Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 7

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Coefficient of $x^{11}$ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$ is

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Important Questions on Binomial Theorem

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 8

The coefficient of

x^{9}

in the expansion of

(1 + x) (1 + x^{2}) (1 + x^{3}) \dots \dots . (1 + x^{100})

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 9

Let m be the smallest positive integer such that the coefficient of

x^{2}

in the expansion of

{(1 + x)}^{2} + {(1 + x)}^{3} + \dots \dots \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}

is

{(3 n + 1)}^{51} C_{3}

for some positive integer

n

. Then the value of

n

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 10

Let

X = {({}^{10}C_{1})}^{2} + 2 {({}^{10}C_{2})}^{2} + 3 {({}^{10}C_{3})}^{2} + \dots + 10 {({}^{10}C_{10})}^{2}

where

{}^{10}C_{r}, r \in {1, 2, \dots, 10}

denote binomial coefficients. Then, the value of

\frac{1}{1430} X

is ..... .

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 11

Let $S_{1} = \sum_{j = 1}^{10} j {(j - 1)}^{10} C_{1}, S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_{3} = \sum_{j = 1}^{10} j^{2} 10 C_{i}$

Statement- $1$ : $S_{3} = 55 \times 2^{9}$ .

Statement- $2$ : $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 12

The coefficient of

x^{7}

in the expansion of

{(1 - x - x^{2} + x^{3})}^{6}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 13

If

n

is

a

positive integer, then

(\sqrt{3} + 1)^{2 n} - (\sqrt{3} - 1)^{2 n}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 14

The term independent of

x

in expansion of

{(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-3>Q 15

If the coefficients of

x^{3}

and

x^{4}

in the expansion of

(1 + a x + b x^{2}) {(1 - 2 x)}^{18}

in powers of

x

are both zero, then

(a, b)

is equal to