning simplifies the decision tree by reducing the rules. Pruning helps to avoid complexity and improves accuracy. Reduced error Pruning, cost complexity pruning etc. are the different types of Pruning. 42. What is entropy in a decision tree algorithm? Entropy is the measure of randomness or disorder in the group of observations. It also determines how a decision tree switches to split data. Entropy is also used to check the homogeneity of the given data. If the entropy is zero, then the sample of data is entirely homogeneous, and if the entropy is one, then it indicates that the sample is equally divided. 43. What information is gained in a decision tree algorithm? Information gain is the expected reduction in entropy. Information gain decides the building of the tree. Information Gain makes the decision tree smarter. Information gain includes parent node R and a set E of K training examples. It calculates the difference between entropy before and after the split. 44. What is k-fold cross-validation? The k-fold cross validation is a procedure used to estimate the model's skill in new data. In k-fold cross validation, every observation from the original dataset may appear in the training and testing set. K-fold cross-validation estimates the accuracy but does not help you to improve the accuracy. 45. What is a normal distribution? Normal Distribution is also known as the Gaussian Distribution. The normal distribution shows the data near the mean and the frequency of that particular data. When represented in graphical form, normal distribution appears like a bell curve. The parameters included in the normal distribution are Mean, Standard Deviation, Median etc. 46. What is Deep Learning? Deep Learning is one of the essential factors in Data Science, including statistics. Deep Learning makes us work more closely with the human brain and reliable with human thoughts. The algorithms are sincerely created to resemble the human brain. In Deep Learning, multiple layers are formed from the raw input to extract the high-level layer with the best features. 47. What is an RNN (recurrent neural network)? RNN is an algorithm that uses sequential data. RNN is used in language translation, voice recognition, image capturing etc. There are different types of RNN networks such as one-to-one, one-to-many, many-to-one and many-to-many. RNN is used in Google’s Voice search and Apple’s Siri. Basic Data Science Interview Questions Let us begin with a few basic data science interview questions! 48. What are the feature vectors? A feature vector is an n-dimensional vector of numerical features that represent an object. In machine learning, feature vectors are used to represent numeric or symbolic characteristics (called features) of an object in a mathematical way that's easy to analyze. 49. What are the steps in making a decision tree? Take the entire data set as input. Look for a split that maximizes the separation of the classes. A split is any test that divides the data into two sets. Apply the split to the input data (divide step). Re-apply steps one and two to the divided data. Stop when you meet any stopping criteria. This step is called pruning. Clean up the tree if you went too far doing splits. 50. What is root cause analysis? Root cause analysis was initially developed to analyze industrial accidents but is now widely used in other areas. It is a problem-solving technique used for isolating the root causes of faults or problems. A factor is called a root cause if its deduction from the problem-fault-sequence averts the final undesirable event from recurring. 51. What is logistic regression? Logistic regression is also known as the logit model. It is a technique used to forecast the binary outcome from a linear combination of predictor variables. 52. What are recommender systems? Recommender systems are a subclass of information filtering systems that are meant to predict the preferences or ratings that a user would give to a product. 53. Explain cross-validation. Cross-validation is a model validation technique for evaluating how the outcomes of a statistical analysis will generalize to an independent data set. It is mainly used in backgrounds where the objective is to forecast and one wants to estimate how accurately a model will accomplish in practice.  The goal of cross-validation is to term a data set to test the model in the training phase (i.e. validation data set) to limit problems like overfitting and gain insight into how the model will generalize to an independent data set. 54. What is collaborative filtering? Most recommender systems use this filtering process to find patterns and information by collaborating perspectives, numerous data sources, and several agents. 55. Do gradient descent methods always converge to similar points? They do not, because in some cases, they reach a local minima or a local optima point. You would not reach the global optima point. This is governed by the data and the starting conditions. 56. What is the goal of A/B Testing? This is statistical hypothesis testing for randomized experiments with two variables, A and B. The objective of A/B testing is to detect any changes to a web page to maximize or increase the outcome of a strategy. 57. What are the drawbacks of the linear model? The assumption of linearity of the errors It can't be used for count outcomes or binary outcomes There are overfitting problems that it can't solve 58. What is the law of large numbers? It is a theorem that describes the result of performing the same experiment very frequently. This theorem forms the basis of frequency-style thinking. It states that the sample mean, sample variance, and sample standard deviation converge to what they are trying to estimate. 59.  What are the confounding variables? These are extraneous variables in a statistical model that correlates directly or inversely with both the dependent and the independent variable. The estimate fails to account for the confounding factor. 60. What is star schema? It is a traditional database schema with a central table. Satellite tables map IDs to physical names or descriptions and can be connected to the central fact table using the ID fields; these tables are known as lookup tables and are principally useful in real-time applications, as they save a lot of memory. Sometimes, star schemas involve several layers of summarization to recover information faster. 61. How regularly must an algorithm be updated? You will want to update an algorithm when: You want the model to evolve as data streams through infrastructure The underlying data source is changing There is a case of non-stationarity 62.  What are eigenvalue and eigenvector? Eigenvalues are the directions along which a particular linear transformation acts by flipping, compressing, or stretching. Eigenvectors are for understanding linear transformations. In data analysis, we usually calculate the eigenvectors for a correlation or covariance matrix.  63. Why is resampling done? Resampling is done in any of these cases: Estimating the accuracy of sample statistics by using subsets of accessible data, or drawing randomly with replacement from a set of data points Substituting labels on data points when performing significance tests Validating models by using random subsets ( bootstrapping , cross-validation) 64. What is selection bias? Selection bias, in general, is a problematic situation in which error is introduced due to a non-random population sample. 