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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Mathematical Induction and Binomial Theorem (Set 2)

Take the ECAT Mathematics — Mathematical Induction and Binomial Theorem (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. The coefficient of x⁵ in the expansion of 1/(1 − x)³ (|x| < 1) is:

    • A. 15
    • B. 35
    • C. 10
    • D. 21 ✓

    Answer: D. 21

    Explanation: (1 − x)⁻³ = Σ ⁽ʳ⁺²⁾C₂ xʳ, so the coefficient of x⁵ is ⁷C₂ = 21.

  2. Q2. In Pascal's triangle, the middle entry of the row giving the coefficients of (a + b)⁶ is:

    • A. 20 ✓
    • B. 15
    • C. 35
    • D. 10

    Answer: A. 20

    Explanation: The row for n = 6 is 1, 6, 15, 20, 15, 6, 1; the middle entry is ⁶C₃ = 20.

  3. Q3. In the binomial series for (1 + x)^(−1/2), what is the coefficient of x²?

    • A. 3/8 ✓
    • B. 3/4
    • C. 1/8
    • D. −3/8

    Answer: A. 3/8

    Explanation: Coefficient = (−1/2)(−3/2)/2! = (3/4)/2 = 3/8.

  4. Q4. To prove 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 by induction, after assuming it for n = k, the term added to both sides is:

    • A. (2k + 1)²
    • B. k²
    • C. (k + 1)
    • D. (k + 1)² ✓

    Answer: D. (k + 1)²

    Explanation: Going from k to k + 1 adds the next term of the series, (k + 1)², to the left-hand side.

  5. Q5. For |x| < 1 and any rational n, the (r + 1)th term of the binomial series (1 + x)ⁿ is:

    • A. [n(n + 1)…(n + r − 1)/r!] xʳ
    • B. [n(n − 1)(n − 2)…(n − r + 1)/r!] xʳ ✓
    • C. [n!/(r!(n − r)!)] xⁿ⁻ʳ
    • D. [n(n − 1)…(n − r)/r!] xʳ

    Answer: B. [n(n − 1)(n − 2)…(n − r + 1)/r!] xʳ

    Explanation: The general term of (1 + x)ⁿ is n(n − 1)…(n − r + 1)xʳ/r!, which reduces to ⁿCᵣxʳ when n is a positive integer.

The other 25 questions

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  1. Q6. The coefficient of x³ in the expansion of (1 + 2x)(1 + x)⁵ is:

    • A. 10
    • B. 40
    • C. 20
    • D. 30

  2. Q7. Let P(k) be the statement 2 + 4 + 6 + … + 2k = k(k + 1). Then P(k + 1) is:

    • A. 2 + 4 + … + 2k = (k + 1)(k + 2)
    • B. 2 + 4 + … + 2(k + 1) = k(k + 1) + 2
    • C. 2 + 4 + … + (2k + 1) = (k + 1)²
    • D. 2 + 4 + … + 2(k + 1) = (k + 1)(k + 2)

  3. Q8. The middle term in the expansion of (2x + 3)⁴ is:

    • A. 96x²
    • B. 36x²
    • C. 54x²
    • D. 216x²

  4. Q9. Find the least positive integer n₀ for which the inequality n! > 2ⁿ is true for all n ≥ n₀.

    • A. 4
    • B. 2
    • C. 3
    • D. 5

  5. Q10. The principle of mathematical induction is used to prove statements that are true for all:

    • A. real numbers
    • B. irrational numbers
    • C. natural numbers (from some starting value onwards)
    • D. complex numbers

  6. Q11. For every positive integer n, the product n(n + 1)(n + 2) is always divisible by:

    • A. 9
    • B. 4
    • C. 5
    • D. 6

  7. Q12. Using the binomial expansion up to two terms, √99 is approximately:

    • A. 9.9
    • B. 9.99
    • C. 9.95
    • D. 9.5

  8. Q13. For which values of x does the binomial series of 1/(2 + x) converge?

    • A. all real x
    • B. |x| < 2
    • C. |x| < 1
    • D. |x| < 1/2

  9. Q14. Using the binomial theorem, the value of (99)³ is:

    • A. 960299
    • B. 970200
    • C. 970299
    • D. 970301

  10. Q15. The coefficient of x² in the expansion of (2 + x)⁵ is:

    • A. 160
    • B. 80
    • C. 40
    • D. 10

  11. Q16. For every positive integer n, 7ⁿ − 3ⁿ is divisible by:

    • A. 3
    • B. 10
    • C. 4
    • D. 7

  12. Q17. When (1 + x)⁵ − (1 − x)⁵ is expanded and simplified, the number of non-zero terms is:

    • A. 2
    • B. 6
    • C. 3
    • D. 5

  13. Q18. If the expansion of (a + b)ⁿ has 21 terms, then n equals:

    • A. 20
    • B. 10
    • C. 22
    • D. 21

  14. Q19. The remainder when 2³⁰ is divided by 7 is:

    • A. 4
    • B. 2
    • C. 1
    • D. 6

  15. Q20. In the expansion of (a − b)¹⁰, the term containing b⁷ is:

    • A. the middle term
    • B. positive
    • C. negative (it has a minus sign)
    • D. zero

  16. Q21. The term independent of x in the expansion of (2x + 1/x²)⁹ is:

    • A. 672
    • B. 2688
    • C. 10752
    • D. 5376

  17. Q22. The expansion of (x + y)¹¹ has:

    • A. two middle terms, T₅ and T₆
    • B. two middle terms, T₆ and T₇
    • C. one middle term, T₆
    • D. one middle term, T₇

  18. Q23. The term independent of x in the expansion of (x − 2/x)⁶ is:

    • A. −160
    • B. −20
    • C. 160
    • D. 240

  19. Q24. The greatest binomial coefficient in the expansion of (1 + x)¹⁰ is:

    • A. 120
    • B. 210
    • C. 252
    • D. 1024

  20. Q25. When n is a negative integer, the binomial expansion of (1 + x)ⁿ (|x| < 1) has:

    • A. n + 1 terms
    • B. |n| + 1 terms
    • C. n terms
    • D. infinitely many terms

  21. Q26. The value of ⁵C₁ + 2·⁵C₂ + 3·⁵C₃ + 4·⁵C₄ + 5·⁵C₅ is:

    • A. 75
    • B. 32
    • C. 80
    • D. 160

  22. Q27. The 5th term in the expansion of (x − 2)⁷ is:

    • A. −560x³
    • B. 560x³
    • C. 280x³
    • D. −280x⁴

  23. Q28. The sum of all the coefficients in the expansion of (3x − 2)⁵ is:

    • A. 243
    • B. 1
    • C. −1
    • D. 32

  24. Q29. If the coefficients of the 2nd and 3rd terms in the expansion of (1 + x)ⁿ are in the ratio 1 : 3, then n equals:

    • A. 9
    • B. 7
    • C. 6
    • D. 5

  25. Q30. Find the coefficient of x² when (1 − 3x)^(1/3) is expanded in ascending powers of x.

    • A. −2
    • B. −1/9
    • C. −1
    • D. 1