ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Mathematical Induction and Binomial Theorem (Set 4): 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. Find the coefficient of x⁴ in (1 + 3x)⁷.
Answer: B. 2835
Explanation: The x⁴ term is C(7,4)(3x)⁴ = 35 × 81x⁴, so the coefficient is 2835.
Q2. Find the term free of x in (x² − 1/x)⁹.
Answer: D. 84
Explanation: Tᵣ₊₁ = C(9,r)(x²)⁹⁻ʳ(−1/x)ʳ has x¹⁸⁻³ʳ, so the free term has r = 6. It equals C(9,6)(−1)⁶ = 84.
Q3. The middle term in the expansion of (x + 2y)⁶ is:
Answer: D. 160x³y³
Explanation: With 7 terms the middle one is the 4th: C(6,3)x³(2y)³ = 20 × 8 x³y³ = 160x³y³.
Q4. The sum C(10,0) + C(10,2) + C(10,4) + … + C(10,10) of the even-placed binomial coefficients equals:
Answer: A. 512
Explanation: The sum of the coefficients with even lower index equals the sum with odd lower index, each being 2ⁿ⁻¹ = 2⁹ = 512.
Q5. Using the binomial theorem, (1.02)¹⁰ correct to three decimal places is:
Answer: B. 1.219
Explanation: (1 + 0.02)¹⁰ = 1 + 10(0.02) + 45(0.0004) + 120(0.000008) + … = 1 + 0.2 + 0.018 + 0.00096 + 0.0000336 ≈ 1.21899, which is 1.219 to three decimals.
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Q6. The coefficient of x³ in the expansion of (1 − 2x)^(−1/2) is:
Q7. In the expansion of √(4 − x) in ascending powers of x, the coefficient of x² is:
Q8. The binomial expansion of (2 + 3x)⁻³ in ascending powers of x is valid for:
Q9. What is the largest integer that divides 8ⁿ − 3ⁿ for every natural number n?
Q10. In proving 1 + 2 + … + n = n(n + 1)/2 by induction, after assuming P(k), the inductive step needs to show that:
Q11. The inequality 2ⁿ > n² is true for n = 1 but false for n = 2, 3, 4. If it is to be proved by induction for all integers n ≥ m, the smallest possible value of m is:
Q12. The value of C(8,0) − C(8,1) + C(8,2) − C(8,3) + … + C(8,8) is:
Q13. The value of C(10,4) + C(10,5) is:
Q14. The binomial coefficients of the 5th, 6th and 7th terms of (1 + x)ⁿ form an arithmetic progression. Find the sum of all admissible values of n.
Q15. The value of Σ r·C(10, r) for r = 0 to 10 is:
Q16. The 6th term in the expansion of (2x − 1/x)⁸ is:
Q17. The expansion of (a + b)¹⁰ + (a − b)¹⁰ has how many terms?
Q18. The expansion of (x + y)⁸(x − y)⁸ has how many terms?
Q19. The row of Pascal's triangle that begins 1, 9, 36, … has entries whose sum is:
Q20. Using the binomial theorem, √99 = 10(1 − 0.01)^(1/2) is approximately:
Q21. The coefficient of x³ in the expansion of (1 + x)⁴(1 + 2x)³ is:
Q22. If the coefficients of x² and x³ in the expansion of (3 + kx)⁹ are equal, the value of k is:
Q23. The largest integer dividing 9ⁿ − 1 for every natural number n is:
Q24. A statement P(n) has P(1) true, and P(k) true implies P(k + 2) true for every k. By induction this establishes P(n) for:
Q25. The statement n! > 2ⁿ is true for all integers n ≥ m. The smallest possible m is:
Q26. The sum C(5,0)·2⁰ + C(5,1)·2¹ + C(5,2)·2² + … + C(5,5)·2⁵ equals:
Q27. The greatest binomial coefficient in the expansion of (1 + x)¹² is:
Q28. In (x² − 2/x)⁶, the term containing x³ has coefficient:
Q29. The value of C(5,0)² + C(5,1)² + C(5,2)² + … + C(5,5)² is:
Q30. Which of the following expressions is divisible by 6 for every natural number n?