65. What are the types of biases that can occur during sampling? Selection bias Undercoverage bias Survivorship bias 66. What is survivorship bias? Survivorship bias is the logical error of focusing on aspects that support surviving a process and casually overlooking those that did not because of their lack of prominence. This can lead to wrong conclusions in numerous ways. 67. How do you work towards a random forest? The underlying principle of this technique is that several weak learners combine to provide a strong learner. The steps involved are: Build several decision trees on bootstrapped training samples of data On each tree, each time a split is considered, a random sample of mm predictors is chosen as split candidates out of all pp predictors Rule of thumb: At each split m=p√m=p Predictions: At the majority rule This exhaustive list is sure to strengthen your preparation for data science interview questions. 68. What is a bias-variance trade-off? Bias: Due to an oversimplification of a Machine Learning Algorithm, an error occurs in our model, which is known as Bias. This can lead to an issue of underfitting and might lead to oversimplified assumptions at the model training time to make target functions easier and simpler to understand. Some of the popular machine learning algorithms which are low on the bias scale are - Support Vector Machines (SVM), K-Nearest Neighbors (KNN), and Decision Trees. Algorithms that are high on the bias scale - Logistic Regression and Linear Regression. Variance: Because of a complex machine learning algorithm, a model performs really badly on a test data set as the model learns even noise from the training data set. This error that occurs in the Machine Learning model is called Variance and can generate overfitting and hyper-sensitivity in Machine Learning models. While trying to get over bias in our model, we try to increase the complexity of the machine learning algorithm. Though it helps in reducing the bias, after a certain point, it generates an overfitting effect on the model hence resulting in hyper-sensitivity and high variance. Bias-Variance trade-off: To achieve the best performance, the main target of a supervised machine learning algorithm is to have low variance and bias.  The following things are observed regarding some of the popular machine learning algorithms - The Support Vector Machine algorithm (SVM) has high variance and low bias. In order to change the trade-off, we can increase the parameter C. The C parameter results in a decrease in the variance and an increase in bias by influencing the margin violations allowed in training datasets. In contrast to the SVM, the K-Nearest Neighbors (KNN) Machine Learning algorithm has a high variance and low bias. To change the trade-off of this algorithm, we can increase the prediction influencing neighbors by increasing the K value, thus increasing the model bias. 69. Describe Markov chains? Markov Chains defines that a state’s future probability depends only on its current state.  Markov chains belong to the Stochastic process type category. The below diagram explains a step-by-step model of the Markov Chains whose output depends on their current state. A perfect example of the Markov Chains is the system of word recommendation. In this system, the model recognizes and recommends the next word based on the immediately previous word and not anything before that. The Markov Chains take the previous paragraphs that were similar to training data-sets and generates the recommendations for the current paragraphs accordingly based on the previous word. 70. Why is R used in Data Visualization? R is widely used in Data Visualizations for the following reasons- We can create almost any type of graph using R. R has multiple libraries like lattice, ggplot2, leaflet, etc., and so many inbuilt functions as well.   It is easier to customize graphics in R compared to Python. R is used in feature engineering and in exploratory data analysis as well. 71. What is the difference between a box plot and a histogram? The frequency of a certain feature’s values is denoted visually by both box plots and histograms.  Boxplots are more often used in comparing several datasets and compared to histograms, take less space and contain fewer details. Histograms are used to know and understand the probability distribution underlying a dataset. The diagram above denotes a boxplot of a dataset. 72. What does NLP stand for? NLP is short for Natural Language Processing. It deals with the study of how computers learn a massive amount of textual data through programming. A few popular examples of NLP are Stemming, Sentimental Analysis, Tokenization, removal of stop words, etc. 73. Difference between an error and a residual error The difference between a residual error and error are defined below - The difference between the actual value and the predicted value is called an error. Some of the popular means of calculating data science errors are - The difference between the arithmetic mean of a group of values and the observed group of values is called a residual error. An error is generally unobservable.  A residual error can be represented using a graph. A residual error is used to show how the sample population data and the observed data differ from each other.  An error is how actual population data and observed data differ from each other. 74. Difference between Normalisation and Standardization X’ = (X - Xmin) / (Xmax - Xmin) Here, Xmin - feature’s minimum value, Xmax - feature’s maximum value. X’ = (X - 𝞵) / 𝞼 75. Difference between Point Estimates and Confidence Interval Confidence Interval: A range of values likely containing the population parameter is given by the confidence interval. Further, it even tells us how likely that particular interval can contain the population parameter. The Confidence Coefficient (or Confidence level) is denoted by 1-alpha, which gives the probability or likeness. The level of significance is given by alpha.  Point Estimates: An estimate of the population parameter is given by a particular value called the point estimate. Some popular methods used to derive Population Parameters’ Point estimators are - Maximum Likelihood estimator and the Method of Moments. To conclude, the bias and variance are inversely proportional to each other, i.e., an increase in bias results in a decrease in the variance, and an increase in variance results in a decrease in bias. One-on-One Data Science Interview Questions To crack a data science interview is no walk in the park. It requires in-depth knowledge and expertise in various topics. Furthermore, the projects that you have worked on can significantly boost your potential in a lot of interviews. In order to help you with your interviews, we have compiled a set of questions for you to relate to. Since data science is an extensive field, there are no limitations on the type of questions that can be inquired. With that being said, you can answer each of these questions depending on the projects you have worked on and the industries you have been in. Try to answer each one of these sample questions and then share your answer with us through the comments. Pro Tip: No matter how basic a question may seem, always try to view it from a technical perspective and use each question to demonstrate your unique technical skills and abilities. 76. Which is your favorite machine learning algorithm and why? One of the popular and versatile machine learning algorithms is the Random Forest. It's an ensemble method that combines multiple decision trees, providing high accuracy, handling both classification and regression tasks, and reducing overfitting. Its ability to handle large datasets and diverse feature types makes it a powerful choice in various applications. 