Binomial theorem for positive integral index - Question Bank

1. What is the (r+1)th term in the expansion of (a+x)^n?
A) C(n, r) * a^(n-r) * x^r
B) C(n, r) * a^r * x^(n-r)
C) C(n, r+1) * a^(n-r-1) * x^(r+1)
D) C(n, r-1) * a^(n-r+1) * x^(r-1)
2. The sum of the coefficients of the expansion of (x + y)^n is obtained by setting x=1 and y=1. What is the result?
A) 2^n
B) n
C) 1
D) 0
3. If the term T_{r+1} in the expansion of (x + a)^n is C(n, r) x^(n-r) a^r, then the term containing x^k will have r equal to:
A) n-k
B) k
C) n+k
D) n/k
4. What is the coefficient of the term with x^p in the expansion of (x+a)^n?
A) C(n, n-p) * a^p
B) C(n, p) * a^(n-p)
C) C(n, p) * a^p
D) C(n, n-p) * a^(n-p)
5. In the expansion of (x + a)^n, the term with the highest power of 'a' is:
A) The last term
B) The first term
C) The middle term
D) The second term
6. What is the coefficient of x^k in the expansion of (x+a)^n, where k is the power of x?
A) C(n, k) * a^(n-k)
B) C(n, n-k) * a^k
C) C(n, k) * a^k
D) C(n, n-k) * a^(n-k)
7. What is the coefficient of x^2 in the expansion of (1 + x)^5?
A) 10
B) 5
C) 4
D) 1
8. Find the term independent of x in the expansion of (x^2 - 2/x)^9.
A) -84 * 2^6
B) 84 * 2^6
C) -126 * 2^3
D) 126 * 2^3
9. What is the coefficient of x^7 y^3 in the expansion of (x + y)^10?
A) C(10, 7)
B) C(10, 3)
C) C(10, 7) and C(10, 3)
D) 10
10. In the expansion of (x + a)^n, the coefficient of x^r a^(n-r) is:
A) C(n, n-r)
B) C(n, r)
C) C(n, r) and C(n, n-r)
D) n
11. What is the value of C(n, r) + C(n, r-1)?
A) C(n+1, r)
B) C(n+1, r-1)
C) C(n, r)
D) C(n+1, r+1)
12. The expansion of (x + a)^n has terms T_1, T_2, ..., T_{n+1}. What is the index of the term that contains a^k?
A) k+1
B) k
C) n-k+1
D) n-k
13. What is the coefficient of the 5th term in the expansion of (a + b)^9?
A) 126
B) 210
C) 252
D) 120
14. Find the term containing x^3 in the expansion of (x + 1/x)^7.
A) 35 x^3
B) 35 x^4
C) 21 x^3
D) 21 x^4
15. What is the sum of the coefficients of the terms in the expansion of (3x - 2y)^4?
A) 1
B) 16
C) 81
D) 256
16. Find the middle term in the expansion of (x + 3y)^10.
A) 4035 * 243 * x^5 y^5
B) 252 * 243 * x^5 y^5
C) 4035 * 243 * x^6 y^4
D) 252 * 243 * x^6 y^4
17. What is the 4th term in the expansion of (2x - 3y)^7?
A) -26460 x^4 y^3
B) 26460 x^4 y^3
C) -13230 x^3 y^4
D) 13230 x^3 y^4
18. Find the term independent of x in the expansion of (x - 2/x)^6.
A) 192
B) -192
C) 24
D) -24
19. What is the coefficient of x^3 in the expansion of (x + 2)^5?
A) 80
B) 40
C) 10
D) 20
20. The (r+1)th term in the expansion of (x + a)^n is T_{r+1} = C(n, r) x^(n-r) a^r. What is the value of r for the term containing a^5 in the expansion of (x+a)^10?
A) 5
B) 4
C) 6
D) 10
21. In the expansion of (x + a)^n, if n is odd, what are the indices of the two middle terms (starting from r=0)?
A) (n-1)/2 and (n+1)/2
B) n/2 and n/2 + 1
C) (n+1)/2 and (n+3)/2
D) n/2 - 1 and n/2
22. In the expansion of (x + a)^n, if n is even, what is the index of the middle term (starting from r=0)?
A) n/2
B) (n+1)/2
C) n/2 + 1
D) n-1
23. What is the value of C(n, n-1)?
A) n
B) 1
C) 0
D) n!
24. What is the value of C(n, 1)?
A) n
B) 1
C) 0
D) n!
25. What is the sum of the binomial coefficients C(n, 0) + C(n, 1) + ... + C(n, n)?
A) 2^n
B) 2^n - 1
C) n
D) 1
26. In the expansion of (x + a)^n, what is the last term (r=n)?
A) C(n, n) x^0 a^n
B) C(n, n-1) x^1 a^(n-1)
C) C(n, 0) x^n a^0
D) C(n, n) x^n a^0
27. The binomial theorem for a positive integral index n states that (x + a)^n = sum_{r=0}^n C(n, r) x^(n-r) a^r. What is the first term (r=0)?
