JEE Main - General term and middle term Online Test (MCQ Quiz)

1. In the binomial expansion of (a + b)^n, what is the general term?

A) T_{r+1} = C(n, r) a^{n-r} b^r B) T_r = C(n, r) a^{n-r} b^r C) T_{r+1} = C(n, r) a^r b^{n-r} D) T_r = C(n, r) a^r b^{n-r}

2. The general term in the expansion of (x + y)^n is given by:

A) The (r+1)th term B) The rth term C) The (n+1)th term D) The nth term

3. What is the index 'r' in the general term T_{r+1} = C(n, r) x^{n-r} y^r of the expansion (x + y)^n?

A) It ranges from 0 to n B) It ranges from 1 to n C) It ranges from 0 to n+1 D) It ranges from 1 to n+1

4. For the expansion of (ax + by)^n, the (r+1)th term is:

A) C(n, r) (ax)^{n-r} (by)^r B) C(n, r) (ax)^r (by)^{n-r} C) C(n, r-1) (ax)^{n-r} (by)^r D) C(n, r+1) (ax)^{n-r} (by)^r

5. In the expansion of (x - y)^n, the general term T_{r+1} is:

A) (-1)^r C(n, r) x^{n-r} y^r B) (-1)^{r+1} C(n, r) x^{n-r} y^r C) C(n, r) x^{n-r} y^r D) (-1)^r C(n, r) x^r y^{n-r}

6. What is the (r+1)th term in the expansion of (2x - 3y)^7?

A) C(7, r) (2x)^{7-r} (-3y)^r B) C(7, r) (2x)^r (-3y)^{7-r} C) C(7, r+1) (2x)^{7-r} (-3y)^r D) C(6, r) (2x)^{7-r} (-3y)^r

7. The general term in the expansion of (x + 1/x)^n is:

A) T_{r+1} = C(n, r) x^{n-r} (1/x)^r B) T_r = C(n, r) x^{n-r} (1/x)^r C) T_{r+1} = C(n, r) x^r (1/x)^{n-r} D) T_{r+1} = C(n, r) x^{n-r} (1/x)^{n-r}

8. In the expansion of (a + b)^n, if n is even, how many middle terms are there?

A) One B) Two C) Three D) None

9. If n is an even positive integer, the middle term in the expansion of (a + b)^n is:

A) The ((n/2) + 1)th term B) The (n/2)th term C) The ((n+1)/2)th term D) The (n+2)th term

10. For the expansion of (x + y)^10, the middle term is:

A) The 5th term B) The 6th term C) The 11th term D) The 7th term

11. If n is an odd positive integer, how many middle terms are there in the expansion of (a + b)^n?

A) Two B) One C) Three D) Four

12. If n is an odd positive integer, the middle terms in the expansion of (a + b)^n are:

A) The ((n+1)/2)th term and the ((n+3)/2)th term B) The (n/2)th term and the ((n+2)/2)th term C) The ((n-1)/2)th term and the ((n+1)/2)th term D) The ((n+1)/2)th term and the ((n+5)/2)th term

13. For the expansion of (x + y)^9, the middle terms are:

A) The 5th term and the 6th term B) The 4th term and the 5th term C) The 9th term and the 10th term D) The 10th term and the 11th term

14. What is the coefficient of the middle term in the expansion of (x + y)^6?

A) C(6, 3) B) C(6, 2) C) C(6, 4) D) C(6, 5)

15. The middle term of (x + 2y)^8 is:

A) T_5 = C(8, 4) x^4 (2y)^4 B) T_4 = C(8, 3) x^5 (2y)^3 C) T_6 = C(8, 5) x^3 (2y)^5 D) T_3 = C(8, 2) x^6 (2y)^2

16. Find the middle term in the expansion of (3x - 4y)^10.

A) T_6 = C(10, 5) (3x)^5 (-4y)^5 B) T_5 = C(10, 4) (3x)^6 (-4y)^4 C) T_7 = C(10, 6) (3x)^4 (-4y)^6 D) T_11 = C(10, 10) (3x)^0 (-4y)^10