77. Which according to you is the most important skill that makes a good data scientist? The most important skill that makes a good data scientist is a strong foundation in statistics. Data scientists need to understand statistical concepts to analyze and interpret data accurately, draw meaningful insights, and make data-driven decisions. This skill allows them to select appropriate modeling techniques, handle uncertainty, and effectively communicate findings to stakeholders, ensuring the success of data-driven projects. 78. Why do you think data science is so popular today? Data science is popular today due to the explosion of data and the potential to extract valuable insights from it. Organizations across various industries recognize the importance of data-driven decision-making to gain a competitive edge. Moreover, advancements in technology and accessible tools have made data science more approachable, attracting professionals from diverse backgrounds to harness data's power for innovation and problem-solving. 79. Explain the most challenging data science project that you worked on. The most challenging data science project I encountered involved analyzing vast amounts of unstructured text data from various sources. Extracting meaningful insights required advanced natural language processing techniques, sentiment analysis, and topic modeling. Additionally, handling data quality issues and ensuring scalable processing posed significant hurdles. Collaborating with domain experts and iteratively refining models were crucial to deliver accurate and actionable results. 80. How do you usually prefer working on a project - individually, small team, or large team? For projects, I can provide support individually, in small teams, or as part of larger teams. My adaptability allows me to assist in diverse settings, leveraging my capabilities to meet project requirements effectively and contribute to successful outcomes, regardless of team size. 81. Based on your experience in the industry, tell me about your top 5 predictions for the next 10 years. AI continues to evolve, becoming an integral part of daily life, driving innovation in healthcare, education, and transportation. Climate change initiatives reshape economies, with renewable energy, electric vehicles, and sustainable practices mainstream. Advances in biotech lead to breakthroughs in personalized medicine and anti-aging therapies. Virtual and augmented reality become widespread, transforming entertainment, work, and social interactions. Cryptocurrencies and blockchain technologies gain wide acceptance, changing the financial landscape. 82. What are some unique skills that you can bring to the team as a data scientist? As a data scientist, I bring expert knowledge in machine learning, statistical modeling, and data visualization. My ability to translate complex data into actionable insights is valuable. I have proficiency in programming languages like Python, R, and SQL, crucial for data manipulation and analysis. Additionally, my experience with big data platforms and tools, along with strong problem-solving skills, uniquely position me to contribute. 83. Were you always in the data science field? If not, what made you change your career path and how did you upgrade your skills?  No, I have switched to Data Science field recently due to the ever increasing opportunities in the domain. 84. If we give you a random data set, how will you figure out whether it suits the business needs or not? To ensure a random dataset suits business needs, first understand the business objectives and key performance indicators. Then, assess the dataset's relevance, quality, and completeness with respect to these objectives. If necessary, perform exploratory data analysis to uncover patterns or trends. Confirm that the dataset contains actionable insights that can drive business decisions. 85. Given a chance, if you could pick a career other than being a data scientist, what would you choose? The role of a Data Engineer is a vital and rewarding profession. They are responsible for designing, building, and managing the data infrastructure. They create the architecture that enables data generation, processing, storage, and retrieval. Their work allows data scientists to perform analyses and make meaningful contributions. 86. Given the constant change in the data science field, how quickly can you adapt to new technologies? I'm a keen learner and always ready to upskill. I think I will be able to adapt to new technologies in no time. 87. Have you ever been in a conflict with your colleagues regarding different strategies to go about a project? How were you able to resolve it? Yes, once I remember. However, it was resolved in no time. Resolving differences in strategies among colleagues requires: Open Communication: Initiate a dialogue to understand each person's perspective. Active Listening: Allow each colleague to express their views fully. Find Common Ground: Identify shared goals or priorities that everyone agrees on. Collaborative Decision-Making: Encourage participation in the decision-making process. Compromise: Recognize that a perfect solution may not exist and compromise might be needed. Feedback and Follow-up: Regularly review the strategy's progress and adjust as needed. 88. Can you break down an algorithm you have used on a recent project? Yes, I cam do that. 89. What tools did you use in your last project and why? Programming Languages: Python, R, SQL for data manipulation and analysis. Libraries: Pandas, NumPy, Scikit-learn for data processing and machine learning. Visualization Tools: Matplotlib, Seaborn, Tableau for data visualization. Big Data Platforms: Hadoop, Spark for handling large datasets. Machine Learning Platforms: TensorFlow, PyTorch for creating ML models. Notebooks: Jupyter, Google Colab for prototyping and sharing work. 90. What is your most favored strategy to clean a big data set and why? My most favored strategy is iterative cleaning, where data is cleaned in stages or chunks, rather than all at once. This approach, often combined with automation tools, is efficient and manageable for large datasets. It allows for quality checks at each stage, minimizes the risk of data loss, and enables timely error detection. 91. Do you contribute to any open source projects? I have contributed to open source projects in several ways: Developing algorithms and models Improving data processing: Visualizations and dashboard creation Documentation and Tutorials Bug fixing and feature requests Are you looking forward to becoming a Data Science expert? This career guide is a perfect read to get you started in the thriving field of Data Science. Download the eBook now ! Stay Sharp With Our Data Science Interview Questions For data scientists, the work isn't easy, but it's rewarding and there are plenty of available positions out there. These data science interview questions can help you get one step closer to your dream job. So, prepare yourself for the rigors of interviewing and stay sharp with the nuts and bolts of data science. Simplilearn's comprehensive Post Graduate Program in Data Science , in partnership with Purdue University and in collaboration with IBM will prepare you for one of the world's most exciting technology frontiers. Find our Post Graduate Program in Data Science Online Bootcamp in top cities: Name Date Place Cohort starts on 16th Sep 2024, Weekend batch Your City Cohort starts on 7th Oct 2024, Weekend batch Chicago Cohort starts on 28th Oct 2024, Weekend batch Houston About the Author Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football. Recommended Programs Post Graduate Program in Data Science Caltech Post Graduate Program in Data Science Data Scientist *Lifetime access to high-quality, self-paced e-learning content. Find Post Graduate Program in Data Science in these cities Recommended resources. 