JEE (Main) Mathematics - Binomial Theorem and Simple Applications

Multiple Choice Questions (MCQs)

46. In the expansion of \( (x + 3)^5 \), the general term is:

    • a) \( \binom{5}{r} x^{5-r} 3^r \)
    • b) \( \binom{5}{r} x^r 3^{5-r} \)
    • c) \( \binom{5}{r} x^{5-r} 3^{5-r} \)
    • d) \( \binom{5}{r} x^r 3^r \)

    Answer: a) \( \binom{5}{r} x^{5-r} 3^r \)

47. The middle term in the expansion of \( (x + y)^6 \) is:

    • a) \( \binom{6}{3} x^3 y^3 \)
    • b) \( \binom{6}{2} x^4 y^2 \)
    • c) \( \binom{6}{3} x^2 y^4 \)
    • d) \( \binom{6}{4} x^2 y^4 \)

    Answer: a) \( \binom{6}{3} x^3 y^3 \)

48. The general term in the expansion of \( (a + b)^7 \) is:

    • a) \( T_r = \binom{7}{r} a^{7-r} b^r \)
    • b) \( T_r = \binom{7}{r} a^r b^{7-r} \)
    • c) \( T_r = \binom{7}{r} a^{r} b^{7-r} \)
    • d) \( T_r = \binom{7}{r} a^{7-r} b^{7-r} \)

    Answer: a) \( T_r = \binom{7}{r} a^{7-r} b^r \)

49. In the expansion of \( (x + 2)^6 \), the term containing \( x^3 \) is:

    • a) \( \binom{6}{3} x^3 2^3 \)
    • b) \( \binom{6}{3} x^2 2^4 \)
    • c) \( \binom{6}{3} x^4 2^2 \)
    • d) \( \binom{6}{4} x^3 2^3 \)

    Answer: a) \( \binom{6}{3} x^3 2^3 \)

50. The coefficient of \( x^4 \) in the expansion of \( (2x + 3)^6 \) is:

    • a) \( \binom{6}{2} 2^4 3^2 \)
    • b) \( \binom{6}{4} 2^2 3^4 \)
    • c) \( \binom{6}{4} 2^4 3^2 \)
    • d) \( \binom{6}{2} 2^3 3^3 \)

    Answer: a) \( \binom{6}{2} 2^4 3^2 \)

51. The middle term in the expansion of \( (x + y)^8 \) is:

    • a) \( \binom{8}{4} x^4 y^4 \)
    • b) \( \binom{8}{3} x^5 y^3 \)
    • c) \( \binom{8}{5} x^3 y^5 \)
    • d) \( \binom{8}{4} x^5 y^3 \)

    Answer: a) \( \binom{8}{4} x^4 y^4 \)

52. The term containing \( x^2 \) in the expansion of \( (x + 3)^5 \) is:

    • a) \( \binom{5}{2} x^2 3^3 \)
    • b) \( \binom{5}{2} x^3 3^2 \)
    • c) \( \binom{5}{3} x^2 3^3 \)
    • d) \( \binom{5}{2} x^4 3^2 \)

    Answer: a) \( \binom{5}{2} x^2 3^3 \)

53. In the expansion of \( (x - 4)^6 \), the coefficient of \( x^2 \) is:

    • a) \( \binom{6}{4} (-4)^2 \)
    • b) \( \binom{6}{2} (-4)^4 \)
    • c) \( \binom{6}{4} (-4)^3 \)
    • d) \( \binom{6}{2} (-4)^2 \)

    Answer: a) \( \binom{6}{4} (-4)^2 \)

54. In the expansion of \( (a + b)^9 \), the term containing \( a^5 b^4 \) is:

    • a) \( \binom{9}{4} a^5 b^4 \)
    • b) \( \binom{9}{5} a^5 b^4 \)
    • c) \( \binom{9}{4} a^4 b^5 \)
    • d) \( \binom{9}{5} a^4 b^5 \)

    Answer: b) \( \binom{9}{5} a^5 b^4 \)

55. The number of terms in the expansion of \( (x + y)^7 \) is:

    • a) 8
    • b) 9
    • c) 7
    • d) 6

    Answer: b) 8

56. The coefficient of \( x^4 \) in the expansion of \( (2x + 1)^6 \) is:

    • a) \( \binom{6}{4} 2^4 \)
    • b) \( \binom{6}{2} 2^2 \)
    • c) \( \binom{6}{4} 2^2 \)
    • d) \( \binom{6}{2} 2^4 \)

    Answer: a) \( \binom{6}{4} 2^4 \)

57. The general term in the expansion of \( (x + 2)^6 \) is:

    • a) \( \binom{6}{r} x^{6-r} 2^r \)
    • b) \( \binom{6}{r} x^r 2^{6-r} \)
    • c) \( \binom{6}{r} x^{6-r} 2^{6-r} \)
    • d) \( \binom{6}{r} x^r 2^r \)

    Answer: a) \( \binom{6}{r} x^{6-r} 2^r \)

58. The number of terms in the expansion of \( (x + y + z)^3 \) is:

    • a) 10
    • b) 6
    • c) 8
    • d) 7

    Answer: b) 6

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