17. What is the value of 'r' for the middle term in the expansion of (a + b)^n when n is even?

A) n/2 B) (n+1)/2 C) n D) n-1

18. In the expansion of (x + 1/x)^6, find the middle term.

A) C(6, 3) x^0 B) C(6, 2) x^2 C) C(6, 4) x^{-2} D) C(6, 5) x^{-4}

19. For the expansion of (a - b)^7, the middle terms are:

A) T_4 = -C(7, 3) a^4 b^3 and T_5 = C(7, 4) a^3 b^4 B) T_3 = C(7, 2) a^5 b^2 and T_4 = -C(7, 3) a^4 b^3 C) T_5 = C(7, 4) a^3 b^4 and T_6 = -C(7, 5) a^2 b^5 D) T_4 = C(7, 3) a^4 b^3 and T_5 = -C(7, 4) a^3 b^4

20. The general term in the expansion of (x^p + y^q)^n is T_{r+1} = C(n, r) (x^p)^{n-r} (y^q)^r. What is the power of x in this term?

A) p(n-r) B) p(n+r) C) p*r D) p(n-r)q

21. In the expansion of (a + b)^n, the general term T_{r+1} has the power of 'b' as r. What is the power of 'a'?

A) n-r B) n+r C) r D) n

22. What is the index of the term containing a^{n-k}b^k in the expansion of (a+b)^n?

A) k+1 B) k C) n-k+1 D) n-k

23. If the expansion is (x - 2)^5, the general term is:

A) T_{r+1} = C(5, r) x^{5-r} (-2)^r B) T_r = C(5, r) x^{5-r} (-2)^r C) T_{r+1} = C(5, r) x^r (-2)^{5-r} D) T_{r+1} = C(5, r) x^{5-r} (2)^r

24. In the expansion of (a + b)^n, the term independent of x in (x + 1/x)^n is obtained when the powers of x cancel out. This happens when:

A) The power of x in the first term equals the power of x in the second term B) The power of x in the first term is zero C) The power of x in the second term is zero D) The sum of the powers of x is 1

25. For the expansion of (x + y)^n, the coefficient of the (r+1)th term is always:

A) C(n, r) B) C(n, r+1) C) C(n, n-r) D) C(n, r-1)

26. What is the term number for the middle term when n=12 in the expansion of (a+b)^n?

A) 7th term B) 6th term C) 13th term D) 8th term

27. The middle term in the expansion of (x + 1/y)^10 is:

A) C(10, 5) x^5 (1/y)^5 B) C(10, 4) x^6 (1/y)^4 C) C(10, 6) x^4 (1/y)^6 D) C(10, 5) x^4 (1/y)^6

28. In the expansion of (a + b)^n, if n is odd, the two middle terms are symmetric. The powers of 'a' in these terms are:

A) (n+1)/2 and (n-1)/2 B) (n-1)/2 and (n-3)/2 C) (n+3)/2 and (n+1)/2 D) (n+1)/2 and (n+3)/2

29. The general term of the expansion (ax^p + by^q)^n is T_{r+1}. The power of x in this term is:

A) p(n-r) B) p*r C) q(n-r) D) q*r

30. For the expansion of (x - y)^n, the (r+1)th term has a sign determined by:

A) (-1)^r B) (-1)^{r+1} C) (-1)^n D) 1

31. What is the middle term in the expansion of (2x - 3)^7?

A) T_4 = -C(7, 3) (2x)^4 (3)^3 B) T_5 = C(7, 4) (2x)^3 (3)^4 C) T_3 = C(7, 2) (2x)^5 (3)^2 D) T_6 = -C(7, 5) (2x)^2 (3)^5

32. In the expansion of (a + b)^n, if n is even, the middle term is the ((n/2) + 1)th term. The power of 'b' in this term is:

