ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Mathematical Induction and Binomial Theorem (Set 6): 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 coefficient of x⁷ in the expansion of (2x² − 1/x)⁸ is:
Answer: D. −1792
Explanation: The general term is C(8, k)(2x²)⁸⁻ᵏ(−1/x)ᵏ, with power of x equal to 16 − 3k. Setting 16 − 3k = 7 gives k = 3, and the coefficient is C(8, 3) · 2⁵ · (−1)³ = 56 · 32 · (−1) = −1792.
Q2. The term independent of x in the expansion of (x³ − 2/x)⁸ is:
Answer: C. 1792
Explanation: The general term C(8, k)(x³)⁸⁻ᵏ(−2/x)ᵏ has power 24 − 4k, which is zero for k = 6. The term is C(8, 6)(−2)⁶ = 28 · 64 = 1792.
Q3. The sum of all the coefficients in the expansion of (1 + 2x − 3x²)²⁰²⁶ is:
Answer: B. 0
Explanation: Putting x = 1 gives the sum of coefficients: (1 + 2 − 3)²⁰²⁶ = 0²⁰²⁶ = 0.
Q4. In the expansion of (a + b)¹¹ the middle terms are:
Answer: A. the 6th and 7th terms
Explanation: The expansion has 12 terms. For an even number of terms there are two middle terms: the 6th and 7th.
Q5. For x = ½, the numerically greatest term in the expansion of (1 + x)¹⁰ is the term number:
Answer: A. 4
Explanation: The ratio Tᵣ₊₁/Tᵣ = ((11 − r)/r)·x = (11 − r)/(2r) ≥ 1 for r ≤ 3.67, so terms increase up to r = 3 (T₄) and then decrease: T₄ = 120/8 = 15 versus T₃ = 11.25 and T₅ = 13.125.
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Q6. The value of C(12, 1) + C(12, 3) + C(12, 5) + … + C(12, 11) is:
Q7. The value of Σ r · C(8, r) for r = 0 to 8 is:
Q8. Using the binomial series for (1 − x)^(1/2), the value of √99 correct to two decimal places is:
Q9. The coefficient of x³ in the expansion of 1/(2 − x) in ascending powers of x is:
Q10. In the expansion of (1 − 3x)^(1/3), the coefficient of x³ is:
Q11. In the expansion of (1 + x)³⁰ the coefficient of x²ᵏ⁺¹ equals the coefficient of xᵏ⁺⁷, where k is a positive integer. Then k equals:
Q12. In the expansion of (1 + x)ⁿ, three consecutive coefficients are in the ratio 1 : 3 : 5. Then n equals:
Q13. The expression n² + n + 41 is prime for n = 1, 2, 3, …, up to a first failure. The smallest positive integer n for which it is not prime is:
Q14. For how many positive integers n is 2ⁿ ≤ n²?
Q15. Which expression is divisible by 6 for every positive integer n?
Q16. The remainder when 7⁹⁹ is divided by 50 is:
Q17. The last two digits of 7²⁰²⁶ are:
Q18. The value of C(6, 0) + 2·C(6, 1) + 4·C(6, 2) + 8·C(6, 3) + … + 64·C(6, 6) is:
Q19. The value of (√2 + 1)⁶ + (√2 − 1)⁶ is:
Q20. The number of dissimilar terms in the expansion of (x + y)¹⁰ + (x − y)¹⁰ is:
Q21. The coefficient of x⁴ in the product (1 + x)⁸(1 − x)⁸ is:
Q22. If the sum of the binomial coefficients in the expansion of (1 + x)ⁿ is 4096, the greatest binomial coefficient is:
Q23. The sum 1·1! + 2·2! + 3·3! + 4·4! + 5·5! equals:
Q24. The largest positive integer that divides 3²ⁿ − 1 for every positive integer n is:
Q25. Bernoulli's inequality (1 + x)ⁿ ≥ 1 + nx is proved for x ≥ −1. Taking x = −3, which lies outside that range, the smallest positive integer n for which the inequality fails is:
Q26. The coefficient of x⁶ in the product (1 + x)⁵(1 + x)⁷ is:
Q27. The number of rational terms in the expansion of (√2 + ∛3)¹² is:
Q28. In the expansion of (1 + x)ⁿ the coefficient of x² is 45. The coefficient of x³ is:
Q29. A student shows that if P(k) is true then P(k + 1) is true, where P(n) is the statement 'n² + n is odd'. Which conclusion is correct?
Q30. Using the binomial theorem on (100 + 1)⁴, the value of 101⁴ is: