ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Mathematical Induction and Binomial Theorem (Set 3): 30 multiple-choice questions (40 minutes) with instant scoring and a full…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The expression n² + n + 41 gives a prime for every n = 1, 2, …, 39. The smallest positive integer n for which it is NOT prime is:
Answer: D. 40
Explanation: At n = 40: 1600 + 40 + 41 = 1681 = 41². This shows that checking many cases does not prove a statement; induction is needed.
Q2. The inequality 2ⁿ > n² is true for n = 1 but fails for n = 2, 3, 4. Starting from which n is it true for all larger n (provable by induction)?
Answer: A. n = 5
Explanation: 2⁴ = 16 = 4² (not strictly greater), 2⁵ = 32 > 25. For n ≥ 5, 2ᵏ⁺¹ = 2·2ᵏ > 2k² ≥ (k + 1)², so it holds for all n ≥ 5.
Q3. The greatest integer that divides 25ⁿ − 1 for every positive integer n is:
Answer: D. 24
Explanation: For n = 1 the value is 24, and 25ⁿ − 1 is divisible by 25 − 1 = 24 for all n. So the greatest such integer is 24.
Q4. The largest integer that always divides the product of four consecutive positive integers is:
Answer: B. 24
Explanation: n(n + 1)(n + 2)(n + 3) = 24·C(n + 3, 4), and n = 1 gives exactly 24.
Q5. To prove 1·2 + 2·3 + … + n(n + 1) = n(n + 1)(n + 2)/3 by induction, the inductive step must show that P(k) implies the statement:
Answer: A. 1·2 + … + k(k + 1) + (k + 1)(k + 2) = (k + 1)(k + 2)(k + 3)/3
Explanation: P(k + 1) replaces n by k + 1 on both sides: the left gains the term (k + 1)(k + 2) and the right becomes (k + 1)(k + 2)(k + 3)/3.
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Q6. The coefficient of x³ in the expansion of (2 + x)⁶ is:
Q7. The coefficient of x⁵ in the expansion of (1 + 2x)⁸ is:
Q8. The term independent of x in the expansion of (x³ + 1/x)⁸ is:
Q9. The term independent of x in (2x − 1/x)⁸ is:
Q10. The middle term in the expansion of (x − 2)⁶ is:
Q11. The sum of all the coefficients in the expansion of (3x + 1)⁴ is:
Q12. The sum C(9,0) + C(9,2) + C(9,4) + C(9,6) + C(9,8) equals:
Q13. In the expansion of (1 + x)ⁿ the coefficient of x² is 21. Then n equals:
Q14. The greatest coefficient in the expansion of (1 + x)⁹ is:
Q15. In the expansion of (1 + x)⁻² (|x| < 1) the coefficient of x⁴ is:
Q16. In the expansion of (1 + x)¹ᐟ² the coefficient of x³ is:
Q17. The coefficient of x² in the expansion of (1 − 3x)⁻¹ᐟ³ is:
Q18. Using the first three terms of the binomial expansion, (1.01)²⁰ is approximately (to two decimal places):
Q19. The expansion of (2 + 3x)⁻¹ in ascending powers of x is valid when:
Q20. The value of ¹²C₅ + ¹²C₆ is:
Q21. In the expansion of (1 + x)⁴³ the coefficients of the (2r + 1)th and (r + 2)th terms are equal, where r > 1. Then r equals:
Q22. The value of (√2 + 1)⁵ + (√2 − 1)⁵ is:
Q23. The remainder when 8⁴⁰ is divided by 49 is:
Q24. The expansion of (a + b)¹⁵ has 16 terms. The two middle terms are the:
Q25. The coefficients of the 5th, 6th and 7th terms in the expansion of (1 + x)ⁿ are in A.P. The sum of the possible values of n is:
Q26. The statement n! > 2ⁿ is false for n = 1, 2, 3. The base case for an induction proof of the statement for all larger n is n =
Q27. The value of 1·1! + 2·2! + 3·3! + 4·4! is:
Q28. Using the binomial theorem, 99³ equals:
Q29. The number of terms in the expansion of (1 + x)ⁿ + (1 − x)ⁿ for even n (after simplification, in powers of x) is:
Q30. The fourth term in the expansion of (1 − 2x)⁷ is: