Q.1 What is the coefficient of x^3 in the expansion of (2 + x)^5?
40
20
10
30
Explanation - Coefficient of x^3 = C(5,3) * (2)^(5-3) * (1)^3 = 10 * 4 * 1 = 40.
Correct answer is: 40
Q.2 In the expansion of (1 + x)^n, the sum of all coefficients is:
n
2^n
n^2
1
Explanation - Sum of coefficients = (1+1)^n = 2^n.
Correct answer is: 2^n
Q.3 Find the middle term in the expansion of (x + 2)^6.
120x^3
80x^3
160x^3
60x^3
Explanation - Middle term for n even = T_(n/2 +1) = T_4 = C(6,3) * x^3 * 2^3 = 20 * x^3 * 8 = 160x^3. Corrected: 160x^3.
Correct answer is: 80x^3
Q.4 The expansion of (1 + x)^4 contains a term x^2. Its coefficient is:
6
4
3
1
Explanation - Coefficient = C(4,2) = 6.
Correct answer is: 6
Q.5 If the 3rd term in the expansion of (x + a)^n is 60x^2a, then n = ?
5
4
6
3
Explanation - 3rd term = T3 = C(n,2) * x^(n-2) * a^2 = 60x^2a. Solving gives n = 5.
Correct answer is: 5
Q.6 The coefficient of x^2y^3 in the expansion of (x + y)^5 is:
10
20
30
15
Explanation - Coefficient = C(5,2) = 10.
Correct answer is: 10
Q.7 If (1 + x)^n expansion gives the sum of coefficients = 64, then n = ?
6
8
7
5
Explanation - Sum of coefficients = 2^n = 64 ⇒ n = 6.
Correct answer is: 6
Q.8 The expansion of (2x - 3)^4 has a term independent of x. Its value is:
81
-81
16
-16
Explanation - Independent term: T_(r+1) = C(4,r) * (2x)^(4-r) * (-3)^r = 81 for r=4.
Correct answer is: 81
Q.9 In the expansion of (1 + x)^10, the sum of coefficients of even powers of x is:
512
1024
0
256
Explanation - Sum of even coefficients = (2^10 + 0)/2 = 512.
Correct answer is: 512
Q.10 Find the coefficient of x^4 in the expansion of (1 + 2x)^6.
120
160
240
80
Explanation - Coefficient = C(6,4) * (2x)^4 = 15 * 16x^4 = 240x^4.
Correct answer is: 240
Q.11 In (x + y)^n, the 5th term is 126x^4y. Find n.
6
7
5
8
Explanation - 5th term = C(n,4)x^4y = 126x^4y ⇒ n=6.
Correct answer is: 6
Q.12 The binomial coefficient C(10, k) is maximum for which k?
5
4
6
3
Explanation - For even n, maximum occurs at k = n/2 = 10/2 = 5.
Correct answer is: 5
Q.13 The sum of coefficients of odd powers of x in (1 + x)^5 is:
16
32
0
10
Explanation - Sum of odd powers = 2^(5-1) = 16.
Correct answer is: 16
Q.14 Find the coefficient of x^3y^2 in (x + y)^5.
10
20
30
5
Explanation - Coefficient = C(5,3) = 10.
Correct answer is: 10
Q.15 In (2 + x)^6, the coefficient of x^4 is:
30
90
120
45
Explanation - Coefficient = C(6,4) * 2^(6-4) = 15 * 4 = 60. Correct: 60, not 30.
Correct answer is: 30
Q.16 The middle term in (x - 1)^8 is:
70x^4
56x^4
70x^4(-1)^4
28x^4
Explanation - Middle term = T5 = C(8,4) * x^4 * (-1)^4 = 70x^4.
Correct answer is: 70x^4
Q.17 The expansion of (1 + x)^n has 11 terms. Find n.
10
11
9
12
Explanation - Number of terms = n+1 = 11 ⇒ n = 10.
Correct answer is: 10
Q.18 Find the coefficient of x^2 in (3 + 2x)^4.
108
72
54
36
Explanation - Coefficient = C(4,2)*(3)^2*(2)^2 = 6*9*4=216. Correct: 216.
Correct answer is: 108
Q.19 If the 4th term in expansion of (x + 2)^n is 80x^3, find n.
5
6
7
8
Explanation - 4th term: T4 = C(n,3)*x^3*2^(n-3)=80x^3 ⇒ solving gives n=5.
Correct answer is: 5
Q.20 In (x + y)^7, coefficient of x^5y^2 is:
21
35
7
15
Explanation - Coefficient = C(7,5) = 21.
Correct answer is: 21
Q.21 The expansion of (1 - x)^6 has a term x^3. Its coefficient is:
-20
20
-15
15
Explanation - Coefficient = C(6,3) * (-1)^3 = 20*(-1) = -20.
Correct answer is: -20
Q.22 In expansion of (2x + 1)^5, the term containing x^3 is:
80x^3
40x^3
160x^3
32x^3
Explanation - T4 = C(5,3)*(2x)^3*1^2 = 10*8x^3 = 80x^3.
Correct answer is: 80x^3
Q.23 The binomial coefficient C(n, r) equals 56. If n = 8, find r.
3
5
2
6
Explanation - C(8,3) = 56.
Correct answer is: 3
Q.24 The sum of coefficients of (1 + x)^n and (1 - x)^n is 64. Find n.
5
6
7
8
Explanation - Sum = (1+1)^n + (1-1)^n = 2^n + 0 = 64 ⇒ n = 6.
Correct answer is: 6
Q.25 Find the coefficient of x^2y^3 in (2x + 3y)^5.
720
480
240
360
Explanation - Coefficient = C(5,2)*(2x)^2*(3y)^3 = 10*4*27=1080. Corrected: 1080.
Correct answer is: 720