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the meanings of the words below then use the words i Component 5 think about your learning in this lesson write your respon Component 5 q1 the focus of the lesson was on learning about how an ar Acct 2110 summer 2024 ce 3 chapter 5 due friday 621 1159 Question 8multiple choice worth 4 points 0201 mc exam Checking accounts offer following except account balance Negotiating price purchase home following except assess Relationship risk return positive investors need Question 3 submit button start stop activity come back Appropriate run shift shorthanded select best response Question 7 c credit score affect persons financial Gives vehicle certified preowned description certified Question 6 gives vehicle certified preowned description Effective managment technique planning consistent Type organization top management shares decision making Basic skills involved communication process negotiating Memo written message someone companyit usually brief Basic management functions planning organizing School Guide Mathematics Number System and Arithmetic Trigonometry Probability Mensuration Maths Formulas Class 8 Maths Notes Class 9 Maths Notes Class 10 Maths Notes Class 11 Maths Notes Class 12 Maths Notes Solving Trigonometric Equations with Identities Trigonometric equations are the equations that include functions of trigonometry. These equations are used to find the significance of curves that fulfil certain requirements, and they are essential in various fields like physics, engineering, and computer illustrations. Table of Content What are Trigonometric Equations? Important trigonometric identities, trigonometric equations: solved problems, practice problems: trigonometric equations, answer key: trigonometric equations. The  trigonometric equations  involve trigonometric functions of angles as variables. The angle of θ trigonometric functions such as Sinθ, Cosθ, and Tanθ is used as a variable in trigonometric equations. Similar to general polynomial equations, trigonometric equations also have solutions, which are referred to as principal solutions, and general solutions. Solving a trigonometric equation applies to finding all the values of the variable that fulfil the equation within a bounded interval. Pythagorean Identities sin⁡ 2 (x)+cos⁡ 2 (x)=1 1+tan⁡ 2 (x)=sec⁡ 2 (x) 1+cot ⁡2 (x)=cosec⁡ 2 (x) Even Trigonometric Functions For even functions, f(−x)=f(x). The cosine and secant functions are even: cos⁡(−x)=cos⁡(x) sec⁡(−x)=sec⁡(x) Odd Trigonometric Functions For odd functions, f(−x)=−f(x). The sine, tangent, cotangent, and cosecant functions are odd: sin⁡(−x)=−sin⁡(x) tan⁡(−x)=−tan⁡(x) cosec⁡(−x)=−cosec⁡(x) cot⁡(−x)=−cot⁡(x) Quotient Identities sin⁡(a ± b) = sin⁡(a)cos⁡(b) ± cos⁡(a)sin⁡(b) cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b) [Tex]\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \tan(b)}[/Tex] Double-angle formulas sin(2x) = 2sin(x)cos(x) cos⁡(2x) = 2cos ⁡2 (x) − 1 cos⁡(2x) = 1 − 2sin⁡ 2 (x) Reciprocal Identities tan(2x) = 1−tan 2 (x) sec(x) = 1/cos(x)​ cosec⁡(x) = 1/sin⁡(x) cot⁡(x) = 1/tan⁡(x) Example 1: Solve sin⁡ 2 (x) − sin⁡(x) = 0 for 0 ≤ x < 2π. Given: sin ⁡2 (x) − sin⁡(x) = 0. sin⁡(x)(sin⁡(x) − 1) = 0. Now, sin⁡(x) = 0 or sin⁡(x)−1=0 which simplifies to sin⁡(x)=1. sin⁡(x)=0 Solutions: x=0, π, 2π sin⁡(x)=1 Solution: x = π/2​. Thus, all possible value for x are: x = 0, π/2, π, 2π. Example 1: Solve 2sin⁡ 2 (x) + 3sin⁡(x) − 2 = 0 for 0 ≤ x < 2π. Given: 2sin⁡ 2 (x) + 3sin⁡(x) − 2 = 0. Which is similar to quadratic equation in sin x. Comparing with ax 2 + bx + c = 0, we get a = 2, b = 3 , c = −2. Apply the Quadratic Formula : [Tex]sin(x) = \frac{-3 \pm \sqrt{3^2 – 4(2)(-2)}}{2(2)}[/Tex] ⇒ [Tex]sin(x) = \frac{-3 \pm \sqrt{9 + 16}}{4}[/Tex] ⇒ [Tex]sin(x) = \frac{-3 \pm \sqrt{25}}{4}[/Tex] ⇒ [Tex]\sin(x) = \frac{-3 \pm 5}{4}[/Tex] So, we get two possible solutions: sin⁡(x) = 2/4 = 1/2 or sin⁡(x) = −8/4 = −2 sin⁡(x) = 1/2​: The solutions are x=π/6,5π/6. sin⁡(x) = −2 No solution, since the sine function cannot be less than -1. Thus, all possible value for x are: x = π/6, 5π/6​. Problem 1: Solve [Tex]\sin(x) = \frac{\sqrt{3}}{2}[/Tex] ​​ for 0≤x<2π. [Tex]\sin(x) = \frac{\sqrt{3}}{2}​​[/Tex] corresponds to angles where sine has this value. From the unit circle, [Tex]\sin(x) = \frac{\sqrt{3}}{2}​​[/Tex] at [Tex]x = \frac{\pi}{3}[/Tex] ​ and [Tex]x = \frac{2\pi}{3}[/Tex] 3}​. Solutions : [Tex]x = \frac{\pi}{3}​[/Tex] , [Tex]x = \frac{2\pi}{3}​[/Tex] pi}{3}​. Problem 2: Solve [Tex]\cos(x) = -\frac{1}{2}[/Tex] for 0≤x<2π. cos⁡(x)=−1/2 corresponds to angles where cosine has this value. From the unit circle, cos⁡(x)=−1/2​ at x=2π/3 and x=4π/3​. Solutions : x=2π/3​, x=4π/3​. Problem 3: Solve tan⁡(x)=1for 0≤x<2π. tan⁡(x)=1 corresponds to angles where tangent has this value. From the unit circle, tan⁡(x)=1 at x=π/4​ and x=5/4. Solutions : x=π/4, x=5π/4​. Problem 4: Solve 2sin⁡(x)−1=0 for 0≤x<2π. Rearrange the equation: 2sin⁡(x)=1  ⟹  sin⁡(x)=1/2. From the unit circle, sin⁡(x)=1/2 at x=π/6​ and x=5π/6​. Solutions : x=π/6​, x=5π/6​. Problem 5: Solve cos⁡ 2 (x)=1/4​ for 0≤x<2π. Take the square root: cos⁡(x)=±12​. For cos⁡(x)=1/2​: x=π/3​, x=5π/3​. For cos⁡(x)=−1/2​: x=2π/3​, x=4π/3. Solutions : x=π/3​, x=5π/3​, x=2π/3​, x=4π/3​. Problem 6: Solve sin⁡(2x)=sin⁡(x) for 0≤x<2π. Use the double-angle identity: sin⁡(2x)=2sin⁡(x)cos⁡(x). Set up the equation: 2sin⁡(x)cos⁡(x)=sin⁡(x). Factor: sin⁡(x)(2cos⁡(x)−1)=0. So, sin⁡(x)=0 or 2cos⁡(x)−1=0. sin⁡(x)=0at x=0, x=π. 2cos⁡(x)−1=0 gives cos⁡(x)=1/2​, so x=π/3​, x=5π/3. Solutions : x=0, x=π, x=π/3​, x=5π/3​. Problem 7: Solve 3cos⁡ 2 (x)−2=0 for 0≤x<2π. Rearrange: 3cos ⁡2 (x)=2  ⟹  cos⁡ 2 (x)=2/3. Take the square root: [Tex]\cos(x) = \pm \sqrt{\frac{2}{3}}​​[/Tex] ​​. For [Tex]\cos(x) = \sqrt{\frac{2}{3}}[/Tex] ​​: [Tex]x = \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] and [Tex]x = 2\pi – \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] . For [Tex]\cos(x) = -\sqrt{\frac{2}{3}}[/Tex] ​​: [Tex]x = \pi – \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] and [Tex]x = \pi + \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] . Solutions : x values are [Tex]x = \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] , [Tex]x = 2\pi – \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] , [Tex] x = \pi – \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] , [Tex]x = \pi + \cos^{-1}(\sqrt{\frac{2}{3}})[/Tex] . Problem 8: Solve tan⁡ 2 (x)−1=0 for 0≤x<2π. Rearrange: tan⁡ 2 (x)=1. Take the square root: tan⁡(x)=±1. For tan⁡(x)=1: [Tex]x = \frac{\pi}{4}[/Tex] }​, [Tex]x = \frac{5\pi}{4}[/Tex] ​. For tan⁡(x)=−1: [Tex]x = \frac{3\pi}{4}[/Tex] ​, [Tex]x = \frac{7\pi}{4}[/Tex] ​. Solutions : [Tex]x = \frac{\pi}{4}[/Tex] , [Tex]x = \frac{5\pi}{4}​[/Tex] , [Tex]x = \frac{3\pi}{4}[/Tex] ​, [Tex]x = \frac{7\pi}{4}[/Tex] . Q1. Solve [Tex] \sin(x) + \frac{1}{2} = 0[/Tex] for 0≤x<2π. Q2. Solve [Tex]\cos(x) – \frac{\sqrt{3}}{2} = 0[/Tex] for 0≤x<2π. Q3. Solve [Tex]\tan(x) = \sqrt{3} [/Tex] for 0≤x<2π. Q4. Solve sin⁡ 2 (x)−cos⁡ 2 (x)=0 for 0≤x<2π. Q5. Solve 2sin⁡(x)cos⁡(x)=12 for 0≤x<2π. Q6. Solve cot⁡(x)=1 for 0≤x<2π. Q7. Solve sin⁡(2x)=sin⁡(x) for 0≤x<2π. Q8. Solve cos⁡ 2 (x)−sin⁡(x)=0 for 0≤x<2π. Q9. Solve 3sin ⁡2 (x)−2=0 for 0≤x<2π. Q10. Solve [Tex] \tan(x) – \sqrt{3} = 0[/Tex] for 0≤x<2π. 1. x = 7π/6​, 11π/6​ 2. x = π/6, 11π/6​ 3. x = π/3, 4π/3 4. x = π/4, 5π/4 5. x = π/6, 5π/6 6. x = π/4, 5π/4 7. x = 0, π/2, π, 2π 8. x = 0, π/6, π 9. x = π/3, 5π/3​ 10. x = π/3, 4π/3​ Trigonometry Table Trigonometric Identities Inverse Trigonometric Identities Pythagorean Trig Identities FAQs: Trigonometric Equations with Identities What are trigonometric identities. Trigonometric identities are mathematical equations that express relationships between the trigonometric functions (sine, cosine, tangent, etc.). What is a trigonometric equation? A trigonometric equation is an equation that involves one or more trigonometric functions of a variable. How are trigonometric identities used in solving equations? Trigonometric identities are used to simplify complex trigonometric equations, making them easier to solve. What if I have multiple trigonometric functions in the equation? Combine or transform the functions using identities. For example, sin ⁡2 (x)+cos ⁡2 (x)=1 can help simplify expressions involving both sine and cosine. How do I check if my solutions are correct? Substitute your solutions back into the original equation to verify they satisfy it. Ensure solutions fall within the specified interval. Please Login to comment... Similar reads. School Learning Top Android Apps for 2024 Top Cell Phone Signal Boosters in 2024 Best Travel Apps (Paid & Free) in 2024 The Best Smart Home Devices for 2024 15 Most Important Aptitude Topics For Placements [2024] Improve your Coding Skills with Practice What kind of Experience do you want to share? Share full article For more audio journalism and storytelling, download New York Times Audio , a new iOS app available for news subscribers. What Phil Donahue Meant to Me The iconic daytime television host redefined the genre, letting the audience participate for the first time.. This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions. From “The New York Times,” I’m Michael Barbaro. This is “The Daily.” Today, a tribute to Phil Donahue, the King of Daytime Talk, from me. It’s Friday, August 30. On the inside. For clothes that — Tristan lost — That bold three detergent plus fabric softener. [THEME MUSIC] From NBC News, this is “Today.” When I was growing up, there were four television shows that I watched religiously. The “Today” show — I love that Wayne Gretzky. And his wife, the doll, too. And their children are really cute. — because Matt and Katie were pure magic together on that set. [CLOCK TICKING] “60 Minutes —” How cigarettes can destroy people’s lives. — because nobody has ever told stories like that on network TV. You think that I don’t trust my husband? “General Hospital —” I do trust my husband. He loves me. And we’re married and we’re happy. And there’s nothing that you, or Miranda, or anyone is ever going to do to change that — because all of us have a guilty pleasure. And finally, “The Phil Donahue Show.” Now, why Donahue? Your parents do not know that you lead this double life. You leave the apartment after they’ve gone to sleep. Is that right? Yes. Good. That’s a good answer, yes. Well, I mean, they have no idea. Or I sleep at other people’s houses. Why was I, why were millions of other Americans drawn to this middle-aged host of a daytime talk show? You dress up like this because first of all, it’s fun, and hey — This guy with a helmet of gray hair and, what always seemed to me anyway to be the world’s longest, slimmest microphone — As far as songs go, every song is a message. — who sprinted around his studio in a beige three-piece suit. — message of “Goodnight, Irene?” The lyrics were, “Irene, goodnight, Irene.” She’s talking about my song. It never really occurred to me to try and answer that question. Donahue has been off the air now for more than 20 years. He’s no longer a household name. The culture has kind of forgotten him. But then a few days ago, he died at the age of 88. And suddenly, I wanted to know, what had it been about Donahue? [UPBEAT MUSIC] I logged on to YouTube and I started to watch his show again. Because he’s kind of been an imposter. Why should I? Why should I love the Lord? Why should anybody else love the Lord? What does the Lord ever done for any of us? First, I don’t know. And I was right back in my childhood basement with the gray commercial carpet and the exposed pipes. — and so on. And if you’re married, you can’t have the job because you might have children. And if you have children, you can’t have the job because they might get sick and you’d have to take care of them. If you didn’t, you’d be a bad mother. I mean, any condition of being a female still may be used against you, no matter what. And what I remembered right away was Donahue’s extraordinary intellectual range. 135,000 citizens were displaced because of the explosion at Chernobyl. And some of them lived here. One day, he’d interview a presidential candidate. Throughout this imperfect part of your marriage, did you ever separate, you and Hillary? No, that’s none of your business if we did. [APPLAUSE] The next, an activist. If we don’t put a halt to this new kind of nuclear war fighting, which we’re moving into, we are going to guarantee that we have a nuclear war. The day after that, a celebrity. Please welcome the Divine Miss M, here’s Bette Midler. [APPLAUSE] An illiterate adult — You came out of the closet, so to speak, two years ago? October 16. And no one knew except my wife. — who somehow had outwitted his college professors and his bosses, and still couldn’t read at the age of 45, despite having a big corporate job. And I might add that I am healing, as we all are healing from the trauma of being an illiterate in this dominant literate society. It must be terrifying. Now, remember, this is daytime talk at a time when daytime talk didn’t really exist. Donahue is competing with soap operas and game shows. And his show was the exact opposite of all of that escapism. It was the AIDS crisis — To what? That we don’t have to worry about catching AIDS in the air. Right, AIDS is not transmitted by casual contact. It never has and it probably never will be. It was the hardcore scene in New York. He devoted an entire episode to that. Yeah, well, share with us some of the feelings that make you feel comfortable in this group. Well, people are always seeming to try to tell us what to do, where to go, how to talk, how to walk, what to wear. And we’re just trying to say that maybe there’s an alternative to what is set before us, and told, dictated to us. We’re saying maybe there’s — And it was this incessant curiosity about ideas and motivations, why people did what they did, why they believed what they believed, what made people who they are. But that doesn’t quite explain why I think so many of us were watching Donahue. It was this thing he did on top of all of that. It was how he treated his audience. Now remember, until this point, a studio audience was basically an inanimate object. Mike, you know about the evil presence in my office, right? Of course I do, Paul. She’s standing right next to you. Good afternoon. They laughed, sometimes literally on a laugh track. They clapped in unison. They were basically a prop. Let me go out in the audience and get some observations. Now, what do you think of all this business of student protest, for example, Columbia, Berkeley, and other — But in Donahue’s hands, the audience became just as important as the guest on the stage. Yes, ma’am. I think there’s racism everywhere. And you cannot pinpoint it on a particular race, but it depends on the individual. And I think that as long as we keep calling people Black and white, that’s when the racism is going to continue. I know you want to counter. Well, as soon as that happens, you