A) C(n, 0) x^n a^0
B) C(n, 1) x^(n-1) a^1
C) C(n, n) x^0 a^n
D) C(n, 0) x^0 a^n
28. What is the term independent of x in the expansion of (x + 1/x)^n?
A) C(n, n/2) if n is even, otherwise 0
B) C(n, n/2)
C) 0
D) 1
29. In the expansion of (x + a)^n, what is the coefficient of the term containing x^(n-r) a^r?
A) C(n, r)
B) C(n, n-r)
C) Both C(n, r) and C(n, n-r)
D) n
30. What is the value of C(n, n)?
A) 1
B) n
C) 0
D) n!
31. What is the value of C(n, 0)?
A) 1
B) n
C) 0
D) n!
32. Consider the expansion of (1 + x)^n. What is the coefficient of x^r?
A) C(n, r)
B) C(n, n-r)
C) C(n, r+1)
D) n
33. What is the sign of the terms in the expansion of (x - a)^n when n is a positive integer?
A) Alternating positive and negative
B) All positive
C) All negative
D) Alternating negative and positive
34. Which property of binomial coefficients is represented by C(n, r) = C(n, n-r)?
A) Pascal's Identity
B) Symmetry Property
C) Binomial Theorem
D) Upper Negation
35. What is the value of C(n, r) + C(n, r+1)?
A) C(n+1, r+1)
B) C(n+1, r)
C) C(n, r+1)
D) C(n+1, r+2)
36. What is the coefficient of a^k in the expansion of (x + a)^n?
A) C(n, k) * x^(n-k)
B) C(n, n-k) * x^k
C) C(n, k) * x^k
D) C(n, n-k) * x^(n-k)
37. What is the coefficient of x^k in the expansion of (x + a)^n?
A) C(n, k) * a^(n-k)
B) C(n, n-k) * a^k
C) C(n, k) * a^k
D) C(n, n-k) * a^(n-k)
38. Which term in the expansion of (x + a)^n is independent of x?
A) The term where the power of x is 0
B) The term where the power of a is n
C) The last term
D) The first term
39. In the expansion of (x + a)^n, the sum of the powers of x and a in each term is always:
A) n
B) n - 1
C) n + 1
D) 1
40. The expansion of (x + a)^n has (n+1) terms. What is the index of the last term (starting from index 0)?
A) n
B) n - 1
C) n + 1
D) 1
41. When n is an even positive integer, how many terms are there in the expansion of (x + a)^n?
A) n
B) n + 1
C) n - 1
D) 2n
42. When n is an odd positive integer, how many terms are there in the expansion of (x + a)^n?
A) n
B) n + 1
C) n - 1
D) 2n
43. What is the value of the middle term in the expansion of (x + a)^n when n is even?
A) C(n, n/2) * x^(n/2) * a^(n/2)
B) C(n, n/2 + 1) * x^(n/2 - 1) * a^(n/2 + 1)
C) C(n, (n+1)/2) * x^((n-1)/2) * a^((n+1)/2)
D) C(n, n/2 - 1) * x^(n/2 + 1) * a^(n/2 - 1)
44. To find the sum of the coefficients in the expansion of (ax + by)^n, we substitute:
A) x=1, y=1
B) x=0, y=0
C) x=1, y=0
D) x=0, y=1
45. What is the sum of the coefficients in the binomial expansion of (x + y)^n?
A) 2^n
B) 2^n - 1
C) n
D) 1
46. In the expansion of (a + b)^n, the coefficient of the term containing a^(n-r) * b^r is:
A) C(n, r)
B) C(n, n-r)
C) Both C(n, r) and C(n, n-r)
D) n
47. For the expansion of (x - a)^n, where n is a positive integer, what is the general term?
A) T_{r+1} = C(n, r) * x^(n-r) * (-a)^r
B) T_{r+1} = C(n, r) * x^r * (-a)^(n-r)
C) T_{r+1} = -C(n, r) * x^(n-r) * a^r
D) T_{r+1} = C(n, r) * x^(n-r) * a^r
48. The binomial expansion of (x + a)^n is given by the sum of terms from r=0 to n. What is the formula for the (r+1)th term?
A) T_{r+1} = C(n, r) * x^r * a^(n-r)
B) T_{r+1} = C(n, r) * x^(n-r) * a^r
C) T_{r+1} = C(n, r+1) * x^(n-r-1) * a^(r+1)
D) T_{r+1} = C(n, r-1) * x^(n-r+1) * a^(r-1)
49. In the binomial expansion of (x + a)^n, what is the (r+1)th term?
A) nCr * x^r * a^(n-r)
B) nCr * x^(n-r) * a^r
C) nC(r+1) * x^(n-r-1) * a^(r+1)
D) nC(r-1) * x^(n-r+1) * a^(r-1)
50. What is the general term in the binomial expansion of (x + a)^n, where n is a positive integer?
A) T_{r+1} = C(n, r) * x^(n-r) * a^r
B) T_r = C(n, r) * x^r * a^(n-r)
C) T_{r+1} = C(n, r+1) * x^(n-r-1) * a^(r+1)
D) T_r = C(n, r-1) * x^(n-r+1) * a^(r-1)
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