ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Mathematical Induction and Binomial Theorem (Set 5): 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 x³ in the expansion of (1 + 2x)⁷ is:
Answer: B. 280
Explanation: The term is ⁷C₃(2x)³ = 35 · 8x³ = 280x³.
Q2. The coefficient of x² in the expansion of (1 + x)⁵(1 − x)³ is:
Answer: C. −2
Explanation: (1 + x)⁵(1 − x)³ = (1 − x²)³(1 + x)². The x² terms are −3x² (from (1 − x²)³ times 1) and +x² (from 1 times (1 + x)²): −3 + 1 = −2.
Q3. The sum of all the coefficients in the expansion of (2x² − 3x + 1)⁵ is:
Answer: D. 0
Explanation: Putting x = 1 gives (2 − 3 + 1)⁵ = 0⁵ = 0.
Q4. The number of terms in the expansion of (a + b)¹⁰ + (a − b)¹⁰ after simplification is:
Answer: D. 6
Explanation: Terms with odd powers of b cancel. Only b⁰, b², …, b¹⁰ survive: 6 terms.
Q5. The two middle terms in the expansion of (x − y)⁷ are:
Answer: C. −35x⁴y³ and 35x³y⁴
Explanation: With 8 terms the middle ones are T₄ = ⁷C₃x⁴(−y)³ = −35x⁴y³ and T₅ = ⁷C₄x³(−y)⁴ = +35x³y⁴.
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Q6. The greatest binomial coefficient in the expansion of (1 + x)¹¹ is:
Q7. The value of Σ k·¹⁰Cₖ for k = 0 to 10, i.e. 1·¹⁰C₁ + 2·¹⁰C₂ + … + 10·¹⁰C₁₀, is:
Q8. The value of (⁵C₀)² + (⁵C₁)² + (⁵C₂)² + (⁵C₃)² + (⁵C₄)² + (⁵C₅)² is:
Q9. Using the first three terms of the binomial expansion, (0.99)⁸ is approximately (to two decimal places):
Q10. The coefficient of x⁴ in the expansion of (1 − x)⁻³ is:
Q11. The coefficient of x² in the expansion of √(1 − 2x) is:
Q12. The coefficient of x² in the expansion of (1 − 4x)^(−1/2) is:
Q13. The coefficient of x³ in the expansion of (1 + x)^(−1/2) is:
Q14. By induction, 1·1! + 2·2! + 3·3! + … + n·n! equals:
Q15. The largest positive integer that divides n³ − n for every integer n ≥ 2 is:
Q16. The largest integer that divides 9ⁿ − 8n − 1 for every positive integer n is:
Q17. By direct checking, the smallest integer n₀ such that 2ⁿ > n² for every n ≥ n₀ is:
Q18. Which statement has a valid inductive step but is false, because its base case fails?
Q19. Using (6 + 1), the remainder when 5⁹⁹ is divided by 6 is:
Q20. The sum of the coefficients of the odd powers of x in the expansion of (1 + x)⁸ is:
Q21. The coefficient of x⁻³ in the expansion of (x − 1/x)⁹ is:
Q22. The term independent of x in the expansion of (x³ − 1/x²)¹⁰ is:
Q23. The coefficient of x²y³ in the expansion of (2x − y)⁵ is:
Q24. In the expansion of (1 + x)ⁿ the coefficients of the 2nd, 3rd and 4th terms are in A.P. Then n equals:
Q25. If three consecutive coefficients ⁿCᵣ₋₁, ⁿCᵣ, ⁿCᵣ₊₁ are in the ratio 1 : 2 : 3, then n equals:
Q26. The value of (2 + √3)⁴ + (2 − √3)⁴ is:
Q27. The value of 1·2 + 2·2² + 3·2³ + … + 6·2⁶ is:
Q28. The sum of the coefficients of the even powers of x in the expansion of (1 + 2x)⁵ is:
Q29. The numerically greatest term in the expansion of (1 + 3x)⁸ when x = 1/2 has the value:
Q30. In Pascal's triangle, the row whose entries add up to 1024 has, as its fourth entry (counting from the left), the number: