Math Trivia 2 Trivia Quiz

Answer: -30

The product of two negative real numbers is a positive real number. The product of a positive real number and a negative real number is a negative real number. So

(-5)*(-3)*(-2) = 15*(-2) = -30.

Answer: -9

By order of operations, you need to do exponents before multiplications (the negative sign in front is the same as multiplying by -1). So -3^2 = -9.

Answer: They are complementary.

The sum of the measures of the angles in any triangle in the plane is 180 degrees. A right triangle is a triangle with a 90 degree angle, hence the other two angles sum to 90 degrees. This means that they are complementary.

Other answers: Supplementary angles are angles whose measures sum to 180 degrees. Congruent angles are angles that have the same measure. Vertical angles are the two angles formed by two intersecting lines, which are necessarily congruent.

Answer: Second quadrant

An angle theta corresponds to a point on the unit circle x^2 + y^2 = 1 having coordinates (cos(theta), sin(theta)). So the cosine is the x-coordinate and the sine is the y-coordinate. The second quadrant consists of the points with negative x-coordinate and positive y-coordinate.

Answer: An ellipse

If you were to hammer two nails into a board and wrap a string around them, you could then trace out an ellipse by the "pin and string" method. Note that the sum of the distances between the two points, called the foci, is always a constant length.

Answer: x*f '(x) + f(x)

g(x) is a product of x and f(x), so we must use the product rule to find its derivative. Below is one satement of the product rule:

(h(x)*f(x))' = h(x)*f '(x) + f(x)*h '(x)

In words, its the first function times the derivative of the second plus the second times the derivative of the first. Here, h(x) = x so h '(x) = 1. So the derivate of x*f(x) is x*f'(x) + f(x)*1.

Answer: 2x - y = 0

Let (x,y) be a point on the line. Since the line passes through the origin (0,0), xi + yj is a vector parallel to the line. Since 2i - j is a normal vector, the dot product of these vectors is 0. This gives 2x - y = 0.

Note that you can read off the normal vectors at a glance:

The line with equation 2x + y = 0 has normal vector 2i + j.

The line with equation x + 2y = 0 has normal vector i + 2j.

The line with equation x - 2y = 0 has normal vector i - 2j.

Note that 2i - j is not a scalar multiple of the above vectors, so there is only one solution.

Answer: 8

The Euler phi function, phi(n), gives the number of positive integers less than n or equal to n that are relatively prime to n. If n = p where p is prime, then phi(p) = p - 1 (since only p is not relatively prime to p). The Euler phi function is multiplicative, meaning that if a and b are relatively prime positive integers, then phi(ab) = phi(a)*phi(b). So in this case,

phi(15) = phi(3) * phi(5) = 2*4 = 8.

In fact, here are all 8 positive integers less than 15 that are relatively prime to 15:

1, 2, 4, 7, 8, 11, 13, 14

Answer: B' is a subset of A'

Let x be an element of the complement of B. Then x is not in B. Since A is a subset of B, x is also not in A. Therefore, x is in A'. So I've shown:

x in B' implies x in A'

This means that B' is a subset of A'. I know this is not intuitive, but draw a Venn diagram and you'll see it! Or, consider the following example:

Let U be the set containing 1, 2, 3
Let A be the subset of U containing 1
Let B be the subset of U containing 1, 2

Then A is a subset of B.
Now A' contains 2, 3
B' contains 3
Therefore, B' is a subset of A'.

Note that the intersection of A' and B' is B' since B' is a subset of A'. Also, the union of A' and B' is A' since B' is a subset of A'. So none of the other answers can be correct.

Answer: T(x,y,z) = (x, y, -x - y)

All of the answers are linear transformations, but only T(x,y,z) = (x,y,-x - y) is well-defined on S. An ordered triple (x,y,z) is an element of S if and only if its components sum to 0 (this is the relation x + y + z = 0). This T is the only one in which the entries of the image sum to 0. Of course, one should verify that S is indeed a subspace (it is) and that T is a linear transformation (it is).

Answer: 60

The smallest nonabelian simple group is A5, the alternating group on 5 letters. A5 is the subgroup of S5 consisting of all even permutations (S5 is the symmetric group on 5 letters. Elements of S5 are permutations on 5 objects, so its order (size) is 5! = 120. Exactly half the permutations of S5 are even).

Simple groups are groups that have no nontrivial normal subgroups. They are theoretically important since by the Jordan/Holder Theorem, a finite group determines a unique list of simple groups. So in principal, if one knows all finite simple groups and knows all the ways to put them together, one knows all groups. Now the classification of all finite simple groups is complete and its proof covers thousands of journal pages! The part about putting them together is probably even more difficult than the classification! In my mind there is nothing simple about nonabelian simple groups! [In fact, there is one that is so big that it is called "The Monster"!]

Answer: (5)

Terminology: An ideal in Z is a subset I which is closed under subtraction of any two elements in Z and also has the property that if x is an element of Z and r is an element of I, then xr is an element of I. The ideal generated by 10 and 15 consists of all numbers of the form 10x + 15y where x and y are integers.

Z is a principal ideal domain, which means that Z is an integral domain (meaning ab = 0 implies a = 0 or b = 0) in which every ideal is generated by one element. In fact, the generator is just the greatest common divisor of the elements. The greatest common divisor of 10 and 15 is 5, so (10, 15) = (5).

Answer: f is continuous for all x

One can show by "the sandwich theorem" that the limit as x approaches 0 of f(x) is 0, which equals f(0), so f is continuous at 0. f is clearly continous everywhere else (since sin(x) and 1/x are continous everywhere else). Therefore, f is continous for all x.

However, f is not differentiable at 0: Consider the difference quotient

(f(x) - f(0))/(x - 0) = x*sin(1/x)/x = sin(1/x)

The limit as x approaches 0 of sin(1/x) does not exist (in any neighborhood of the origin, each value from -1 to 1 is attained by the function infinitely many times). So f cannot be continuously differentiable either (this means that the first derivative is continuous).

Answer: 5

The modulus of a + bi is just the square root of a^2 + b^2. It represents the distance from a + bi to the origin in the complex plane. So in this case, the modulus is the square root of (16 + 9) = sqrt(25) = 5.

Answer: S is open in X but not open in R

Open sets in X are of the form X intersected with U where U is an open set in R (this is the subspace topology). Now S = (1,3) intersected with X, and (1,3) is open in R. Therefore, S is an open set in X. However, S is not open in R: 2 is an element of S, but there is no open set V of R that contains 2 with V a subset of S.

I hope you enjoyed my quiz! Thanks for playing!