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Quiz about To All You MathletesPart II
Quiz about To All You MathletesPart II

To All You Mathletes...Part II Quiz


A follow-up to part I of this (could-be) series. A mix of 6 problems and 4 conceptual questions. You can do all of these without a calculator. Good luck!

A multiple-choice quiz by redsoxfan325. Estimated time: 7 mins.
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Author
redsoxfan325
Time
7 mins
Type
Multiple Choice
Quiz #
298,681
Updated
Dec 03 21
# Qns
10
Difficulty
Very Difficult
Avg Score
4 / 10
Plays
710
Question 1 of 10
1. To begin with, let's try some arithmetic probability. If you're playing a game where you have two tries to guess a number between 1 and 10 (inclusive), what are the odds that you guess the number correctly in one of those two tries (assuming you don't guess the same number twice)? Hint


Question 2 of 10
2. Let's continue with some Algebra I. If you have a rectangular swimming pool of volume 105 m³ whose length is 2 meters longer than its width and whose depth is 2 meters shorter than its width, what is the perimeter (around the top) of the rectangular pool? Hint


Question 3 of 10
3. Let's move on to a little geometry. If you connect the midpoints of all four sides of a quadrilateral to create a new quadrilateral, that new quadrilateral will always be a parallelogram.


Question 4 of 10
4. Next up, Algebra II: What would the set of all points satisfying the equation: √(y²) = √(x²)

(In the answers, abs(x) denotes the absolute value of x.)
Hint


Question 5 of 10
5. Let's do some Precalculus. You've traveled in a motorboat 8 miles away from the dock (the origin) at a 30° angle above the shoreline (the x-axis). You then make a turn back toward the shore and travel exactly 5 miles so that your boat stops exactly on the shoreline.

Given the above information, you could be at exactly two different points on the shore (the x-axis) right now: (A,0) and (B,0). What's the difference (in miles) between these two points?
Hint


Question 6 of 10
6. Differential Calculus: You are blowing up a perfectly spherical balloon. The radius (in centimeters) of the balloon after t seconds is given by r(t)=2√t. At what rate is the volume of the balloon changing at time t=4?

(Remember that the volume of a sphere is (4/3)πr3.)
Hint


Question 7 of 10
7. On to Integral Calculus now. Of the following, which would you want to integrate the least?

(Hint: It's the only one that you can't integrate.)
Hint


Question 8 of 10
8. Uh-oh. It's going to get worse from here. Next up is...gulp...Multivariable Calculus.

Let's say that m is the number of movies you see each month and s is the number of songs you buy on iTunes (you'd never illegally download, of course). Your happiness H is represented by the function H = 10m0.8s0.2. Each movie costs $8 and each song costs $1. You have a $50 budget. Assuming you're spending all of your money, how many movies should you watch and how many songs should you buy to maximize your happiness?
Hint


Question 9 of 10
9. A little different: Linear Algebra. (I'll make this conceptual to give you a break.)

If you know that an nxn matrix A is invertible, what can you NOT also say about it?
Hint


Question 10 of 10
10. Let's get Complex to end this quiz...

This isn't really practical at all, but it's pretty cool. What is the principal value of ii?

The letter i denotes the square root of -1.
Hint



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Quiz Answer Key and Fun Facts
1. To begin with, let's try some arithmetic probability. If you're playing a game where you have two tries to guess a number between 1 and 10 (inclusive), what are the odds that you guess the number correctly in one of those two tries (assuming you don't guess the same number twice)?

Answer: 1/5

The first guess is a 1/10 probability.

The second guess is conditional on your missing the first guess. Thus it is 1/9*9/10 (the probability of guessing incorrectly the first time). 1/9*9/10 = 1/10.

1/10 + 1/10 = 1/5.

Thanks to Rodney_indy and Wesley Crusher for pointing out the condition on the second guess.
2. Let's continue with some Algebra I. If you have a rectangular swimming pool of volume 105 m³ whose length is 2 meters longer than its width and whose depth is 2 meters shorter than its width, what is the perimeter (around the top) of the rectangular pool?

Answer: 24 meters

Your equation is w(w+2)(w-2)=105 which gives w³-4w=105 (where w is the width). Solving for w here gives you w=5. (The other two solutions for w are imaginary, so we can discard them.) Given that the width is 5 meters, the length is 7 meters, and the depth is 3 meters, we know that the perimeter around the top is 2*7+2*5 = 24 meters.
3. Let's move on to a little geometry. If you connect the midpoints of all four sides of a quadrilateral to create a new quadrilateral, that new quadrilateral will always be a parallelogram.

Answer: True

It always works! Try it out with a few different quadrilaterals and you'll see.

The proof is a little hard to explain without diagrams, but it goes generally like this:

Draw the diagonals to the original quadrilateral. By the Midpoint Theory, two sides of the new quadrilateral are parallel to one of the diagonals of the original quadrilateral, and are thus parallel to each other. The same argument applies to the other two sides of the new quadrilateral. Thus the new quadrilateral is a parallelogram.
4. Next up, Algebra II: What would the set of all points satisfying the equation: √(y²) = √(x²) (In the answers, abs(x) denotes the absolute value of x.)

