Mathematical Induction and Binomial Theorem - Multiple Choice Questions
Instructions: This quiz contains 55 multiple-choice questions on mathematical induction and binomial theorem. Click the "Show Answers" button to highlight the correct answers in yellow.
| 01 | The first step in mathematical induction is called: (A) Base case (B) Inductive step (C) Hypothesis (D) Conclusion |
| 02 | The general term in the binomial expansion of (a+b)n is: (A) nCr ar bn (B) nCr an-r br (C) nCr ar bn-r (D) nCr an br |
| 03 | In mathematical induction, we assume P(k) is true for some k ∈ N. This is called: (A) Base case (B) Inductive hypothesis (C) Inductive step (D) Conclusion |
| 04 | The middle term in the expansion of (x + y)8 is: (A) 4th term (B) 5th term (C) 6th term (D) 4th and 5th terms |
| 05 | If P(n): 1 + 2 + 3 + ... + n = n(n+1)/2, then P(1) is: (A) 1 = 1 (B) 1 = 1(2)/2 (C) 1 = 1(1)/2 (D) 1 = 2/2 |
| 06 | The coefficient of x5 in (1+x)8 is: (A) 8 (B) 56 (C) 70 (D) 28 |
| 07 | Mathematical induction is used to prove statements about: (A) Real numbers (B) Natural numbers (C) Complex numbers (D) All numbers |
| 08 | The number of terms in the expansion of (a+b)n is: (A) n (B) n+1 (C) n-1 (D) 2n |
| 09 | In the inductive step of mathematical induction, we prove: (A) P(1) is true (B) P(k) ⇒ P(k+1) (C) P(n) is true for all n (D) P(k) is true |
| 10 | The term independent of x in (x + 1/x)6 is: (A) 6th term (B) 4th term (C) 3rd term (D) 2nd term |
| 11 | Which of the following can be proved by mathematical induction? (A) 2n > n for all n ∈ N (B) 2n > n for all n ∈ N (C) √2 is irrational (D) There are infinite primes |
| 12 | The coefficient of x3 in (2-x)5 is: (A) -40 (B) -40 (C) 40 (D) -80 |
| 13 | The statement "P(n) is true for all n ∈ N" is called: (A) Base case (B) Inductive hypothesis (C) Conclusion (D) Assumption |
| 14 | The greatest coefficient in the expansion of (1+x)10 is: (A) 10C4 (B) 10C5 (C) 10C6 (D) 10C3 |
| 15 | If P(n): n2 + n is even, then P(1) is: (A) 1 + 1 = 2 (even) (B) 1 + 1 = 2 (even) (C) 1² + 1 = 2 (even) (D) 1² + 1 = 2 (even) |
| 16 | The expansion of (1-x)-1 is: (A) 1 + x + x² + x³ + ... (B) 1 + x + x² + x³ + ... (C) 1 - x + x² - x³ + ... (D) 1 - x + x² - x³ + ... |
| 17 | In strong mathematical induction, we assume: (A) P(k) is true (B) P(1), P(2), ..., P(k) are all true (C) P(n) is true for all n (D) P(k+1) is true |
| 18 | The coefficient of x4 in (1+2x)7 is: (A) 35 (B) 560 (C) 280 (D) 1120 |
| 19 | Which step is crucial in mathematical induction? (A) Both base case and inductive step (B) Only base case (C) Only inductive step (D) Neither |
| 20 | The general term in (x² - 1/x)9 is: (A) 9Cr x18-2r (-1/x)r (B) 9Cr x18-3r (-1)r (C) 9Cr x18-2r (D) 9Cr x9-r (-1)r |
| 21 | If P(n): 32n - 1 is divisible by 8, then P(1) is: (A) 3² - 1 = 8 (divisible by 8) (B) 3² - 1 = 8 (divisible by 8) (C) 3 - 1 = 2 (D) 9 - 1 = 8 |
| 22 | The term containing x10 in (x² + 1/x)12 is: (A) 3rd term (B) 5th term (C) 7th term (D) 9th term |
| 23 | Mathematical induction was first used by: (A) Euclid (B) Francesco Maurolico (C) Pascal (D) Newton |
| 24 | The coefficient of x5 in (1+x)n is 126. Then n = ? (A) 7 (B) 9 (C) 8 (D) 10 |
| 25 | In the principle of mathematical induction, the statement P(n) is proved for: (A) n = 1 only (B) Some particular n (C) All natural numbers n (D) Real numbers |
| 26 | The sum of coefficients in (1+x)n is: (A) n (B) 2n (C) 2n-1 (D) 0 |
| 27 | If P(n): 7n - 3n is divisible by 4, then for P(k+1), we consider: (A) 7k - 3k (B) 7k+1 - 3k+1 (C) 7k + 3k (D) 7k-1 - 3k-1 |
