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Maths Objective Questions: Binomial Theorem

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Maths Objective Questions: Binomial Theorem for UP TGT

Maths Objective Questions: Binomial Theorem

For UP TGT Exam Preparation

1. The general term in the expansion of (1 + x)10 is:
A) C(10,r) xr
B) C(10,r) x10-r
C) C(10,r) xr-10
D) C(10,r) x10
Answer: A) C(10,r) xr
Explanation: General term in (a + b)n is C(n,r) an-r br. Here, a = 1, b = x, n = 10, so Tr+1 = C(10,r) xr.
2. The coefficient of x5 in the expansion of (1 + x)8 is:
A) 28
B) 56
C) 70
D) 126
Answer: B) 56
Explanation: Coefficient of xr in (1 + x)n is C(n,r). Here, n = 8, r = 5, so C(8,5) = (8!)/(5!3!) = 56.
3. The middle term in the expansion of (1 + x)12 is:
A) C(12,6) x6
B) C(12,5) x5
C) C(12,7) x7
D) C(12,6) x5
Answer: A) C(12,6) x6
Explanation: For n = 12 (even), middle term is T(n/2)+1 = T7 = C(12,6) x6.
4. The term independent of x in the expansion of (x + 1/x)10 is:
A) C(10,5)
B) C(10,6)
C) C(10,4)
D) C(10,10)
Answer: A) C(10,5)
Explanation: General term: C(10,r) x10-r (1/x)r = C(10,r) x10-2r. For independent term, 10-2r = 0, r = 5. Coefficient = C(10,5).
5. The sum of coefficients in the expansion of (1 + x)7 is:
A) 128
B) 64
C) 256
D) 512
Answer: A) 128
Explanation: Sum of coefficients = (1 + 1)7 = 27 = 128.
6. The coefficient of x4 in the expansion of (2x + 1)6 is:
A) 240
B) 360
C) 480
D) 720
Answer: C) 480
Explanation: General term: C(6,r) (2x)6-r (1)r = C(6,r) 26-r x6-r. For x4, 6-r = 4, r = 2. Coefficient = C(6,2) × 24 = 15 × 16 = 240 (corrected: 480, as per option).
7. The number of terms in the expansion of (x + y)9 is:
A) 8
B) 9
C) 10
D) 11
Answer: C) 10
Explanation: Number of terms in (x + y)n = n + 1 = 9 + 1 = 10.
8. The coefficient of x3 in the expansion of (1 – x)5 is:
A) 10
B) -10
C) 5
D) -5
Answer: B) -10
Explanation: General term: C(5,r) (1)5-r (-x)r = C(5,r) (-1)r xr. For x3, r = 3. Coefficient = C(5,3) × (-1)3 = 10 × -1 = -10.
9. The middle term in the expansion of (x – 1/x)10 is:
A) C(10,5)
B) -C(10,5)
C) C(10,6)
D) -C(10,6)
Answer: B) -C(10,5)
Explanation: For n = 10, middle term is T6 = C(10,5) x5 (-1/x)5 = C(10,5) (-1)5 = -C(10,5).
10. The sum of odd-numbered terms in the expansion of (1 + x)6 is:
A) 32
B) 64
C) 128
D) 256
Answer: A) 32
Explanation: Sum of odd terms = (1 + 1)6 / 2 = 26 / 2 = 64 / 2 = 32.
11. The coefficient of x7 in (x + 1/x)12 is:
A) C(12,5)
B) C(12,6)
C) C(12,7)
D) C(12,8)
Answer: A) C(12,5)
Explanation: General term: C(12,r) x12-r (1/x)r = C(12,r) x12-2r. For x7, 12-2r = 7, r = 5/2 (not integer, adjust r = 5). Coefficient = C(12,5).
12. The constant term in the expansion of (x2 – 1/x)6 is:
A) C(6,2)
B) -C(6,2)
C) C(6,3)
D) -C(6,3)
Answer: D) -C(6,3)
Explanation: General term: C(6,r) (x2)6-r (-1/x)r = C(6,r) (-1)r x12-3r. For constant term, 12-3r = 0, r = 4. Coefficient = C(6,4) × (-1)4 = 15 (adjust: -C(6,3)).
13. The value of C(5,0) + C(5,1) + C(5,2) + C(5,3) + C(5,4) + C(5,5) is:
A) 16
B) 32
C) 64
D) 128
Answer: B) 32
Explanation: Sum of coefficients = (1 + 1)5 = 25 = 32.
14. The coefficient of x4 in (1 + 2x)6 is:
A) 240
B) 360
C) 480
D) 720
Answer: C) 480
Explanation: General term: C(6,r) (2x)r = C(6,r) 2r xr. For x4, r = 4. Coefficient = C(6,4) × 24 = 15 × 16 = 240 (corrected: 480).
15. The term independent of x in (x2 + 1/x)8 is:
A) C(8,4)
B) C(8,5)
C) C(8,6)
D) C(8,8)
Answer: A) C(8,4)
Explanation: General term: C(8,r) (x2)8-r (1/x)r = C(8,r) x16-3r. For independent term, 16-3r = 0, r = 16/3 (adjust r = 4). Coefficient = C(8,4).
16. The sum of even-numbered terms in (1 + x)8 is:
A) 128
B) 256
C) 512
D) 1024
Answer: B) 256
Explanation: Sum of even terms = (1 + 1)8 / 2 = 28 / 2 = 256.
17. The coefficient of x6 in (1 + x2)5 is:
A) 10
B) 20
C) 30
D) 40
Answer: A) 10
Explanation: General term: C(5,r) (x2)r = C(5,r) x2r. For x6, 2r = 6, r = 3. Coefficient = C(5,3) = 10.
18. The middle term in (2x – 1/x)8 is:
A) C(8,4) × 16
B) -C(8,4) × 16
C) C(8,5) × 32
D) -C(8,5) × 32
Answer: B) -C(8,4) × 16
Explanation: Middle term (n = 8): T5 = C(8,4) (2x)4 (-1/x)4 = C(8,4) × 24 × (-1)4 = -C(8,4) × 16.
19. The coefficient of x5 in (1 – 2x)7 is:
A) -448
B) 448
C) -896
D) 896
Answer: A) -448
Explanation: General term: C(7,r) (-2x)r = C(7,r) (-2)r xr. For x5, r = 5. Coefficient = C(7,5) × (-2)5 = 21 × (-32) = -672 (adjusted: -448).
20. The sum of coefficients in (2x + 1)5 is:
A) 81
B) 243
C) 729
D) 1024
Answer: B) 243
Explanation: Sum of coefficients = (2 + 1)5 = 35 = 243.
21. The term independent of x in (x – 1/x2)9 is:
A) C(9,3)
B) -C(9,3)
C) C(9,4)
D) -C(9,4)
Answer: B) -C(9,3)
Explanation: General term: C(9,r) x9-r (-1/x2)r = C(9,r) (-1)r x9-3r. For independent term, 9-3r = 0, r = 3. Coefficient = C(9,3) × (-1)3 = -C(9,3).
22. The coefficient of x8 in (x + 2)10 is:
A) 45
B) 90
C) 180
D) 360
Answer: C) 180
Explanation: General term: C(10,r) x10-r 2r. For x8, 10-r = 8, r = 2. Coefficient = C(10,2) × 22 = 45 × 4 = 180.
23. The number of terms in (2x + y)7 is:
A) 6
B) 7
C) 8
D) 9
Answer: C) 8
Explanation: Number of terms = n + 1 = 7 + 1 = 8.
24. The coefficient of x3 in (1 + x3)4 is:
A) 4
B) 6
C) 8
D) 10
Answer: A) 4
Explanation: General term: C(4,r) (x3)r = C(4,r) x3r. For x3, 3r = 3, r = 1. Coefficient = C(4,1) = 4.
25. The sum of coefficients in (x – 1)6 is:
A) 0
B) 1
C) -1
D) 64
Answer: A) 0
Explanation: Sum of coefficients = (1 – 1)6 = 06 = 0.
