JEE Main - Binomial theorem for positive integral index Online Test (MCQ Quiz)

1. 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)

2. 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)

3. 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)

4. 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

5. 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

6. 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

7. 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

8. 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)

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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)

15. 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)

16. 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)

17. 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

18. 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

19. 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

20. What is the value of C(n, 0)?

A) 1 B) n C) 0 D) n!

21. What is the value of C(n, n)?

A) 1 B) n C) 0 D) n!

22. 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

23. 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

24. 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

25. 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

26. 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

27. What is the value of C(n, 1)?

A) n B) 1 C) 0 D) n!

28. What is the value of C(n, n-1)?

A) n B) 1 C) 0 D) n!

29. 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

30. 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

31. 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

32. What is the coefficient of x^3 in the expansion of (x + 2)^5?

A) 80 B) 40 C) 10 D) 20

33. Find the term independent of x in the expansion of (x - 2/x)^6.

A) 192 B) -192 C) 24 D) -24

34. 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

35. 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

36. 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

37. 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

38. What is the coefficient of the 5th term in the expansion of (a + b)^9?

A) 126 B) 210 C) 252 D) 120

39. 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

40. 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)

41. 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

42. 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

43. 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

44. What is the coefficient of x^2 in the expansion of (1 + x)^5?

A) 10 B) 5 C) 4 D) 1

45. 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)

46. 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

47. 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)

48. 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

49. 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

50. 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)

Binomial theorem for positive integral index - Online Test

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