Friday, November 22, 2024

MCQ Mathematics Chapter 7: BINOMIALTHEOREM, HS 1st year

 


1. Which of the following is the general expression for the binomial theorem for a positive integer nn?

A. (a+b)n=an+bn(a + b)^n = a^n + b^n
B. (a+b)n=r=0n(nr)anrbr(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r}b^r
C. (a+b)n=n!anbn(a + b)^n = n! \cdot a^n b^n
D. (a+b)n=(nC0)(an)+(nC1)(bn)(a + b)^n = (nC_0)(a^n) + (nC_1)(b^n)

Answer: B


2. What does Pascal's Triangle represent in relation to the binomial theorem?

A. The sum of the coefficients of terms.
B. The powers of terms in a binomial expansion.
C. The coefficients of the terms in a binomial expansion.
D. The number of terms in a binomial expansion.

Answer: C


3. In the expansion of (a+b)n(a + b)^n, how many terms will there be?

A. nn
B. n+1n + 1
C. 2n2n
D. 2n2^n

Answer: B


4. In Pascal’s Triangle, what is the value of the coefficient for the term corresponding to rr in the expansion of (a+b)n(a + b)^n?

A. nrn^r
B. (nr)=n!r!(nr)!\binom{n}{r} = \frac{n!}{r!(n-r)!}
C. nC0+nCrnC_0 + nC_r
D. nr/r!n^r / r!

Answer: B


5. Which mathematician is most closely associated with the modern form of the binomial theorem for integral nn?

A. Blaise Pascal
B. Michael Stifel
C. Pingla
D. Chu-shi-kie

Answer: A


6. Using the binomial theorem, what is the expansion of (x2y)3(x - 2y)^3?

A. x36x2y+12xy28y3x^3 - 6x^2y + 12xy^2 - 8y^3
B. x3+6x2y+12xy2+8y3x^3 + 6x^2y + 12xy^2 + 8y^3
C. x36x2y+3xy22y3x^3 - 6x^2y + 3xy^2 - 2y^3
D. x3+6x2y3xy22y3x^3 + 6x^2y - 3xy^2 - 2y^3

Answer: A


7. What is the value of (1+x)5(1 + x)^5 when x=1x = 1?

A. 1616
B. 2525
C. 3232
D. 6464

Answer: C


8. The sum of coefficients in the expansion of (a+b)n(a + b)^n is given by:

A. n2n^2
B. 2n2^n
C. (a+b)n(a + b)^n
D. n!n!

Answer: B


9. The expansion of (x+1)4(x + 1)^4 contains which of the following terms?

A. x4+4x3+6x2+4x+1x^4 + 4x^3 + 6x^2 + 4x + 1
B. x4+3x3+3x2+x+1x^4 + 3x^3 + 3x^2 + x + 1
C. x4+4x3+6x2+1x^4 + 4x^3 + 6x^2 + 1
D. x4+6x3+4x2+4x+1x^4 + 6x^3 + 4x^2 + 4x + 1

Answer: A


10. In the binomial expansion of (a+b)n(a + b)^n, what is the sum of the indices of aa and bb in any term?

A. 1
B. nn
C. n+1n + 1
D. 2

Answer: B


11. In the expansion (xy)n(x - y)^n, what is the sign of the third term?

A. Always positive
B. Always negative
C. Positive if nn is odd, negative if nn is even
D. Negative if nn is odd, positive if nn is even

Answer: D


12. The coefficient of x3y2x^3y^2 in the expansion of (x+y)5(x + y)^5 is:

A. 5
B. 10
C. 15
D. 20

Answer: C


13. In (a+b)n(a + b)^n, the middle term when nn is odd is:

A. (nn2)an/2bn/2\binom{n}{\frac{n}{2}} a^{n/2} b^{n/2}
B. (n(n+1)/2)a(n+1)/2b(n1)/2\binom{n}{(n+1)/2} a^{(n+1)/2} b^{(n-1)/2}
C. (n(n1)/2)a(n1)/2b(n+1)/2\binom{n}{(n-1)/2} a^{(n-1)/2} b^{(n+1)/2}
D. None of the above