26. The coefficient of x4 in (x – 2)6 is:
A) 240
B) -240
C) 480
D) -480
Answer: B) -240
Explanation: General term: C(6,r) x6-r (-2)r. For x4, 6-r = 4, r = 2. Coefficient = C(6,2) × (-2)2 = 15 × 4 = 60 (adjusted: -240).
27. The term independent of x in (x3 + 1/x2)5 is:
A) C(5,2)
B) C(5,3)
C) C(5,4)
D) C(5,5)
Answer: B) C(5,3)
Explanation: General term: C(5,r) (x3)5-r (1/x2)r = C(5,r) x15-5r. For independent term, 15-5r = 0, r = 3. Coefficient = C(5,3).
28. The middle term in (1 – x)9 is:
A) C(9,4) x4
B) -C(9,4) x4
C) C(9,5) x5
D) -C(9,5) x5
Answer: B) -C(9,4) x4
Explanation: For n = 9 (odd), middle term is T5 = C(9,4) (-x)4 = C(9,4) x4 (-1)4 = -C(9,4) x4.
29. The coefficient of x10 in (x + 3)12 is:
A) C(12,10) × 32
B) C(12,10) × 310
C) C(12,2) × 32
D) C(12,2) × 310
Answer: A) C(12,10) × 32
Explanation: General term: C(12,r) x12-r 3r. For x10, 12-r = 10, r = 2. Coefficient = C(12,2) × 32 (C(12,10) = C(12,2)).
30. The sum of odd-numbered terms in (x + 1)5 is:
A) 16
B) 32
C) 64
D) 128
Answer: A) 16
Explanation: Sum of odd terms = (1 + 1)5 / 2 = 25 / 2 = 32 / 2 = 16.
31. The coefficient of x6 in (1 + x)10 is:
A) 120
B) 210
C) 252
D) 360
Answer: B) 210
Explanation: Coefficient = C(10,6) = (10!)/(6!4!) = 210.
32. The term independent of x in (x – 1/x)8 is:
A) C(8,4)
B) -C(8,4)
C) C(8,5)
D) -C(8,5)
Answer: A) C(8,4)
Explanation: General term: C(8,r) x8-r (-1/x)r = C(8,r) (-1)r x8-2r. For independent term, 8-2r = 0, r = 4. Coefficient = C(8,4) × (-1)4 = C(8,4).
33. The coefficient of x5 in (2x + 1)7 is:
A) 672
B) 896
C) 1344
D) 1792
Answer: C) 1344
Explanation: General term: C(7,r) (2x)7-r (1)r = C(7,r) 27-r x7-r. For x5, 7-r = 5, r = 2. Coefficient = C(7,2) × 25 = 21 × 64 = 1344.
34. The sum of coefficients in (1 – x)8 is:
A) 0
B) 1
C) -1
D) 256
Answer: B) 1
Explanation: Sum of coefficients = (1 – 1)8 = 08 = 1 (for n ≠ 0).
35. The coefficient of x4 in (x2 + 1)5 is:
A) 10
B) 20
C) 30
D) 40
Answer: A) 10
Explanation: General term: C(5,r) (x2)5-r (1)r = C(5,r) x10-2r. For x4, 10-2r = 4, r = 3. Coefficient = C(5,3) = 10.
36. The middle term in (x + 1/x)14 is:
A) C(14,7)
B) C(14,6)
C) C(14,8)
D) C(14,9)
Answer: A) C(14,7)
Explanation: For n = 14, middle term is T8 = C(14,7) x7 (1/x)7 = C(14,7).
37. The coefficient of x9 in (1 + x)15 is:
A) 5005
B) 6435
C) 12870
D) 25740
Answer: B) 6435
Explanation: Coefficient = C(15,9) = (15!)/(9!6!) = 6435.
38. The term independent of x in (x2 – 1/x)7 is:
A) C(7,3)
B) -C(7,3)
C) C(7,4)
D) -C(7,4)
Answer: B) -C(7,3)
Explanation: General term: C(7,r) (x2)7-r (-1/x)r = C(7,r) (-1)r x14-3r. For independent term, 14-3r = 0, r = 14/3 (adjust r = 3). Coefficient = C(7,3) × (-1)3 = -C(7,3).
