FSc Part 1 Maths chapter 8: Mathematical induction and binomial theorem MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

28 exam-style questions on Mathematical induction and binomial theorem, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

28 questions · 3 topics · 10 per test

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What chapter 8 covers

Ten Mathematical induction and binomial theorem questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. The term independent of x in the expansion of (x + 1/x)6 is

  1. A
    15
  2. B
    20
  3. C
    6
  4. D
    1
Show answer

Answer: B. Tr+1 = 6Crx6−rx−r = 6Crx6−2r. For no x, 6 − 2r = 0, so r = 3 and the term is 6C3 = 20.

2. For the formula 12 + 22 + ... + n2 = n(n + 1)(2n + 1)/6, both sides at n = 3 are equal to

  1. A
    12
  2. B
    9
  3. C
    14
  4. D
    36
Show answer

Answer: C. Left: 1 + 4 + 9 = 14. Right: 3 × 4 × 7/6 = 84/6 = 14. 9 is only the last term and 36 is (1 + 2 + 3)2.

3. The coefficient of x2 in the expansion of (1 − x)−2 is

  1. A
    3
  2. B
    −3
  3. C
    2
  4. D
    1
Show answer

Answer: A. n = −2 and the x term is −x: n(n − 1)/2! × (−x)2 = (−2)(−3)/2 × x2 = 3x2.

4. In proving a statement P(n) for all natural numbers n by mathematical induction, the first step is to

  1. A
    show that P(n) is true for every n
  2. B
    assume that P(k) is true
  3. C
    show that P(k + 1) is true
  4. D
    show that P(1) is true
Show answer

Answer: D. The basis step checks the first case, P(1). The inductive step then assumes P(k) and proves P(k + 1).

5. In the inductive step of mathematical induction, we

  1. A
    assume P(1) is true and prove P(2) is true
  2. B
    assume P(k + 1) is true and prove P(k) is true
  3. C
    assume P(k) is true and prove P(k + 1) is true
  4. D
    assume P(k) is true and prove P(2k) is true
Show answer

Answer: C. If truth for k always forces truth for k + 1, then P(1) true gives P(2), P(3), ... in turn.

6. The sum of all the binomial coefficients in the expansion of (1 + x)n is

  1. A
    n2
  2. B
    2n−1
  3. C
    n!
  4. D
    2n
Show answer

Answer: D. Put x = 1: (1 + 1)n = nC0 + nC1 + ... + nCn = 2n.

7. The number of terms in the expansion of (a + x)10 is

  1. A
    12
  2. B
    10
  3. C
    11
  4. D
    20
Show answer

Answer: C. (a + x)n has n + 1 terms, from nC0 to nCn. So there are 11 terms.

8. For every natural number n, n3 − n is divisible by

  1. A
    6
  2. B
    4
  3. C
    5
  4. D
    9
Show answer

Answer: A. n3 − n = (n − 1)n(n + 1), a product of three consecutive integers, which always contains a multiple of 2 and of 3. n = 2 gives 6, which rules out 4, 5 and 9.

9. For |x| < 1, the (r + 1)th term of the expansion of (1 − x)−1 is

  1. A
    (r + 1)xr
  2. B
    (−1)rxr
  3. C
    rxr
  4. D
    xr
Show answer

Answer: D. (1 − x)−1 = 1 + x + x2 + ..., so the (r + 1)th term is xr. The alternating (−1)rxr belongs to (1 + x)−1.

10. The expansion of (1 − 2x)−1/2 in ascending powers of x is valid when

  1. A
    |x| < 2
  2. B
    |x| < 1
  3. C
    |x| < 1/2
  4. D
    |x| > 1/2
Show answer

Answer: C. The binomial series for (1 + y)n needs |y| < 1. Here y = −2x, so |2x| < 1, i.e. |x| < 1/2.

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Ways to practise this chapter

Mathematical induction and binomial theorem: questions students ask

How many Mathematical induction and binomial theorem MCQs are there?

28, each with the answer and a short explanation, across 3 topics: Mathematical induction, Binomial theorem for positive integral index, Binomial series. Every chapter test draws ten of them, so you can take it many times.

Is the FSc Part 1 Maths chapter 8 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

Can I read the explanations in Urdu?

Yes. In practice mode and on your results you can switch the explanations to Urdu.

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