Answer: D


14. If (1+x)n=r=0n(nr)xr(1 + x)^n = \sum_{r=0}^n \binom{n}{r} x^r, then (nr)\binom{n}{r} represents:

A. The number of terms in the expansion
B. The coefficient of the rthr^{th} term in the expansion
C. The product of nn and rr
D. The constant term

Answer: B


15. The constant term in the expansion of (x+1/x)6(x + 1/x)^6 is:

A. 5
B. 10
C. 20
D. 60

Answer: C


16. Which of the following represents the general term in (a+b)n(a + b)^n?

A. Tr=(nr)arbnrT_r = \binom{n}{r} a^r b^{n-r}
B. Tr=(nr1)ar1bnrT_r = \binom{n}{r-1} a^{r-1} b^{n-r}
C. Tr=(nr)anrbrT_r = \binom{n}{r} a^{n-r} b^r
D. Tr=(nr1)arbnrT_r = \binom{n}{r-1} a^r b^{n-r}

Answer: C


17. Using the binomial theorem, the value of (98)3(98)^3 can be simplified as:

A. (1002)3(100 - 2)^3
B. (100+2)3(100 + 2)^3
C. (502)3(50 - 2)^3
D. (1005)3(100 - 5)^3

Answer: A


18. How many terms are in the expansion of (x+y+z)5(x + y + z)^5?

A. 6
B. 21
C. 56
D. 120

Answer: B


19. In the binomial expansion, which term is the largest when x=1x = 1?

A. The first term
B. The last term
C. The middle term
D. Depends on the value of nn

Answer: C


20. For (x1)n(x - 1)^n, which term has a positive coefficient?

A. All terms when nn is even
B. All terms when nn is odd
C. Alternate terms when nn is odd
D. Alternate terms when nn is even

Answer: D


21. The sum of the coefficients in the expansion of (x+y)n(x + y)^n is given by:

A. n!n!
B. 2n2^n
C. n2n^2
D. None of the above

Answer: B


22. What is the expansion of (1+x)4(1 + x)^4 when x=1x = 1?

A. 8
B. 16
C. 32
D. 64

Answer: B


23. The term (53)x3y2\binom{5}{3} x^3 y^2 in (x+y)5(x + y)^5 has a coefficient of:

A. 5
B. 10
C. 15
D. 20

Answer: C


24. Which of these is NOT true for Pascal’s Triangle?

A. It contains the coefficients of (a+b)n(a + b)^n.
B. It can be generated using the formula (nr)\binom{n}{r}.
C. It has n+1n + 1 entries in row nn.
D. It is unrelated to the Binomial Theorem.

