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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. 1. The value of \( \log_e e \) is:
    • a) 0
    • b) 1
    • c) 2
    • d) e
    Answer: b) 1
  2. 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. 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. 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. 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. 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. 7. The value of \( \log_{10} 1000 \) is:
    • a) 2
    • b) 3
    • c) 4
    • d) 5
    Answer: b) 3
  8. 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. 9. If \( a = 2 \), \( b = 8 \), and \( x = 3 \), what is \( \log_2 8 \)?
    • a) 3
    • b) 2
    • c) 8
    • d) 1
    Answer: b) 3
  10. 10. The value of \( \ln(e^x) \) is:
    • a) x
    • b) \( \log_e x \)
    • c) \( e^x \)
    • d) \( \ln x \)
    Answer: a) x
  11. 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. 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. 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. 14. The value of \( \log_e 1 \) is:
    • a) 0
    • b) 1
    • c) e
    • d) Undefined
    Answer: a) 0
  15. 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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