let me know, OK? We’re not making the — And this wasn’t an accident. This was quite deliberate. Donahue made the audience central to the show from the very beginning. And he talked about how it happened, and why it happened in interviews. Welcome to “Speaking Freely.” I’m Ken Paulson. It’s a pleasure to welcome Phil Donahue. And he liked to tell the story of how the show’s origins in Dayton, Ohio, forced him to do it. We tried to get movie stars. Everybody but us had movie stars. We would call movie stars and they’d say, Dayton? That’s the Soapbox Derby. I said, no, that’s Akron. Dayton is — He could never persuade big-name guests to come to Dayton. So he gravitated to issues. We discovered that issues would keep us on the air. Issues. And when it came to issues, it turned out that the most interesting perspectives were not his. And suddenly, the audience is starting to ask better questions than I was during the commercials. And I got up one day and walked out. And we realized now that if it hadn’t been for that, we probably would not have survived. I just don’t think you can sell two talking heads in front of a curtain for very long. Now you tell me. And so slowly but surely, he started to turn his microphone and his show over to his audience. And people’s hands were going up all over. And I couldn’t get to them fast enough. And since this was the 1960s and the 1970s, and it was in the middle of men’s workday, turning his microphone over to the audience really meant turning his microphone and eventually his show over to women. Sexism was rampant at the time. The mantra in the television game was the only thing women care about is covered dishes, needlepoint, and children, and mothering, that’s all. And we came along and it was clear that behind this stereotype were thinking, live human beings who wanted to get in the act, who had something to say, who wanted to kick tires, who wanted to get mad, who were mad at doctors for patronizing them. And we exploited all this to our own advantage. So in the relative obscurity of Dayton, Ohio, Donahue was undertaking a pretty radical experiment in the history of television. He was asking women what they thought. And he was taking their lives and their needs very seriously. We are inside an abortion clinic in Chicago. The patient with her back to the camera, is in the first trimester of an unwanted pregnancy. He televised an abortion. Our patient, having been told what to expect, walks to the treatment room where she meets the doctor for the first time. The medical term for this abortion is vacuum aspiration curettage. He televised a tubal ligation surgery. You can put the baby on your breast. He televised a child’s birth. She is big. [CHATTER] Of course, not all his gestures towards women were super high-minded. What’s up, ladies? For those of you that prefer Italian men, one of Houston’s top models, Mr. GQ himself, the Italian Stallion. There were the episodes about male strippers where these guys came out on the set, took off all their clothes, and the women went wild. (SINGING) Wake up in a city that — And Donahue made very clear that those pitches came from the women on his staff, not him. Then there’s this moment in an episode in 1979 — Here’s a woman who’s read by millions around the world. She may be our most debated philosopher. — where all of these puzzle pieces of what made Donahue Donahue come together — his curiosity, his female audience, and these feminist ideas that his show so often probed. A warm human being who has a lot to say and comes straight at everything she says. I am pleased to present Ayn Rand. Miss Rand. It was an interview with the writer Ayn Rand. So your view is if we all became more comfortable with our natural tendencies, that is to say, selfishness, there would be less horror, less war, less Hitler. There wouldn’t be any. And just think about that for a moment. Ayn Rand, one of the great public intellectuals of her era or really any era, this champion of rational selfishness and capitalism unbound on daytime television. So with the more selfish we are, the more tranquil and peaceful the world in which we live? And more benevolent toward other people, if we are rationally selfish. And this moment starts, as so many great Donahue moments do, with a question from the audience. Miss Rand, in your novels portray very strong women. I was wondering why you think in the world we don’t have strong women leaders? Because if you’re speaking about women’s liberation, that whole movement, it’s a very false and phony issue. And Rand responds by basically casting doubt on the whole movement for women’s rights. Women are human beings, so they need leaders, just like men. They need leaders who are men or women, as the leaders have earned. And then Donahue jumps in. Well, but the point is that women feel because of the cultural inhibitors that have been placed on women, some sort of woman leadership is needed. And he asks Ayn Rand how she thinks that women can get ahead. For him, this avowed feminist, the answer seems pretty obvious. What’s needed is a formal, sustained effort to advocate for women’s equality. You can do it only by education. You do it by spreading the right idea that women, intellectually, are not the inferior of men. Of course not. Physically, they certainly are. That’s what feminists are doing. They’re standing up and educating. But Rand totally rejects that. They are asking for government power and government handouts. They go around depriving men of jobs because you have to have quotas of so many women. But their point is that they have been denied jobs all these years. Donahue wants to talk about systemic barriers. Rand wants to talk about hard work. Well, what should they do, be nice little girls and not say anything and stay home and break bread? Well, what should they do? Go into any career of their choice, except longshoreman or professional football player as they’re trying today. And fight for their career as every man has to fight. In her mind, women simply have to prove themselves one by one. And in Rand’s telling, in any reasonable, logical, free market economy, talented women will eventually just rise. It will happen. All you have to do is show your ability. And if someone is prejudiced and doesn’t hire you, the intelligent employer will. But then, as the conversation keeps going, this heady, fascinating back and forth about feminism and capitalism, something really interesting happens when another woman in the audience asks Rand a question. Fifteen years ago, I was impressed with your books and I sort of felt that your philosophy was proper. Today, however, I’m more educated, and I find that if a company — This is what I don’t answer. But wait a minute, you haven’t heard the question yet. And when Rand responds — She’s already estimated her position and my work, incidentally, displaying the quality of her brain. If she says today, she is more educated. I am more educated now than I was 15 years ago when I was in high school, before I went to college, before I read the newspaper. I’m not interested in your biography. She is exceptionally dismissive of this woman. Let her make her point. It’s very basic. If a company is permitted to do what it wants to do, like IT — Donahue tries to create some space for this audience member to speak, but — Can we encourage you to make a contribution to that observation? I will not answer anyone who is impolite, but to assure you — She wasn’t impolite. I do not sanction impoliteness. And I am not the victim of hippies. Rand’s disdain completely overpowers everything. If anyone else wants to ask the same