BLEPT Reviewer 8 - 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 \( f(x) = x^3 + 2x^2 - x + 5 \)?

A) \( 3x^2 - 2x - 1 \)
B) \( 3x^2 + 4x - 1 \)
C) \( 3x^2 + 2x + 1 \)
D) \( x^2 + 4x - 1 \)

2. Evaluate the limit: \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \)

A) 0
B) 1
C) 4
D) 2

3. What is the integral of \( \int 2x \, dx \)?

A) \( x^2 + 2 \)
B) \( x^2 + C \)
C) \( 2x + C \)
D) \( 2x^2 + C \)

4. What is the value of the determinant of the matrix \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)?

A) 10
B) -2
C) -2
D) 1

5. Solve for \( x \): \( e^x = 5 \)

A) \( x = \ln(10) \)
B) \( x = \ln(5) \)
C) \( x = 5 \ln(e) \)
D) \( x = \log(5) \)

6. What is the second derivative of \( f(x) = x^4 \)?

A) \( 4x \)
B) \( 12x^2 \)
C) \( 8x^2 \)
D) \( 6x \)

7. What is the domain of the function \( f(x) = \frac{1}{x-3} \)?

A) All real numbers except 3
B) All real numbers
C) \( x \geq 3 \)
D) \( x \neq 0 \)

8. Simplify the expression: \( \ln(e^3) \)

A) \( e^3 \ln(e) \)
B) 3
C) \( \ln(3e) \)
D) \( e^3 \)

9. Which of the following is NOT a vector quantity?

A) Velocity
B) Temperature
C) Acceleration
D) Force

10. Find the value of the expression: \( \log_{2}(8) \)

A) 4
B) 1
C) 3
D) 2

11. What is the solution to the inequality \( 2x - 5 > 9 \)?

A) \( x > 5 \)
B) \( x > 7 \)
C) \( x < 7 \)
D) \( x < 5 \)

12. What is the solution to \( \sin(x) = \frac{\sqrt{2}}{2} \) in the interval \( [0, 2\pi] \)?

A) \( \frac{\pi}{2}, \frac{3\pi}{2} \)
B) \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4} \)
C) \( \frac{\pi}{4}, \frac{3\pi}{4} \)
D) \( \frac{5\pi}{4}, \frac{7\pi}{4} \)

13. Find the inverse of the function \( f(x) = 2x + 1 \)

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

14. Which of the following functions is NOT continuous at \( x = 2 \)?

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

15. What is the sum of the first 10 terms of the arithmetic sequence: 2, 5, 8, ...?

A) 120
B) 135
C) 155
D) 180

16. If \( \log_a (x) = 2 \), what is \( x \) in terms of \( a \)?

A) \( a^2 \)
B) \( a^2 \)
C) \( 2a \)
D) \( \log_a 2 \)

17. Solve for \( x \): \( 3^{2x} = 81 \)

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

18. Find the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \).

A) \( 1, 6 \)
B) \( 2, 3 \)
C) \( -2, -3 \)
D) \( 3, 4 \)

19. What is the Taylor series expansion of \( e^x \) at \( x=0 \)?

A) \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
B) \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
C) \( x + x^2 + x^3 + \cdots \)
D) \( 1 + \frac{x}{2} + \frac{x^2}{3!} + \cdots \)

20. If \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), what is \( \det(A) \)?

A) 11
B) 5
C) 10
D) 14

21. Evaluate the integral \( \int_0^1 (3x^2) dx \).

A) 1
B) 1
C) 3
D) 0.5

22. What is the binomial coefficient \( \binom{5}{2} \)?

A) 10
B) 5
C) 20
D) 15

23. The equation \( \cos^2 \theta + \sin^2 \theta = ? \) equals:

A) 0
B) 1
C) \( \cos \theta \)
D) \( \sin \theta \)

24. Find the sum of the infinite geometric series \( 3 + 1.5 + 0.75 + \cdots \)

A) 9
B) 6
C) 12
D) 3

25. If \( f(x) = \ln(x) \), what is \( f'(x) \)?

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

26. Find the value of \( \lim_{x \to 0} \frac{\sin x}{x} \).

A) 1
B) 0
C) Does not exist
D) Infinity

27. What is the solution for \( x \) in the system \( \begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases} \)?

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

28. The eigenvalues of the matrix \( \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \) are:

A) 5 and 2
B) 4 and 3
C) 6 and 1
D) 7 and 0

29. If \( z = 3 + 4i \), find the modulus \( |z| \).

A) 7
B) 5
C) 1
D) 12

30. Solve \( \frac{d}{dx} (\sin x \cdot \cos x) \).

A) \( \cos^2 x - \sin^2 x \)
B) \( \cos^2 x - \sin^2 x \)
C) \( \sin^2 x + \cos^2 x \)
D) \( \sin 2x \)

Result