EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 25 June 2022 Shift 1>Q 89

HARD

JEE Main

IMPORTANT

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Let $C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of ${(1 + x)}^{10}$ . If for $α, β \in R$ , $C_{1} + 3 \cdot 2 C_{2} + 5 \cdot 3 C_{3} + \dots . . 19 \cdot 10 \cdot C_{10}$ $= \frac{α \times 2^{11}}{2^{β} - 1}$ $(C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \dots . . \frac{C_{10}}{11})$ then the value of $α + β$ is equal to _____.

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Important Questions on Binomial Theorem

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 25 June 2022 Shift 2>Q 90

The coefficient of

x^{101}

in the expression

{(5 + x)}^{500} + x {(5 + x)}^{499} + x^{2} {(5 + x)}^{498} + \dots \dots + x^{500}, x > 0

is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 25 June 2022 Shift 2>Q 91

If the sum of the co-efficients of all the positive even powers of

x

in the binomial expansion of

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

is

5^{10} - β \cdot 3^{9}

, then

β

is equal to _____.

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 26 June 2022 Shift 1>Q 92

The remainder when

{(2021)}^{2023}

is divided by

7

is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 27 June 2022 Shift 1>Q 93

If the coefficient of

x^{10}

in the binomial expansion of

{(\frac{\sqrt{x}}{5^{\frac{1}{4}}} + \frac{\sqrt{5}}{x^{\frac{1}{3}}})}^{60}

is

5^{k} l

, where

l, k \in N

and

l

is coprime to

5

, then

k

is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 27 June 2022 Shift 2>Q 94

If the sum of the coefficients of all the positive powers of

x

, in the binomial expansion of

{(x^{n} + \frac{2}{x^{5}})}^{7}

is

939

, then the sum of all the possible integral values of

n

is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 28 June 2022 Shift 1>Q 95

If

\sum_{k = 1}^{31} ({}^{31}C_{k}) ({}^{31}C_{k - 1}) - \sum_{k = 1}^{30} ({}^{30}C_{k}) ({}^{30}C_{k - 1}) = \frac{α (60!)}{(30!) (31!)}

, where

α \in R

, then the value of

16 α

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 28 June 2022 Shift 1>Q 96

The number of positive integers

k

such that the constant term in the binomial expansion of

{(2 x^{3} + \frac{3}{x^{k}})}^{12}, x \neq 0

is

2^{8} \cdot l

, where

l

is an odd integer, is ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 28 June 2022 Shift 2>Q 97

The term independent of

x

in the expression of

(1 - x^{2} + 3 x^{3}) {(\frac{5}{2} x^{3} - \frac{1}{5 x^{2}})}^{11}, x \neq 0

is