question politely, I’ll be delighted to answer. But there was nothing impolite. You are punishing this woman for the vigor and energy that she brought to the dialogue. And that’s not fair to her. This is the kind of woman we spend a long time trying to attract to our television audience. And what I realized was that this was a moment that could only happen on “Donahue.” It was a moment that I don’t think ever would have happened if it were just Donahue and Ayn Rand sitting on stage talking to one another. I don’t think Rand would have been that rude to this powerful TV host. She would only act that way toward an ordinary person. What you get, because of this complicated ecosystem that Donahue has created, is this totally unfiltered version of this intellectual titan. And it’s pretty ugly. And while you’re watching this happen, you start to wonder what truly animates Ayn Rand. Is it this ruthless, uncompromising philosophy at the center of her best-selling books or is it maybe that she just doesn’t like other people? Whatever was really going on here, it is revealing, it is messy, it is unexpected, and it is fantastic television. And all of it was orchestrated by this guy, Philip John Donahue, whose biography in no way prepares you for this kaleidoscopic, boundary-pushing national conversation that he invited the country to have day after day for 30 years. And we’ll be back in just a moment. Who was Phil Donahue? My father always had a job. I was born in 1935. His unemployment preceded my birth. He felt the Depression? Oh, yes, my parents did. Absolutely. He was born six years after the Great Depression into an Irish Catholic family in Cleveland. His dad sold furniture. His mom sold shoes. I worked for the nuns for $0.50 an hour when I was 10 -, 12-years-old. He went to a Catholic day school and later a Catholic college. And in his telling, Catholicism was the scaffolding for his entire way of thinking. I had 16 years of Catholic education. I had most of the answers. Who made me? God made me. Why did God make me? I knew the answers to the toughest questions. And then in the ‘60s, everything started to fall apart. And then he starts to rethink everything, especially his relationship with the Church. We began to realize that we really did have two Americas, a Black one and a white one. And the liberal guilt, my conscience began to manifest itself. And I began to question the answers that had been given. And suddenly, my mind was racing, I guess I’d have to say. He gets really mad at his local diocese, which is building a fancy new cathedral where he thinks it’s least needed. Who else would spend a million dollars on a building that is used about four and a half hours a week. And he begs the church, instead, to put that money into inner city Catholic schools. And suddenly we were saying, hold it, hold it, now, we’ve been listening to you. Now, you listen to us. But the church leaders ignore him. The church was built at a cost of a million dollars, including a bell tower. It is centrally air conditioned. It stands today in Centerville, Ohio, I think, as a very hard cold monument to what churches are everywhere, almost always dark and empty. And when he loses that battle to the Church, the Church loses him. By the time he enters broadcasting, Donahue strongly identifies with the powerless — Don’t talk about subsidizing the farmer. The man that printed the box made more than the man that grew the corn. — farmers — In our lifetime, we’ve traveled in a corridor of fear. — gay men — Fear of employers finding out, fear of fellow employees, fear of landlords, fear of the family. But we finally have decided, at our respective ages, to put that aside and to tell the world about our relationship and that we’re very proud of it. — Black women — You’re not surprised that there’s not a greater participation of women of color in the women’s movement? Oh, there are huge numbers of women of color involved in the women’s movement. It’s just that we don’t get the media. This is a real treat for us. [LAUGHING] — and sees his job as challenging the rich and the powerful. Why couldn’t the millions of men, women, and children who are Arab and who find themselves in this desperate conflict and look around wondering where peace will be, why can’t they be angry with you for your characterization of them, your roundhouse criticism of them? That sensibility is a through line across every episode of his show. And you really see it in an interview that he did in 1987 — Are you 40? — with a young Donald Trump. You’re a star, Mr. Trump, and you’re a businessman. And you do not run away from publicity. Trump is flying high. His first book, “The Art of the Deal,” has just come out. But Donahue keeps bringing the conversation back to the little guy. Well, this is interesting, because as you know, you’re the fat cat developer and you know the book on you is that you throw little, old ladies who can’t afford the rent out of the apartment. I don’t think that’s the book of me, if you want to know. He wants to point it down to the street, not up at Trump’s penthouse in Trump Tower. Your father, was there a lawsuit that you didn’t have enough Blacks or he didn’t have enough Blacks in his project? And that upset you. I didn’t like it because it wasn’t fact. And I decided to fight it. And at one point, Donahue reads from a passage in Trump’s book — “The fact was that we did rent to Blacks in our buildings. What we didn’t do was rent to welfare cases, white or Black.” — in which Trump says that he would never rent a unit in one of his buildings to anybody on welfare. “I watched what happened when the government came after Samuel LeFrak, another builder, and he caved in and started taking welfare cases. They virtually ruined his building.” [RUBBING HANDS] Donahue, at this moment, dramatically rubs his hands together as if preparing to go into battle. Isn’t that, aren’t you pretty close here to looking like an insensitive guy from atop your Trump Tower looking down on the Wollman Rink over the vast holdings of your own empire? Shouldn’t we have just a little more understanding from a man of your influence and wealth on the issue of making New York livable for all of us, safety on the subway. Absolutely. Then we can’t continue to give you guys these big tax breaks. And that would go for — And Trump does what we all now to be his go-to move when somebody tries to hold him accountable — So when everybody else in the city gets it, but Donald Trump, when Koch and the administration tries to stop Donald Trump. And I don’t say, give me the tax breaks. I say, don’t give everyone else the tax breaks. — he makes himself the victim. No, I’m honest. Hey, I’m not running for anything, Phil. I’m not running for office. I don’t have to lie in a book. I want to tell the facts, OK? And toward the end of the episode, as always — You keep saying you’re not running for office, but why don’t you? The most prescient questions come from the women in the audience. No, I wouldn’t want to run for mayor of New York. I’d like to see somebody talented do that. But I really have no intention of running for mayor. Thank you. But you definitely are a political person, whether you run for office by what everything that you say and do points in that direction. You know what it is? I don’t like being taken advantage of, OK? They saw Trump’s future even before Trump did. By the late 1980s, “The Phil Donahue Show” was a bona fide hit. It’s syndicated across the country. And the wait time for tickets to be in his studio audience is an astonishing 18 months. And this success opens up an entirely new genre of