Mathematical Mastery for College Aspirants

Evaluate the integral of \( f(x) = 3x^2 \) from \( x = 1 \) to \( x = 4 \).

• 21

• 45

• 63

• 81

Find the determinant of the matrix \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \).

• -2

• 2

• -1

• 1

A fair coin is flipped three times. What is the probability of getting exactly two heads?

• 1/8

• 1/4

• 3/8

• 1/2

The mean of a data set is 20, and the standard deviation is 5. What is the z-score of a data point that is 25?

• 0

• 1

• 5

• 10

Solve the differential equation \( \frac{dy}{dx} = 3y \) with the initial condition \( y(0) = 2 \).

• \( y = 2e^{3x} \)

• \( y = 6e^{2x} \)

• \( y = 2e^{x} \)

• \( y = 3e^{2x} \)

How many different ways can you arrange the letters in the word "STATISTICS"?

• 10!

• \( \frac{10!}{3!3!2!} \)

• \( \frac{10!}{3!2!} \)

• \( \frac{9!}{2!2!} \)

Find the eigenvalues of the matrix \( \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \).

• 2 and 5

• 1 and 6

• 3 and 4

• -1 and 8

If \( F(x) \) is the antiderivative of \( f(x) = \frac{1}{x} \) with \( F(1) = 0 \), what is \( F(e) \)?

• 0

• 1

• e

• \( 1/e \)

What is the solution to the initial value problem \( \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0 \) with \( y(0) = 1 \) and \( y'(0) = 0 \)?

• \( y = e^{2x} \)

• \( y = xe^{2x} \)

• \( y = (1 + x)e^{2x} \)

• \( y = e^{4x} \)

A set contains the elements {1, 2, 3, 4, 5}. How many subsets of this set contain exactly three elements?

• 5

• 10

• 15

• 20