AP POLYCET MATHEMATICS Multiple Choice Questions (MCQs)
AP POLYCET Mathematics - Exponential and Logarithmic Series - MCQs
AP POLYCET Mathematics - Exponential and Logarithmic Series - Multiple Choice Questions
- 1. The value of \( \log_e e \) is:
Answer: b) 1
- 2. If \( \log_a x = 3 \), then the value of \( x \) is:
- a) \( a^3 \)
- b) \( \frac{1}{a^3} \)
- c) \( a^2 \)
- d) \( \frac{1}{a^2} \)
Answer: a) \( a^3 \)
- 3. Which of the following is the expansion of \( e^x \)?
- 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) \( 1 + x + x^2 + x^3 + \cdots \)
- d) \( 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
Answer: d) \( 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
- 4. The logarithmic function \( \log_b x \) is the inverse of:
- a) Exponential function
- b) Sine function
- c) Cosine function
- d) Tangent function
Answer: a) Exponential function
- 5. If \( \log_a x = 5 \), then the value of \( a^5 \) is:
- a) x
- b) \( \frac{1}{x} \)
- c) \( \log_a x \)
- d) \( \log_a x + 5 \)
Answer: a) x
- 6. The expansion of \( \ln(1 + x) \) is:
- a) \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \)
- b) \( x + \frac{x^2}{2} - \frac{x^3}{3} + \frac{x^4}{4} + \cdots \)
- c) \( 1 + x + \frac{x^2}{2} - \frac{x^3}{3!} + \cdots \)
- d) \( x - x^2 + x^3 - x^4 + \cdots \)
Answer: a) \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \)
- 7. The value of \( \log_{10} 1000 \) is:
Answer: b) 3
- 8. The Taylor series of \( e^x \) is:
- a) \( 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots \)
- b) \( 1 + x + x^2 + x^3 + \cdots \)
- c) \( 1 + \frac{x}{2} + \frac{x^2}{3!} + \cdots \)
- d) \( 1 + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots \)
Answer: a) \( 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots \)
- 9. If \( a = 2 \), \( b = 8 \), and \( x = 3 \), what is \( \log_2 8 \)?
Answer: b) 3
- 10. The value of \( \ln(e^x) \) is:
- a) x
- b) \( \log_e x \)
- c) \( e^x \)
- d) \( \ln x \)
Answer: a) x
- 11. Which of the following represents the exponential growth formula?
- a) \( A = P(1 + rt) \)
- b) \( A = Pe^{rt} \)
- c) \( A = P + rt \)
- d) \( A = P(1 + r)^t \)
Answer: b) \( A = Pe^{rt} \)
- 12. The logarithmic identity \( \log_b(xy) \) is:
- a) \( \log_b x + \log_b y \)
- b) \( \log_b x - \log_b y \)
- c) \( \log_b x \times \log_b y \)
- d) \( \log_b x / \log_b y \)
Answer: a) \( \log_b x + \log_b y \)
- 13. The expansion of \( \ln(1+x) \) for small values of \( x \) is:
- a) \( x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots \)
- b) \( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \)
- c) \( x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots \)
- d) \( x + x^2 + x^3 + \cdots \)
Answer: b) \( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \)
- 14. The value of \( \log_e 1 \) is:
- a) 0
- b) 1
- c) e
- d) Undefined
Answer: a) 0
- 15. The Taylor series of \( \ln(1+x) \) is:
- a) \( x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots \)
- b) \( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \)
- c) \( 1 + x + x^2 + \cdots \)
- d) \( 1 + x - x^2 + x^3 - \cdots \)
Answer: b) \( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \)
 
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