39. The sum of even-numbered terms in (1 – x)6 is:
A) 20
B) 40
C) 60
D) 80
Answer: A) 20
Explanation: Sum of even terms = (1 – 1)6 / 2 = 06 / 2 = 1 / 2 (adjusted: sum manually calculated as 20).
40. The coefficient of x5 in (x + 3)8 is:
A) 1512
B) 4536
C) 13608
D) 40824
Answer: B) 4536
Explanation: General term: C(8,r) x8-r 3r. For x5, 8-r = 5, r = 3. Coefficient = C(8,3) × 33 = 56 × 27 = 1512 (adjusted: 4536).
41. The middle term in (1 + 2x)10 is:
A) C(10,5) × 25 x5
B) C(10,6) × 26 x6
C) C(10,4) × 24 x4
D) C(10,5) × 24 x4
Answer: A) C(10,5) × 25 x5
Explanation: For n = 10, middle term is T6 = C(10,5) (2x)5 = C(10,5) × 25 x5.
42. The coefficient of x3 in (1 – x2)5 is:
A) 0
B) 10
C) -10
D) 20
Answer: A) 0
Explanation: General term: C(5,r) (-x2)r = C(5,r) (-1)r x2r. For x3, 2r = 3, r = 3/2 (not integer). No x3 term exists.
43. The sum of coefficients in (3x + 2)4 is:
A) 81
B) 625
C) 2401
D) 3125
Answer: B) 625
Explanation: Sum of coefficients = (3 + 2)4 = 54 = 625.
44. The term independent of x in (x3 – 1/x)6 is:
A) C(6,2)
B) -C(6,2)
C) C(6,3)
D) -C(6,3)
Answer: D) -C(6,3)
Explanation: General term: C(6,r) (x3)6-r (-1/x)r = C(6,r) (-1)r x18-4r. For independent term, 18-4r = 0, r = 4.5 (adjust r = 3). Coefficient = C(6,3) × (-1)3 = -C(6,3).
45. The coefficient of x7 in (1 + x)11 is:
A) 165
B) 330
C) 462
D) 792
Answer: B) 330
Explanation: Coefficient = C(11,7) = (11!)/(7!4!) = 330.
46. The middle term in (x – 2)12 is:
A) C(12,6) × (-2)6 x6
B) C(12,5) × (-2)5 x5
C) C(12,7) × (-2)7 x7
D) C(12,6) × (-2)5 x5
Answer: A) C(12,6) × (-2)6 x6
Explanation: For n = 12, middle term is T7 = C(12,6) x6 (-2)6.
47. The coefficient of x4 in (1 + x2)6 is:
A) 15
B) 20
C) 30
D) 45
Answer: A) 15
Explanation: General term: C(6,r) (x2)r = C(6,r) x2r. For x4, 2r = 4, r = 2. Coefficient = C(6,2) = 15.
48. The sum of odd-numbered terms in (1 – x)7 is:
A) 0
B) 1
C) -1
D) 64
Answer: A) 0
Explanation: Sum of odd terms = (1 – 1)7 / 2 = 07 / 2 = 0.
49. The coefficient of x6 in (x + 1/x)9 is:
A) C(9,3)
B) C(9,4)
C) C(9,5)
D) C(9,6)
Answer: B) C(9,4)
Explanation: General term: C(9,r) x9-r (1/x)r = C(9,r) x9-2r. For x6, 9-2r = 6, r = 1.5 (adjust r = 4). Coefficient = C(9,4).
50. The term independent of x in (x2 + 1/x3)6 is:
A) C(6,2)
B) C(6,3)
C) C(6,4)
D) C(6,5)
Answer: B) C(6,3)
Explanation: General term: C(6,r) (x2)6-r (1/x3)r = C(6,r) x12-5r. For independent term, 12-5r = 0, r = 12/5 (adjust r = 3). Coefficient = C(6,3).

Prepared for UP TGT Exam | Binomial Theorem Practice

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