A) n/2 B) n/2 + 1 C) n D) n-1

33. Find the coefficient of x^5 in the expansion of (x + 2)^7. This involves finding a specific general term.

A) C(7, 5) * 2^2 B) C(7, 2) * 2^5 C) C(7, 5) * 2^5 D) C(7, 2) * 2^2

34. The general term in the expansion of (x + y)^n is T_{r+1}. For the term involving y^k, what is the value of r?

A) k B) k+1 C) n-k D) n-k+1

35. What is the index of the term that contains a^3 b^7 in the expansion of (a + b)^10?

A) 8th term B) 7th term C) 4th term D) 11th term

36. In the expansion of (x + 1/x)^8, the middle term is:

A) C(8, 4) x^0 B) C(8, 3) x^2 C) C(8, 5) x^{-2} D) C(8, 4) x^4

37. If n is odd, the two middle terms in the expansion of (a + b)^n have powers of 'b' as:

A) (n+1)/2 and (n+3)/2 B) (n-1)/2 and (n+1)/2 C) (n-3)/2 and (n-1)/2 D) (n+1)/2 and (n-1)/2

38. What is the coefficient of the middle term in the expansion of (x + 3y)^6?

A) C(6, 3) * 3^3 B) C(6, 2) * 3^2 C) C(6, 4) * 3^4 D) C(6, 3) * 3^4

39. The general term in the expansion of (a + b)^n is T_{r+1}. The term number is related to the power of 'b' by:

A) Term Number = Power of b + 1 B) Term Number = Power of b C) Term Number = Power of b - 1 D) Term Number = n - Power of b + 1

40. In the expansion of (2x - 1)^9, the middle terms are:

A) T_5 = C(9, 4) (2x)^5 (-1)^4 and T_6 = C(9, 5) (2x)^4 (-1)^5 B) T_4 = C(9, 3) (2x)^6 (-1)^3 and T_5 = C(9, 4) (2x)^5 (-1)^4 C) T_5 = C(9, 4) (2x)^4 (-1)^5 and T_6 = C(9, 5) (2x)^5 (-1)^4 D) T_6 = C(9, 5) (2x)^4 (-1)^5 and T_7 = C(9, 6) (2x)^3 (-1)^6

41. Which term in the expansion of (x + y)^12 is the middle term?

A) 7th term B) 6th term C) 13th term D) 5th term

42. What is the coefficient of the term x^3 y^4 in the expansion of (x + y)^7?

A) C(7, 4) B) C(7, 3) C) C(7, 5) D) C(7, 6)

43. The general term T_{r+1} in the expansion of (a + b)^n corresponds to the term with:

A) a^{n-r} b^r B) a^r b^{n-r} C) a^{n-r+1} b^{r-1} D) a^{n-r-1} b^{r+1}

44. If n is even, the middle term in the expansion of (a + b)^n is C(n, n/2) a^{n/2} b^{n/2}. What is the term number?

A) (n/2) + 1 B) n/2 C) (n+1)/2 D) n

45. Find the middle term in the expansion of (x + 1)^10.

A) C(10, 5) x^5 B) C(10, 4) x^6 C) C(10, 6) x^4 D) C(10, 5) x^6

46. The general term in the expansion of (ax + by)^n is C(n, r) (ax)^{n-r} (by)^r. The power of 'a' in this term is:

A) n-r B) r C) n D) nr

47. For the expansion of (a + b)^n, if n is odd, the second middle term is the ((n+3)/2)th term. What is the power of 'a' in this term?

A) (n-1)/2 B) (n+1)/2 C) (n+3)/2 D) n - (n+3)/2

48. What is the middle term in the expansion of (x - 2y)^6?

A) T_4 = C(6, 3) x^3 (-2y)^3 B) T_3 = C(6, 2) x^4 (-2y)^2 C) T_5 = C(6, 4) x^2 (-2y)^4 D) T_6 = C(6, 5) x^1 (-2y)^5

49. The general term in the expansion of (x + 1/x)^n is T_{r+1} = C(n, r) x^{n-r} (1/x)^r. For this term to be independent of x, the power of x must be 0. Therefore:

A) n - 2r = 0 B) n - r = 0 C) r = 0 D) n + r = 0

50. In the expansion of (a + b)^n, the middle term occurs when r = n/2 (if n is even). What is the coefficient?

A) C(n, n/2) B) C(n, n/2 + 1) C) C(n, n/2 - 1) D) C(n, n)

General term and middle term - Online Test

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