ECAT Engineering Entry Test 2021 Mathematics Induction and Binomial Theorem MCQS Online Preparation Sample Paper Questions with Answer

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Mathematics Induction and Binomial Theorem MCQS Online Preparation Sample Paper Questions with Answer

Assessment of comprehension of mathematical proof techniques and binomial expansions is done through Mathematics Induction and Binomial Theorem MCQs. These queries, commonly found in exams such as the ECAT (Engineering College Admission Test), encompass subjects like mathematical induction, binomial coefficients, and applications of the binomial theorem. Platforms like Nokryan.com provide extensive resources for preparation, including sample papers, practice questions, and in-depth solutions. These materials aid in effective learning and mastery of concepts related to Induction and Binomial Theorem, empowering students with the necessary skills to succeed in the mathematics section of the ECAT exam.

Induction and Binomial Theorem by Nokryan.com

1. (1-3y) ⁴ =

  • 1+y+y²+y³+y⁴
  • 1-4y+6y²-4y³+y⁴
  • 1-12y+54y²-108y³+81y⁴
    correct
  • 1+4y+6y²+4y³+y⁴

2. (a² -b²)³ =

  • a³-3a²b+3ab²-b³
  • a⁶-3a⁴b²+3a²b⁴-b⁶
    correct
  • a³+3a²b+3ab²+b³
  • a⁶+3a⁴b²+3a²b⁴+b⁶

3. (x-1) ³ =

  • x³- 3x² + 3x – 1
    correct
  • 1 – 3x + 3x² – x³
  • x³+ x² + x +1
  • x³+ 3x² + 3x + 1

4. (x+1/x) ⁴ =

  • 1+x⁴+4x² + 4/x²+1/x⁴
  • 1-x⁴ +4x² + 4/x² + 1/x⁴
  • x⁴+4x² +6+ 4/x²+1/x⁴
    correct
  • x⁴-4x² -6- 4/x² – 1/x⁴

5. If a statement P(n) is true for n=1 and the truth of P(n) for n = k implies the truth of P(n) for n = k + 1, then P(n) is true for all

  • Positive integers n
    correct
  • Positive real numbers n
  • Real numbers n
  • Integers n

6. (1+√2) ³ =

  • 7-5√2
  • 5+7√2
  • 5-7√2
  • 7+5√2
    correct

7. If n is any positive integer, then 1³ + 2³ + 3³ + … + n =

  • n(n+1)/2
  • n²(n+1)²/4
    correct
  • n(n+1)²/4
  • n²(n+1)/4

8. (1-x) ³=

  • 1+3x+3x²+x³
  • 1-x+ x²-x³
  • 1+x+x²+x³
  • 1-3x+ 3x²-x³
    correct

9. In the expansion of (a – 2b)³ the coefficient of b² is

  • -8a
  • -2a²
  • -4a
  • 12acorrect

10. If n is any positive integer, then 1 + 2 + 3 +…+ n =

  • n/n+2
  • n/n+1
  • n!
  • n(n+1)/2
    correct

11. If n is any positive integer, then 3 + 6 + 9 + …. + 3n =

  • 3n(n+1)
  • 2n(n+1)/3
  • 3n(n+1)/4
  • 3n(n+1)/2correct

12. If n is any positive integer, then n ! 3ⁿ¯¹ is true for all

  • n > 5
  • n > 3
  • n ≥ 5
    correct
  • n ≥ 3

13. If n is any positive integer, then 1/1.2+1/2.3+1/3.4 +…+1/n(n+1)=

  • n!
  • n/n+1
    correct
  • n/2(n+1)
  • n/n+2

14. (1+x)⁷ =

  • 1+x+x²+x³+x⁴+x⁵+x⁶+x⁷
  • 1-7x+21x²+35x³+35x⁴-21x⁵+7x⁶-x⁷
  • 7+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷
  • 1+7x+21x²+35x³+35x⁴21x⁵+7x⁶+x⁷
    correct

15. IF n is any positive integer, then 2ⁿ > 2 (n+1) is true for all

  • n>3
    correct
  • n≤3
  • n≥3
  • n<3

16. If a statement P(n) is true for n = m, where m is some given natural number, and the truth of P(n) for n = k > m implies the truth of P(n) for n = k + 1, then P(n) is true for all positive integers

  • n ≥ m
    correct
  • m ≥ n
  • n > m
  • m > n

17. In the expansion of (a + b)⁷, the 2nd term is

  • a⁷
  • 7a⁶b
    correct
  • None of these
  • 7ab⁶

18. If n is any positive integer, then 1/3 + 1/9 + …..+ 1/3ⁿ =

  • ½(1- 1/2ⁿ)
  • ½ (1-1/3ⁿ)
    correct
  • 1/3 (1-1/3ⁿ)
  • 1/3(1-1/2ⁿ)

19. IF n is any positive integer, then 1/1.3+1/3.5+1/5.7 + …+ 1/(2n – 1)(2n + 1) =

  • n/2(n+1)
  • n/2n+1
    correct
  • 2n/n+1
  • n/n+2

20. If n is any positive integer, then 2¹ + 2²+ 2³ + … + 2ⁿ =

  • 2(2ⁿ¯¹ -1)
  • 2(2 ⁿ⁺¹ – 1)
  • 2(2 ⁿ – 1)
    correct
  • 2(3 ⁿ – 1)

21. If n is any positive integer, then 1 + 3 + 5 + … + (2n – 1) =

  • n+1
  • 2n+1
  • n

  • correct

22. IF n is any positive integer, then 4ⁿ > 3ⁿ + 4 is true for all

  • n<2
  • n≥2
    correct
  • n>2
  • n≤2

23. (x-1/x) ³ =

  • x³+x + 1/x + 1/x³
  • x³-3x + 3/x – 1/x³correct
  • x³+3x+3/x + 1/x³
  • none of these

24. If n is any positive integer, then 1² + 2² + 3² + ……+ n² =

  • n(n+1)(2n+1)/2
  • n(n+1)(2n+1)/6
    correct
  • n(n+1)(2n+1)/3
  • (n+1)(2n+1)/6

25. (1+2x)⁴ =

  • 1-4x+6x²-4x³+ x⁴
  • 1+4x+6x²+4x³+x⁴
  • 1-8x+24x²-32x³+16x⁴
  • 1+8x+24x²+32x³+16x⁴
    correct

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