JEE MATH APEX

Binomial Theorem-JEE-Main-PYQ’s

Mathematics Questions

Given below are two statements:

Statement I: 25^{13} + 20^{13} + 8^{13} + 3^{13} is divisible by 7.

Statement II: The integral part of (7 + 4\sqrt{3})^{25} is an odd number.

In the light of the above statements, choose the correct answer from the options given below:
(A) Statement I is false but Statement II is true
(B) Both Statement I and Statement II are false
(C) Both Statement I and Statement II are true
(D) Statement I is true but Statement II is false
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Correct Answer: [ ]
The sum of the coefficients of x^{499} and x^{500} in (1 + x)^{1000} + x(1 + x)^{999} + x^{2}(1 + x)^{998} + \dots + x^{1000} is:
(A) {}^{1002}C_{501}
(B) {}^{1001}C_{501}
(C) {}^{1000}C_{501}
(D) {}^{1002}C_{500}
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Let S = \frac{1}{25!} + \frac{1}{3!23!} + \frac{1}{5!21!} + \dots up to 13 terms. If 13 S = \frac{2^k}{n!}, k \in \mathbb{N}, then n + k is equal to:
(A) 50
(B) 52
(C) 49
(D) 51
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The sum of all possible values of n \in \mathbb{N}, so that the coefficients of x, x^2 and x^3 in the expansion of (1 + x^2)^2(1 + x)^n, are in arithmetic progression is:
(A) 12
(B) 9
(C) 3
(D) 7
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The value of \frac{{}^{100}C_{50}}{51} + \frac{{}^{100}C_{51}}{52} + \dots + \frac{{}^{100}C_{100}}{101} is:
(A) \frac{2^{101}}{101}
(B) \frac{2^{100}}{101}
(C) \frac{2^{100}}{100}
(D) \frac{2^{101}}{100}
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Let C_r denote the coefficient of x^r in the binomial expansion of (1 + x)^n, n \in \mathbb{N}, 0 \leq r \leq n. If P_n = C_0 - C_1 + \frac{2^2}{3}C_2 - \frac{2^3}{4}C_3 + \dots + \frac{(-2)^n}{n+1}C_n, then the value of \sum_{n=1}^{25} \frac{1}{P_{2n}} equals:
(A) 675
(B) 580
(C) 525
(D) 650
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The coefficient of x^{48} in (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \dots + 100(1 + x)^{100} is equal to:
(A) 100 \cdot {}^{100}C_{49} - {}^{100}C_{48}
(B) 100 \cdot {}^{101}C_{49} - {}^{101}C_{50}
(C) {}^{100}C_{50} + {}^{101}C_{49}
(D) 100 \cdot {}^{100}C_{49} - {}^{100}C_{50}
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If the coefficient of x in the expansion of (ax^2 + bx + c)(1 - 2x)^{26} is -56 and the coefficients of x^2 and x^3 are both zero, then a + b + c is equal to:
(A) 1483
(B) 1300
(C) 1500
(D) 1403
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The sum of the coefficient of x^{2 / 3} and x^{-2 / 5} in the binomial expansion of \left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^{9} is
(A) 19 / 4
(B) 69 / 16
(C) 63 / 16
(D) 21 / 4
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The coefficient of x^{70} in x^{2}(1+x)^{98}+x^{3}(1+x)^{97}+x^{4}(1+x)^{96}+\ldots+x^{54}(1+x)^{46} is { }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}. Then a possible value of \mathrm{p}+\mathrm{q} is :
(A) 61
(B) 83
(C) 55
(D) 68
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Correct Answer: [ ]

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