BLEPT Reviewer 5 - Advanced Mathematics

BLEPT Reviewer

Instructions:

Please answer each question to the best of your ability. Each question is multiple choice, and only one answer per question is correct. Select the most appropriate answer from the options provided.

When you have completed all questions, click the "Submit" button at the bottom of the page to see your score. Good luck!

1. What is the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \)?

A) \( 9x^2 - 10x + 2 \)
B) \( 6x^2 - 10x + 2 \)
C) \( 9x^2 - 10x + 2 \)
D) \( 3x^2 - 5x + 2 \)

2. Solve the integral: \( \int x \ln x \, dx \)

A) \( \frac{1}{2}x^2 \ln x - x^2 \)
B) \( \frac{x^2}{2} \ln x - \frac{x^2}{4} + C \)
C) \( x^2 \ln x + C \)
D) \( x \ln x + C \)

3. What is the limit: \( \lim_{x \to 0} \frac{\sin x}{x} \)?

A) 0
B) ∞
C) 1
D) Undefined

4. What is the eigenvalue of the matrix \( \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \)?

A) 1 and 5
B) 2 and 3
C) -2 and -3
D) 0 and 1

5. Which of the following series converges?

A) \( \sum_{n=1}^{\infty} \frac{1}{n} \)
B) \( \sum_{n=1}^{\infty} \frac{1}{n^2} \)
C) \( \sum_{n=1}^{\infty} \frac{n}{n+1} \)
D) \( \sum_{n=1}^{\infty} \frac{n+1}{n} \)

6. If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), what is the determinant of A?

A) -2
B) 10
C) 2
D) -10

7. Solve for \( x \) in the equation \( e^x = 5 \)

A) \( x = 5 \)
B) \( x = e^5 \)
C) \( x = \ln 5 \)
D) \( x = \frac{1}{5} \)

8. What is the solution to the differential equation \( \frac{dy}{dx} = 3y \)?

A) \( y = 3x \)
B) \( y = Ce^{3x} \)
C) \( y = e^{x^3} \)
D) \( y = \ln(3x) \)

9. What is the Laplace transform of \( f(t) = e^{2t} \)?

A) \( \frac{1}{s} \)
B) \( \frac{1}{s+2} \)
C) \( \frac{1}{s-2} \)
D) \( s+2 \)

10. Find the general solution to \( y'' + y = 0 \)

A) \( y = Ce^x \)
B) \( y = C_1 e^x + C_2 e^{-x} \)
C) \( y = C_1 \cos x + C_2 \sin x \)
D) \( y = C_1 x + C_2 \)

11. What is the Maclaurin series expansion of \( e^x \)?

A) \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)
B) \( \sum_{n=1}^{\infty} \frac{x^n}{n} \)
C) \( \sum_{n=1}^{\infty} (-1)^n \frac{x^n}{n!} \)
D) \( \sum_{n=0}^{\infty} \frac{x^{2n}}{n!} \)

12. Compute the Fourier series of \( f(x) = x \) on \( [-\pi, \pi] \).

A) \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx) \)
B) \( \sum_{n=1}^{\infty} \frac{1}{n} \cos(nx) \)
C) \( \sum_{n=1}^{\infty} \frac{x^n}{n!} \)
D) \( \sum_{n=1}^{\infty} e^{-n} \sin(nx) \)

13. If \( z = 3 + 4i \), what is \( |z| \)?

A) 3
B) 5
C) 7
D) 4

14. Find the inverse of the matrix \( A = \begin{bmatrix} 4 & 3 \\ 3 & 2 \end{bmatrix} \).

A) \( \frac{1}{-1} \begin{bmatrix} -2 & 3 \\ 3 & -4 \end{bmatrix} \)
B) \( \begin{bmatrix} 2 & -3 \\ -3 & 4 \end{bmatrix} \)
C) \( \begin{bmatrix} 4 & -3 \\ -3 & 2 \end{bmatrix} \)
D) No inverse exists.

15. What is the Taylor series expansion of \( \cos x \) centered at \( x = 0 \)?

A) \( \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \)
B) \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \)
C) \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{n}}{n!} \)
D) \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)

16. Solve the second-order differential equation \( y'' + 4y = 0 \).

A) \( y = C_1 e^{2x} + C_2 e^{-2x} \)
B) \( y = C_1 \cos 2x + C_2 \sin 2x \)
C) \( y = C_1 x^2 + C_2 x \)
D) \( y = C_1 e^x + C_2 e^{-x} \)

17. What is the sum of the infinite geometric series \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... \)?

A) 2
B) 1
C) \( \frac{2}{3} \)
D) \( \frac{3}{4} \)

18. What is the determinant of a diagonal matrix with entries 2, 3, and 5?

A) 30
B) 10
C) 6
D) 1

19. Solve for \( x \): \( \log_2(x) = 5 \).

A) 10
B) 32
C) 5
D) 16

20. What is the solution to the system of equations: \( 3x + 2y = 5 \) and \( 4x - y = 7 \)?

A) \( (1,1) \)
B) \( (2,1) \)
C) \( (3,-2) \)
D) \( (0,5) \)

21. What is the value of the definite integral \( \int_0^1 x^2 \, dx \)?

A) \( \frac{1}{3} \)
B) \( \frac{1}{2} \)
C) 1
D) \( \frac{2}{3} \)

22. If \( A \) is an invertible matrix, then \( AA^{-1} \) is equal to:

A) 0
B) \( A^2 \)
C) \( A \)
D) Identity matrix

23. Evaluate the limit \( \lim_{x \to 0} \frac{\sin x}{x} \).

A) 1
B) 0
C) Undefined
D) Infinity

24. Which of the following is a solution to the equation \( x^2 + 4x + 3 = 0 \)?

A) -1 and -4
B) -1 and -3
C) 1 and 3
D) 1 and -3

25. What is the derivative of \( \ln(x^2 + 1) \)?

A) \( \frac{2}{x^2 + 1} \)
B) \( \frac{2x}{x^2 + 1} \)
C) \( \frac{1}{x^2 + 1} \)
D) \( 2 \ln(x^2 + 1) \)

26. What is the Laplace Transform of \( f(t) = e^{3t} \)?

A) \( \frac{1}{s + 3} \)
B) \( \frac{1}{s - 3} \)
C) \( \frac{3}{s} \)
D) \( \frac{3}{s - 1} \)

27. Which of the following is the inverse function of \( f(x) = 3x + 2 \)?

A) \( f^{-1}(x) = \frac{x - 2}{3} \)
B) \( f^{-1}(x) = \frac{x + 2}{3} \)
C) \( f^{-1}(x) = 3x - 2 \)
D) \( f^{-1}(x) = \frac{3}{x + 2} \)

28. What is the eigenvalue of the matrix \( \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \)?

A) 1
B) 2 and 3
C) -2 and -3
D) 0

29. The domain of the function \( f(x) = \sqrt{x - 2} \) is:

A) All real numbers
B) \( x < 2 \)
C) \( x \geq 2 \)
D) \( x \neq 2 \)

30. What is the binomial coefficient \( \binom{7}{3} \)?

A) 20
B) 35
C) 15
D) 21

Result