EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 29 January 2023 Shift 2>Q 119

HARD

JEE Main

IMPORTANT

Earn 100

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of ${(1 + x)}^{99}$ . Let a be the middle term in the expansion of ${(2 + \frac{1}{\sqrt{2}})}^{200}$ . If $\frac{{}^{200}C_{99} K}{a} = \frac{2^{l} m}{n}$ , where $m$ and $n$ are odd numbers, then the ordered pair $(l, n)$ 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 - 30 January 2023 Shift 1>Q 120

If the coefficient of

x^{15}

in the expansion of

{(a x^{3} + \frac{1}{b x^{\frac{1}{3}}})}^{15}

is equal to the coefficient of

x^{- 15}

in the expansion of

{(a x^{\frac{1}{3}} - \frac{1}{b x^{3}})}^{15}

, where

a

and

b

are positive real numbers, then for each such ordered pair

(a, b)

:

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 30 January 2023 Shift 1>Q 121

The coefficient of

x^{301}

in

{(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + \dots . . + x^{500}

is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 30 January 2023 Shift 2>Q 122

Let

x = {(8 \sqrt{3} + 13)}^{13}

and

y = {(7 \sqrt{2} + 9)}^{9}

. If

[t]

denotes the greatest integer

\leq t

, then

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 30 January 2023 Shift 2>Q 123

50^{th}

root of a number

x

is

12

and

50^{th}

root of another number

y

is

18

. Then the remainder obtained on dividing

(x + y)

by

25

is ________.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 31 January 2023 Shift 1>Q 124

Let

α > 0

, be the smallest number such that the expansion of

{(x^{\frac{2}{3}} + \frac{2}{x^{3}})}^{30}

has a term

β x^{- α}, β \in N

. Then

α

is equal to _____ .

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 31 January 2023 Shift 1>Q 125

The remainder on dividing

5^{99}

by

11

is _____ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 31 January 2023 Shift 2>Q 126

Let

A = [a_{\hat{i} \hat{j}}] \forall a_{i j} \in Z \cap [0, 4], 1 \leq i, j \leq 2

. The number of matrices

A

such that the sum of all entries is

a

prime number

p \in (2, 13)

is _____ .

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 31 January 2023 Shift 2>Q 127

The coefficient of

x^{- 6}

, in the expansion of

{(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}

, is