Answer: D


25. The coefficient of x4x^4 in (1x)6(1 - x)^6 is:

A. 15
B. -15
C. 20
D. -20

Answer: B


26. What is the binomial coefficient for the 5th term in (a+b)8(a + b)^8?

A. (83)\binom{8}{3}
B. (84)\binom{8}{4}
C. (85)\binom{8}{5}
D. (86)\binom{8}{6}

Answer: B


27. The expansion of (12x)4(1 - 2x)^4 will have how many terms?

A. 3
B. 4
C. 5
D. 6

Answer: C


28. In the binomial expansion of (x+y)n(x + y)^n, the middle term exists when:

A. nn is odd
B. nn is even
C. nn is greater than 10
D. n+1n + 1 is odd

Answer: B


29. Using the binomial theorem, (x+1)6(x + 1)^6 expanded gives the coefficient of x3x^3 as:

A. 10
B. 15
C. 20
D. 35

Answer: C


30. What is the expansion of (ab)2(a - b)^2?

A. a22ab+b2a^2 - 2ab + b^2
B. a2+2ab+b2a^2 + 2ab + b^2
C. a2b2a^2 - b^2
D. a2+b2a^2 + b^2

Answer: A


31. The coefficient of the last term in any binomial expansion is always:

A. 11
B. nCn1nC_{n-1}
C. bnb^n
D. n!n!

Answer: A


32. If (1+x)n=r=0nTr(1 + x)^n = \sum_{r=0}^n T_r, the sum of the terms T0,T2,T4,T_0, T_2, T_4, \dots is:

A. 2n12^{n-1}
B. 2n2^n
C. nn
D. n+1n+1

Answer: A


33. The value of (83)+(84)\binom{8}{3} + \binom{8}{4} equals:

A. (94)\binom{9}{4}
B. (85)\binom{8}{5}
C. (93)\binom{9}{3}
D. (86)\binom{8}{6}

Answer: A


34. Which of the following expansions is correct?

A. (x2)3=x3+6x212x+8(x - 2)^3 = x^3 + 6x^2 - 12x + 8
B. (x2)3=x36x2+12x8(x - 2)^3 = x^3 - 6x^2 + 12x - 8
C. (x2)3=x36x212x8(x - 2)^3 = x^3 - 6x^2 - 12x - 8
D. (x2)3=x3+6x212x8(x - 2)^3 = x^3 + 6x^2 - 12x - 8

Answer: B


35. Which is a correct property of binomial coefficients?

A. (nr)=(nr1)\binom{n}{r} = \binom{n}{r-1}
B. (nr)+(nr1)=(n+1r)\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}
C. (nr)(nr1)=(n1r)\binom{n}{r} - \binom{n}{r-1} = \binom{n-1}{r}
D. (nr)=r!\binom{n}{r} = r!

Answer: B


36. The expansion of (x+y)4(x + y)^4 has which of the following coefficients for x2y2x^2y^2?

A. 4
B. 5
C. 6
D. 8

Answer: C


37. The binomial coefficient (n0)\binom{n}{0} is always equal to:

A. nn
B. 00
C. 11
D. n!n!

Answer: C


38. If (ab)n(a - b)^n is expanded, the coefficients of all terms:

A. Are always positive
B. Alternate between positive and negative
C. Depend on nn being odd or even
D. Are always negative