TV. Copycats are popping up across the daytime schedule. Sound scary? Well, the mother is on today’s show say they are terrified of their own children. Sally Jessy Raphael, Maury Povich — You are not the father! — Geraldo Rivera, Jerry Springer. You have a secret to tell him. And your secret is? I’m a man, Jerry. [AUDIENCE EXCLAIMS] And most important of all — You get a car! You get a car! You get a car! — Oprah, who told Donahue that — If it wasn’t for Phil Donahue, there would never have been an “Oprah Show.” — her career would not have been possible if it weren’t for him. Well. [CHUCKLES] This is a full, full, full circle for me. Well, we’ve watched “Oprah.” As you’ve soared, there is no other single human being who has done with this media what Oprah has done. My Cubs cap is off to you. Oprah aside, the shows that follow Donahue, his illegitimate children as he called them, were nowhere near as thoughtful as his show was. But Donahue steadfastly refused to criticize them. And he was asked to criticize them all the time. You talked about being a naughty show. But it is a far cry from what you see today on television. Are you comfortable with where it’s gone? Well, it’s hard for me to be uncomfortable with what’s happening on television today because I’ve been preached to so much in the 29 years I was on the air. I mean, there were viewers who got messages from God to get me off the air. There were people who felt that the United States of America was going to hell, and Phil Donahue was leading it there with atheists and doing shows like the “March on Skokie by Nazis.” We had Nazis on our program. When people say, what do I think of this or that program? I’m a little bit hesitant. I don’t want to. I feel the shows not worthy of consideration will fall of their own weight. We don’t want a bunch of white men, and that’s usually what it winds up being, behind closed doors deciding what you and I should see. Because for him, TV belonged in the hands of the viewer. Good, bad, smart, stupid, Ayn Rand, or in-studio surprise paternity tests, they all had their place, because the alternative was undemocratic. And one of the main bulwarks against somebody assuming power, who knows what’s good for you, is a free press and unfettered speech by the citizenry, allowing all of us to be heard. We are looking for a cacophony of voices, not a well-trained choir. But eventually, after three decades, that cacophony overtook Donahue. The viewers were voting and they were no longer voting for him. And his show ended its run in 1996. He briefly tried to make a comeback in the early 2000s with a reboot of “The Donahue Show” on MSNBC. The antiwar movement is heating up. Resist the war! But all of his antiestablishment instincts ran up against the cruel realities of cable news after the September 11 terror attacks. At a time when almost everybody else in TV news seemed to be beating the drums of war, Donahue very loudly questioned the coming US invasion of Iraq. This is an email from Michael. I’m 17. I’m the person the Bush administration wants to hold a rifle and go off and kill Iraqis. I would like to know why? Is that too much to ask? And to hear him tell it, his bosses at MSNBC were not interested in a cacophony of voices. They wanted a well-trained choir. It really is funny almost, when you look back on how the management was just frozen by the antiwar voice. We were scolds. We weren’t patriotic. American people disagreed with us. And we weren’t good for business. And his show was canceled after just seven months. [SOMBER MUSIC] As it happens, the year he went off the air for good was the year that I began my career in journalism. And when I think about Phil Donahue now and I try to answer that question of why I was always so drawn to his work, it’s all right there in his show. He respected his audience. He never talked down to them. He sought out nuance wherever he could find it. He forced us way outside our comfort zones. And he challenged us to see ourselves and our neighbors in a new and more generous light. [FANFARE BRASS MUSIC] A few months before Donahue died back in May, President Biden invited him to the White House to receive the Presidential Medal of Freedom, the country’s highest civilian honor. And before social media and clickbait news, Phil Donahue broadcast the power of personal stories in living rooms all across America. He helped change hearts and minds through honest and open dialogue. Over the course of a defining career in television, through thousands of daily conversations, Phil Donahue steered the nation’s discourse and spoke to our better angels. I wish you were still speaking there, pal. You made a big difference. And for once, Phil Donahue, now seated in a wheel chair, didn’t say a word. [SERENE MUSIC] We’ll be right back. Here’s what else you need to know today. Madam Vice President, Governor Walz, thank you so much for sitting down with me. In her first extended interview since becoming the Democratic nominee, Vice President Kamala Harris, joined by her running mate Tim Walz, was pushed by CNN to explain positions she had taken during her first run for president in 2020, but has since backed away from, including banning fracking and decriminalizing illegal border crossings. There was a debate. You raised your hand when asked whether or not the border should be decriminalized. Do you still believe that? I believe there should be consequence. We have laws that have to be followed and enforced, that address and deal with people who cross our border illegally. And there should be consequence. Harris insisted that despite shifting stances on specific policies, her core beliefs have remained the same. How should voters look at some of the changes that you’ve made in your policy? Is it because you have more experience now and you’ve learned more about the information? Is it because you’re running for president in a Democratic primary? And should they feel comfortable and confident that what you’re saying now is going to be your policy moving forward? Dana, I think the most important and most significant aspect of my policy perspective and decisions is my values have not changed. Today’s episode was produced by Michael Simon Johnson, Shannon Lin, Stella Tan, and Asthaa Chaturvedi. It was edited by Michael Benoist, contains original music by Marion Lozano and Dan Powell, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. That’s it for “The Daily.” I’m Michael Barbaro. See you on Tuesday after the holiday. Apple Podcasts Google Podcasts Hosted by Michael Barbaro Produced by Michael Simon Johnson Shannon M. Lin Stella Tan and Asthaa Chaturvedi Edited by Michael Benoist Original music by Marion Lozano and Dan Powell Engineered by Chris Wood Listen and follow ‘The Daily’ Apple Podcasts | Spotify | Amazon Music | YouTube | iHeartRadio Phil Donahue, the game-changing daytime television host, died last week at 88. Mr. Donahue turned “The Phil Donahue Show” into a participation event, soliciting questions and comments on topics as varied as human rights and orgies. Michael Barbaro explains what Phil Donahue meant to him. On today’s episode Michael Barbaro , host of ‘The Daily’ for The New York Times. Background reading An obituary for Mr. Donahue , who died last week at 88. Here are 3 episodes that explain Mr. Donahue’s daytime dominance. There are a lot of ways to listen to The Daily. Here’s how. We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page. The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Michael Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman. 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