JEE (Main) Mathematics - Binomial Theorem and Its Simple Applications - MCQs

61. In the expansion of \( (x + y)^7 \), the general term is given by:
    • a) \( \binom{7}{r} x^{7-r} y^r \)
    • b) \( \binom{7}{r} x^r y^{7-r} \)
    • c) \( \binom{7}{r} x^{r} y^{7-r} \)
    • d) \( \binom{7}{r} x^{7-r} y^{r} \)
    Answer: a) \( \binom{7}{r} x^{7-r} y^r \)
62. 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) \( \binom{8}{4} x^3 y^5 \)
    Answer: a) \( \binom{8}{4} x^4 y^4 \)
63. The coefficient of \( x^2 \) in the expansion of \( (2x + 3)^4 \) is:
    • a) \( \binom{4}{2} 2^2 3^2 \)
    • b) \( \binom{4}{2} 2^4 3^2 \)
    • c) \( \binom{4}{2} 2^2 3^4 \)
    • d) \( \binom{4}{3} 2^2 3^2 \)
    Answer: a) \( \binom{4}{2} 2^2 3^2 \)
64. The general term in the expansion of \( (x + 2)^5 \) is:
    • a) \( \binom{5}{r} x^{5-r} 2^r \)
    • b) \( \binom{5}{r} x^r 2^{5-r} \)
    • c) \( \binom{5}{r} x^{r} 2^{5-r} \)
    • d) \( \binom{5}{r} x^{5-r} 2^{r} \)
    Answer: d) \( \binom{5}{r} x^{5-r} 2^r \)
65. The term containing \( x^3 \) in the expansion of \( (x + 4)^6 \) is:
    • a) \( \binom{6}{3} x^3 4^3 \)
    • b) \( \binom{6}{2} x^5 4^3 \)
    • c) \( \binom{6}{3} x^3 4^2 \)
    • d) \( \binom{6}{2} x^2 4^4 \)
    Answer: a) \( \binom{6}{3} x^3 4^3 \)
66. In the expansion of \( (x + 3)^6 \), the coefficient of \( x^2 \) is:
    • a) \( \binom{6}{4} 3^2 \)
    • b) \( \binom{6}{2} 3^2 \)
    • c) \( \binom{6}{4} 3^4 \)
    • d) \( \binom{6}{2} 3^4 \)
    Answer: b) \( \binom{6}{2} 3^2 \)
67. The general term in the expansion of \( (a + b)^n \) is:
    • a) \( T_r = \binom{n}{r} a^{n-r} b^r \)
    • b) \( T_r = \binom{n}{r} a^r b^{n-r} \)
    • c) \( T_r = \binom{n}{r} a^r b^r \)
    • d) \( T_r = \binom{n}{r} a^{n-r} b^{n-r} \)
    Answer: a) \( T_r = \binom{n}{r} a^{n-r} b^r \)
68. The middle term in the expansion of \( (x + y)^9 \) is:
    • a) \( \binom{9}{4} x^5 y^4 \)
    • b) \( \binom{9}{5} x^4 y^5 \)
    • c) \( \binom{9}{4} x^4 y^5 \)
    • d) \( \binom{9}{5} x^5 y^4 \)
    Answer: a) \( \binom{9}{4} x^5 y^4 \)
69. The number of terms in the expansion of \( (x + y)^6 \) is:
    • a) 6
    • b) 7
    • c) 8
    • d) 9
    Answer: b) 7
70. The coefficient of \( x^3 \) in the expansion of \( (x + 2)^7 \) is:
    • a) \( \binom{7}{3} 2^4 \)
    • b) \( \binom{7}{3} 2^3 \)
    • c) \( \binom{7}{4} 2^3 \)
    • d) \( \binom{7}{4} 2^4 \)
    Answer: a) \( \binom{7}{3} 2^4 \)
71. The general term in the expansion of \( (a + b)^6 \) is:
    • a) \( \binom{6}{r} a^{6-r} b^r \)
    • b) \( \binom{6}{r} a^{r} b^{6-r} \)
    • c) \( \binom{6}{r} a^{r} b^r \)
    • d) \( \binom{6}{r} a^{6-r} b^{6-r} \)
    Answer: a) \( \binom{6}{r} a^{6-r} b^r \)
72. In the expansion of \( (x + y)^7 \), the number of terms is:
    • a) 7
    • b) 8
    • c) 9
    • d) 10
    Answer: b) 8
73. The coefficient of \( x^4 \) in the expansion of \( (3x - 2)^5 \) is:
    • a) \( \binom{5}{4} 3^4 (-2)^1 \)
    • b) \( \binom{5}{4} 3^1 (-2)^4 \)
    • c) \( \binom{5}{4} 3^4 (-2)^4 \)
    • d) \( \binom{5}{1} 3^4 (-2)^1 \)
    Answer: a) \( \binom{5}{4} 3^4 (-2)^1 \)
74. In the expansion of \( (x + y)^5 \), the coefficient of \( x^2 y^3 \) is:
    • a) \( \binom{5}{2} \)
    • b) \( \binom{5}{3} \)
    • c) \( \binom{5}{2} 1^3 \)
    • d) \( \binom{5}{3} 1^2 \)
    Answer: a) \( \binom{5}{2} \)

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