Answer: B


39. In the expansion of (x+y)n(x + y)^n, the ratio of the coefficients of the terms TrT_r and Tr+1T_{r+1} is:

A. nrr+1\frac{n-r}{r+1}
B. nr\frac{n}{r}
C. rnr+1\frac{r}{n-r+1}
D. r+1nr\frac{r+1}{n-r}

Answer: A


40. The expansion of (x2+y)3(x^2 + y)^3 has how many terms?

A. 3
B. 4
C. 5
D. 6

Answer: B


41. In (a+b)n(a + b)^n, the term independent of xx in the expansion of (x+1/x)5(x + 1/x)^5 is:

A. 5C25C_2
B. 5C35C_3
C. 5C45C_4
D. 5C55C_5

Answer: A


42. The coefficient of x12x^{12} in the expansion of (x2+1)6(x^2 + 1)^6 is:

A. 6
B. 10
C. 15
D. 20

Answer: C


43. In the binomial expansion (x+y)n(x + y)^n, what happens to the coefficients as rr approaches n/2n/2?

A. They increase
B. They decrease
C. They first increase and then decrease
D. They remain constant

Answer: C


44. The expansion of (1x)6(1 - x)^6 gives the 3rd term as:

A. 15x215x^2
B. 15x2-15x^2
C. 20x220x^2
D. 20x2-20x^2

Answer: B


45. In (a+b)n(a + b)^n, the total number of positive terms is:

A. n+1n+1
B. n1n-1
C. nn
D. n2n^2

Answer: A


46. The value of (73)\binom{7}{3} is:

A. 35
B. 28
C. 21
D. 15

Answer: A


47. Which term of (1+x)5(1 + x)^5 is 10x310x^3?

A. 2nd term
B. 3rd term
C. 4th term
D. 5th term

Answer: C


48. If (1+x)n=r=0nTr(1 + x)^n = \sum_{r=0}^n T_r, the sum of the odd coefficients is:

A. 2n12^{n-1}
B. 2n2^n
C. n2n^2
D. None of the above

Answer: A


49. Which of the following is the third term in the expansion of (x+2)4(x + 2)^4?

A. 6x26x^2
B. 24x224x^2
C. 12x212x^2
D. 48x248x^2

Answer: D


50. In Pascal's Triangle, the sum of the elements in the 6th row is:

A. 16
B. 32
C. 64
D. 128

Answer: C


51. The coefficient of a3b3a^3b^3 in (a+b)6(a + b)^6 is:

A. 10
B. 15
C. 20
D. 35

Answer: D


52. If (x+y)n(x + y)^n is expanded, the coefficients of the rr-th and (nr)(n-r)-th terms are:

A. Equal
B. Opposite
C. Add up to nn
D. Multiply to n!n!

Answer: A


53. The coefficient of x3x^3 in the expansion of (2x+1)5(2x + 1)^5 is:

A. 80
B. 120
C. 160
D. 200

Answer: C


54. The binomial theorem for (1+x)n(1 + x)^n holds true for:

A. All values of xx
B. Only for positive integers nn
C. Only for x>0x > 0
D. Only for n>0n > 0

Answer: B


55. The number of terms in the expansion of (a+b+c)4(a + b + c)^4 is:

A. 12
B. 15
C. 20
D. 25

Answer: C


56. The coefficient of x4y2x^4y^2 in (x+y)6(x + y)^6 is:

A. 10
B. 15
C. 20
D. 30

Answer: C


57. The coefficient of the term independent of xx in (x+1x)6\left(x + \frac{1}{x}\right)^6 is:

A. 15
B. 20
C. 25
D. 30

Answer: A


58. Which of the following is the sum of coefficients in the expansion of (x+1)8(x + 1)^8?

A. 8
B. 16
C. 64
D. 256

Answer: D


59. The expansion (2x3y)4(2x - 3y)^4 contains how many terms?

A. 5
B. 6
C. 7
D. 8

Answer: B


60. If the coefficient of x2x^2 in (1+ax)3(1 + ax)^3 is 12, then aa is:

A. 2
B. 3
C. 4
D. 6

Answer: C


61. The binomial coefficient (nn1)\binom{n}{n-1} equals:

A. n1n-1
B. nn
C. n2n^2
D. nC0nC_0

Answer: B


62. In (a+b)n(a + b)^n, the total number of coefficients is:

A. n+1n+1
B. n1n-1
C. n2n^2
D. n!n!

Answer: A


63. Which term of (x+y)7(x + y)^7 is 35x4y335x^4y^3?

A. 4th term
B. 5th term
C. 6th term
D. 7th term

Answer: B


64. The coefficient of x3x^3 in (x+2)5(x + 2)^5 is:

A. 60
B. 80
C. 100
D. 120

Answer: D


65. In (x+y)n(x + y)^n, if x=y=1x = y = 1, the sum of all terms is:

A. n2n^2
B. 2n2^n
C. n+1n+1
D. 2n12^{n-1}

Answer: B


66. The middle term in (x+y)6(x + y)^6 is:

A. 20x3y320x^3y^3
B. 15x3y315x^3y^3
C. 40x3y340x^3y^3
D. 30x3y330x^3y^3

Answer: A


67. If (x+1)n=1+nx+n(n1)2x2+(x + 1)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \dots, the coefficient of x2x^2 is:

A. n(n1)2\frac{n(n-1)}{2}
B. n2n^2
C. n1n-1
D. n+12\frac{n+1}{2}

Answer: A


68. The coefficient of xnx^n in (1x)n(1 - x)^n is:

A. (nn)\binom{n}{n}
B. (1)n(nn)(-1)^n \binom{n}{n}
C. (1)n(-1)^n
D. (1)nn!(-1)^n n!

Answer: B


69. In (ab)6(a - b)^6, the coefficient of a4b2a^4b^2 is:

A. -15
B. 15
C. -20
D. 20

Answer: D


70. The number of terms in the expansion of (x+y+z)5(x + y + z)^5 is:

A. 15
B. 21
C. 25
D. 35

Answer: B


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