Ten Mathematical Excursions Trivia Quiz

Answer: 3

To know the distance from the chord to the center, you need to know how big the circle is. The provided information describes a house shape inscribed in a circle. Mark the center of the circle (also the center of the square). Draw radii from the center to the two corners of the square that are also corners of the triangle. Draw a radius from the center, straight through the middle of the triangle.

These three lines are equal, because they are all radii. We know the upper part of the center radius, because it's given. the lower part is equal to r/sqrt(2) because there is a 45-45-90 triangle inside the square. so, add these together and set it equal to r, and solve for r and you get r=5. Draw the chord, and draw a radius bisecting it. Label the chord 8 and draw two more radii going to the ends of the chord. Now you should be able to use trigonometry to solve the rest.

Answer: 2i

When raising a complex number to a power, it is easiest in polar form: r*cis(theta). In this form you just raise r to the power and multiply theta by the power. R^(1/3)=2, and theta (in this case, 90 degrees) *(1/3)=30 degrees. But don't forget the extra revolutions of theta, 450 and 810 degrees, which yield 150 and 270 degrees when multiplied by (1/3). Only one answer was left in polar form.

Answer: 45 degrees

This is how I solved it:

I used a parametric equation for the path of the balls. When shot, at angle theta, the horizontal component of it's motion is P*cos(theta), where P is the power of the machine. The vertical component is P*sin(theta). It doesn't matter how strong gravity is, because it would always pull equally on the tennis balls, regardless of their trajectory. Now, placing the machine at the origin, we have:
x=t*P*cos(theta)
y=t*P*sin(theta)-t^2
The minus t squared is for gravity. Now, we want to know when the ball's y-coordinate is zero, so insert 0 for y and solve for t.
0=t*P*sin(theta)-t^2
Factor out t.
t(P*sin(theta)-t)=0
Therefore, either t=0 or (P*sin(theta)-t)=0. Obviously t=0, because the balls are in the machine when t=0, so we are concerned with the second equation. Continuing to solve for t, we have:
(P*sin(theta)-t)=0
t=P*sin(theta)
Now that we know t, we can insert t into the x equation to find how far, depending on theta, the balls will be when they hit the ground.
x=t*P*cos(theta)
x=(P*sin(theta))*P*cos(theta)
x=P^2*sin(theta)cos(theta)
Multiply both sides by 2, and use the sin(2theta) equation.
2x=P^2*(2sin(theta)cos(theta))
2x=P^2*sin(2theta)
x=(P^2/2)*sin(2theta)
Now we have an equation that takes the angle, theta, and tells you how far the balls will go. Replace theta with x, and the x with f(x).
f(x)=(P^2/2)*sin(2x).
This is a modified sine equation. It is multiplied by a variable, (P^2/2), but since multiplying a sine equation by a number won't affect the locations of the maxima or minima, we can disregard that.
f(x)=sin(2x)
You can easily see that the balls will go the farthest at the maximum, 45 degrees (or 135 degrees, but that is essentially the same thing), and that the power of the machine does not affect this answer.

Answer: 5985

Don't forget about Pascal's Triangle!
The coefficients of the expansion of (x+y)^z are given by the nth row of Pascal's Triangle. Specifically,
nC0, nC1, nC2, nC3, ... , nC(n-1), nCn.

Answer: 6000

There are 9000 four-digit numbers. The first digit can be 1-9, and the other three can be 0-9. That leaves 9 choices for the first digit, and 10 for the other three. 9*10*10*10=9000. There are two ways to solve this:

In every twelve numbers, there are four that are divisible by 4 or 6. This pattern fits evenly into the nine thousand 750 times. This means that there are 750*4=3000 numbers that ARE divisible by 4 or 6. 9000-3000=6000 numbers that are NOT.

Another way to solve it is this: one in four numbers is divisible by four. This pattern fits evenly into the nine thousand 2250 times. 2250*1=2250 divisible by four. The same method gives 1500 divisible by six. The same method tells that there are one in twelve numbers divisible by 4 AND 6 (divisible by twelve, the least common denominator). 2250+1500-the number divisible by both (750)=3000. 9000-3000=6000.

Answer: No

Considering that black holes have infinite density, the closer he gets to the black hole, the faster time passes, and so he will see the entire future of the universe in increasing speed, even if the universe lasts forever. From the other person's viewpoint, the person who is falling will fall at a slower and slower speed until his speed appears to be, but is not and will never quite be, zero.

Answer: This statement is false.

"Precisely one of the other three options is legitimate." is false, which is easy to see since the question itself states that only one is not legitimate, and the question would not be allowed to lie by FunTrivia.

"Question seven is impossible to answer." is false because, again, FunTrivia would not allow an unanswerable question. Since this option is false, it is legitimate.

"This is the correct answer to question seven." is either true or false, since there absolutely must be an answer to question seven. We don't even need to know whether it is true or false to know if it is legitimate, but now that we know it is legitimate, we also know that it isn't the answer and is therefore false.

"This statement is false."

Let's assume it is true.

"This statement is false." is true. By its own claim, it is false. This contradicts the assumption.

Let's assume it is false.

"This statement is false." is false. Therefore "This statement is not false" is true. By its own claim, it is not false. This contradicts the assumption.

We know that it is not true. We know it is not false. Therefore we know it is neither true nor false. By now, we know that it is absolutely not legitimate. Just for fun, let's check this conclusion.

Let's assume it is neither true nor false.

"This statement is false." is neither true nor false. It claims to be false, but it is neither true nor false. Therefore its claim is false, therefore it is false, but our assumption was that it was neither true nor false. This contradicts the assumption.

We have logically concluded that the statement is "neither true nor false". But we also concluded that the statement is "not neither true nor false". I don't know what to say to that...

We could guess that this question is unanswerable. It probably is. But if you answer the question by saying that it's unanswerable, you just answered it, therefore it is not unanswerable. But if it is not unanswerable, then your answer was wrong and it is once again unanswerable.

Answer: 32 square units

The first square's area is 16, the second's can be found using the 45-45-90 triangle in the corner of the first square. It turns out to be 8. The third's is 4. The sequence of areas is 16, 8, 4, 2, 1, .5, ... continuing forever. This is a geometric sequence, and the ratio is 1/2. But it is the sum we are concerned with. The sums are given by the series 16, 24, 28, 30, 31, 31.5, ... continuing forever. This is given by the equation:
(T1(1-r^n))/(1-r) where r is the ratio, n is the term you want to find, and T1 is the first term. We want the last term, so we'll apply limits. The limit of (16(1-(1/2)^n))/(1-(1/2)) with n approaching infinity will give us the answer.
lim(n>&) of (16(1-(1/2)^n))/(1-(1/2))=
(16(1-0))/(1/2)=
16/(1/2)=
32
It might seem, to some people, that the combined area is infinity, since there are an infinite number of squares, but each square is twice as small as the previous square. This is similar to walking halfway to a destination, then walking halfway from where you are, which is one quarter the total distance, the walking halfway again, and again forever. If you did indeed do this forever, you would get to the destination. You would NOT walk an infinite distance.

Answer: 36

there are 3 arrangements in terms of color. b is blue, r is red. They are:
rrrbb
brrrb
bbrrr
For each arrangement, there are six options for the reds, and two options for the blues. 6*2*3=36.

Answer: No

You just can't. There are corners, edges, and centers on a Rubik's Cube. The three types of pieces can't interchange, so this pattern is impossible. In fact, the centers can never change at all (in relation to each other).