Answer: y=±abs(x)

One thing to note is that √(x²) is |x|, not just x. A way to see this is to note that no matter what value you plug in for x, x² will always be positive, and the square root of a positive number is positive. The same idea holds true for √(y²). Now the equation looks like this: |y|=|x|. To remove the absolute value brackets from y, we can put a ± sign on the other side of the equation. Doing this yields y=±|x|.
5. Let's do some Precalculus. You've traveled in a motorboat 8 miles away from the dock (the origin) at a 30° angle above the shoreline (the x-axis). You then make a turn back toward the shore and travel exactly 5 miles so that your boat stops exactly on the shoreline. Given the above information, you could be at exactly two different points on the shore (the x-axis) right now: (A,0) and (B,0). What's the difference (in miles) between these two points?

Answer: 6 miles

This is the ambiguous case of the triangle problem. You are given two sides and an angle. The first step to solving these problems is to find the distance to the x-axis from the point 8 miles from the origin at a 30° angle. Call the point where this segment intersects the x-axis C. The x-axis, the 8-mile segment and the distance we are trying to find make a 30-60-90 triangle, so we know that the distance we are trying to find is half of 8, or 4 miles. Now that we know this, we can use the Pythagorean theorem to find the distance from A to C and C to B. Segment AC = CB = √(5²-4²) = 3 miles. Segment AB equals segment AC plus segment CB (because C is between A and B), so segment AB equals 6 miles.

Sorry if that explanation was confusing. It's hard to show without pictures.
6. Differential Calculus: You are blowing up a perfectly spherical balloon. The radius (in centimeters) of the balloon after t seconds is given by r(t)=2√t. At what rate is the volume of the balloon changing at time t=4? (Remember that the volume of a sphere is (4/3)πr3.)

Answer: 32π cm³/s

We are looking for dV/dt. We know V(r) and r(t). So we can represent dV/dt as dV/dr * dr/dt. dV/dr is 4πr² and dr/dt is 1/√t. Thus, dV/dt is 4πr²/√t. At t=4, r=4 (from the equation r(t)=2√t). Plugging into dV/dt, we have 4π(4)²/√4 = 32π cm³/s.
7. On to Integral Calculus now. Of the following, which would you want to integrate the least? (Hint: It's the only one that you can't integrate.)

Answer: x/ln(x)

x*ln(x) → This is a one-step integration by parts. Integral: (x²/4)(2*ln(x)-1)
ln(x)/x → This is a simple u-substitution. Let u=ln(x) and du=1/x. Answer: ½(ln(x))²
1/(x*ln(x)) → This is a simple u-substitution. Let u=ln(x) and du=1/x. Answer: ln(ln(x))

x/ln(x) → It is impossible to find a formula for this indefinite integral. You can do a numeral approximation if you are given bounds, but without them, this integral is impossible.
8. Uh-oh. It's going to get worse from here. Next up is...gulp...Multivariable Calculus. Let's say that m is the number of movies you see each month and s is the number of songs you buy on iTunes (you'd never illegally download, of course). Your happiness H is represented by the function H = 10m0.8s0.2. Each movie costs $8 and each song costs $1. You have a $50 budget. Assuming you're spending all of your money, how many movies should you watch and how many songs should you buy to maximize your happiness?

Answer: 5 movies, 10 songs

Using Lagrange Multipliers, we can say that the (components of) gradient vectors* of the function and the constraint should be constant multiples of each other. We can set up three equations:

8m^(-0.2)*s^(0.2) = 8λ
2m^(0.8)*s(-0.8) = λ
8m+s = 50

Solving these three equations simultaneously for m and s yields m=5 and s=10.

* The gradient vector is a vector whose components are the partial derivatives of the functions. Ex: Gradient of f(x,y)=xy+y² is [∂f/∂x, ∂f/∂y]=[y,x+2y].
9. A little different: Linear Algebra. (I'll make this conceptual to give you a break.) If you know that an nxn matrix A is invertible, what can you NOT also say about it?

Answer: There is at least one non-zero vector in the kernel of A.

If an nxn Matrix A is invertible, you can say the following:

1. There exists a unique solution x such that Ax=b, where b is a vector.
2. The reduced-row echelon form of A is the identity matrix.
3. The column vectors of A are linearly independent.
4. There are no non-zero vectors in the kernel of A.
5. The image of A spans Rⁿ.
6. The nullity of A is 0.
7. The rank of A is n.
8. The determinant of A does not equal 0.
9. 0 fails to be an eigenvalue for A.
10. Let's get Complex to end this quiz... This isn't really practical at all, but it's pretty cool. What is the principal value of ii? The letter i denotes the square root of -1.

Answer: e-π/2

If you rewrite i^i as e^(i*ln(i)), you can use the formula for logarithms of imaginary numbers:

Given z = x+iy,
ln(z) = ln(√(x²+y²)) + i*arctan(y/x)

Plugging in from above, ln(i) = ln(√(0+1))+i*arctan(1/0) = 0+iπ/2 = iπ/2.
Now, i*ln(i) = i*iπ/2 = -π/2

Plugging into the original formula, we have i^i = e^(-π/2).

Thanks for taking this quiz. Please rate, and feel free to send me any compliments or corrections.
Source: Author redsoxfan325

This quiz was reviewed by FunTrivia editor crisw before going online.
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