| 28 | The middle term in (x - 1/x)10 is: (A) -252 (B) -252 (C) 252 (D) -210 |
| 29 | Which of the following cannot be proved by mathematical induction? (A) Sum of first n natural numbers (B) 2n > n (C) The area of a circle is ฯr² (D) n! > 2n for n ≥ 4 |
| 30 | The coefficient of x3 in (1+2x+3x²)5 is: (A) 120 (B) 320 (C) 240 (D) 160 |
| 31 | In mathematical induction, if P(1) is true and P(k) ⇒ P(k+1), then: (A) P(2) is true (B) P(3) is true (C) P(n) is true for all n ∈ N (D) P(k) is true |
| 32 | The term independent of x in (√x + 1/√x)12 is: (A) 12C4 (B) 12C6 (C) 12C5 (D) 12C7 |
| 33 | If P(n): n(n+1)(n+2) is divisible by 6, then P(1) is: (A) 1×2×3 = 6 (divisible by 6) (B) 1×2×3 = 6 (divisible by 6) (C) 1+2+3=6 (D) 1×2=2 |
| 34 | The expansion of (1+x)-2 is: (A) 1 - 2x + 3x² - 4x³ + ... (B) 1 - 2x + 3x² - 4x³ + ... (C) 1 + 2x + 3x² + 4x³ + ... (D) 1 + x + x² + x³ + ... |
| 35 | The second step in mathematical induction is called: (A) Base case (B) Inductive step (C) Hypothesis (D) Verification |
| 36 | The coefficient of x7 in (1+x)11 is equal to coefficient of xr. Then r = ? (A) 3 (B) 4 (C) 5 (D) 6 |
| 37 | If P(n) is not true for n=1, then: (A) P(n) may be true for some n (B) Mathematical induction fails (C) P(n) is true for all n (D) Try n=2 |
| 38 | The greatest term in the expansion of (1+3x)8 when x=1/3 is: (A) 4th term (B) 5th term (C) 6th term (D) 7th term |
| 39 | In the inductive step, we prove that: (A) P(1) is true (B) If P(k) is true, then P(k+1) is true (C) P(k) is true (D) P(n) is true |
| 40 | The sum of coefficients of odd powers of x in (1+x)n is: (A) 2n (B) 2n-1 (C) 2n+1 (D) 0 |
| 41 | If P(n): 1² + 2² + ... + n² = n(n+1)(2n+1)/6, then P(k+1) is: (A) 1²+2²+...+k² = k(k+1)(2k+1)/6 (B) 1²+2²+...+(k+1)² = (k+1)(k+2)(2k+3)/6 (C) k(k+1)(2k+1)/6 (D) (k+1)(k+2)(2k+3)/6 |
| 42 | The coefficient of xn in (1+x)2n is: (A) 2nCn (B) 2nCn (C) nCn (D) 2nC2n |
| 43 | Mathematical induction is based on: (A) The well-ordering principle (B) The well-ordering principle (C) Axiom of choice (D) Completeness axiom |
| 44 | The term independent of x in (x + 1/x²)15 is: (A) 15C5 (B) 15C10 (C) 15C15 (D) Does not exist |
| 45 | If P(1) is true and P(k) ⇒ P(k+1) for all k ≥ 1, then: (A) P(2) is true (B) P(100) is true (C) P(n) is true for all n ∈ N (D) P(k) is true |
| 46 | The numerically greatest term in (1+2x)9 when x=3/2 is: (A) 4th term (B) 6th term (C) 7th term (D) 8th term |
| 47 | Which of the following is essential in mathematical induction? (A) Proving P(1) is true (B) Proving P(2) is true (C) Proving P(k) is true (D) Proving P(n) is true |
| 48 | The coefficient of x4 in (1-x)-3 is: (A) 15 (B) 15 (C) 10 (D) 20 |
| 49 | If P(n) fails for some n, then: (A) P(n) is false for all n (B) The induction proof is invalid (C) Try strong induction (D) Change the statement |
| 50 | The sum of coefficients in (1+x-3x²)5 is: (A) -1 (B) -1 (C) 1 (D) 0 |
| 51 | In the binomial expansion, nC0 + nC1 + ... + nCn equals: (A) n! (B) 2n (C) 2n-1 (D) n² |
| 52 | The inductive step proves that the statement is true for: (A) n = 1 (B) n = k+1 assuming n=k is true (C) All n (D) Some particular n |
| 53 | The coefficient of x5 in (1+x)21 + (1+x)22 + ... + (1+x)30 is: (A) 31C6 - 21C6 (B) 31C6 - 21C6 (C) 30C5 - 20C5 (D) 31C5 - 21C5 |
| 54 | Mathematical induction can be used to prove inequalities involving: (A) Real numbers (B) Natural numbers (C) Complex numbers (D) All numbers |
| 55 | The term independent of x in (1+x)n(1+1/x)n is: (A) nC0² + nC1² + ... + nCn² (B) nC0² + nC1² + ... + nCn² (C) 2nCn (D) 2n |
