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

EASY

JEE Main

IMPORTANT

Earn 100

The remainder when ${(2021)}^{2023}$ is divided by $7$ is

72.73% studentsanswered this correctly

Important Questions on Binomial Theorem

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

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 29 June 2022 Shift 1>Q 98

If the constant term in the expansion of

{(3 x^{3} - 2 x^{2} + \frac{5}{x^{5}})}^{10}

is

2^{k} . l

, where

l

is an odd integer, then the value of

k

is equal to

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 29 June 2022 Shift 2>Q 99

Let $n \geq 5$ be an integer. If $9^{n} - 8 n - 1 = 64 α$ and $6^{n} - 5 n - 1 = 25 β$ , then $α - β$ is equal to:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 29 June 2022 Shift 2>Q 100

Let the coefficients of

x^{- 1}

and

x^{- 3}

in the expansion of

{(2 x^{\frac{1}{5}} - \frac{1}{x^{\frac{1}{5}}})}^{15}, x > 0

, be

m

and

n

respectively. If

r

is a positive integer such

m n^{2} = {}^{15}C_{r} \cdot 2^{r}

, then the value of

r

is equal to ______.