EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main 18 March 2021 Shift 2>Q 31

HARD

JEE Main

IMPORTANT

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Let ${}^{n}C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1 + x)^{n}$ . If $\sum_{k = 0}^{10} (2^{2} + 3 k) {}^{n}C_{k} = α . 3^{10} + β \cdot 2^{10}, α, β \in R$ , then $α + β$ 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 - 24 Feb 2021 Shift 1>Q 32

The value of

- {}^{15}C_{1} + 2 \cdot {}^{15}C_{2} - 3 \cdot^{15} C_{3} + . . . . . - 15 \cdot {}^{15}C_{15} + {}^{14}C_{1} + {}^{14}C_{3} + {}^{14}C_{5} + . . . . + {}^{14}C_{11}

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 24 Feb 2021 Shift 2>Q 33

If

n ⩾ 2

is a positive integer, then the sum of the series

{}^{n + 1}C_{2} + 2 ({}^{2}C_{2} + {}^{3}C_{2} + {}^{4}C_{2} + \dots + {}^{n}C_{2})

is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 24 Feb 2021 Shift 2>Q 34

For integers $n$ and $r,$ let $(\begin{array}{l} n \\ r \end{array}) = \{\begin{cases} {}^{n}C_{r}, & if n \geq r \geq 0 \\ 0, & otherwise \end{cases}$ . The maximum value of $k$ for which the sum $\sum_{i = 0}^{k} (\begin{matrix} 10 \\ i \end{matrix}) (\begin{matrix} 15 \\ k - i \end{matrix}) + \sum_{i = 0}^{k + 1} (\begin{matrix} 12 \\ i \end{matrix}) (\begin{matrix} 13 \\ k + 1 - i \end{matrix})$ is maximum, is equal to _________.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 25 Feb 2021 Shift 2>Q 35

If the remainder when

x

is divided by

4

is

3

, then the remainder when

{(2020 + x)}^{2022}

is divided by

8

is ___ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 26 Feb 2021 Shift 1>Q 36

Let

m, n \in N

and

\gcd (2, n) = 1

. If

30 (\begin{matrix} 30 \\ 0 \end{matrix}) + 29 (\begin{matrix} 30 \\ 1 \end{matrix}) + \dots . . + 2 (\begin{matrix} 30 \\ 28 \end{matrix}) + 1 (\begin{matrix} 30 \\ 29 \end{matrix}) = n . 2^{m}

, then

n + m

is equal to _______. (Here

(\begin{array}{l} n \\ k \end{array}) = {}^{n}C_{k}

)

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 2 September 2020 Shift 1>Q 37

Let

α > 0, β > 0

be such that

α^{3} + β^{2} = 4

. If the maximum value of the term independent of

x

in the binomial expansion of

{(α x^{\frac{1}{9}} + β x^{- \frac{1}{6}})}^{10}

is

10 k

, then

k

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 2 September 2020 Shift 2>Q 38

For a positive integer

n, {(1 + \frac{1}{x})}^{n}

is expanded in increasing powers of

x

. If three consecutive coefficients in this expansion are in the ratio,

2 : 5 : 12,

then

n

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Binomial Theorem>JEE Main - 3 September 2020 Shift 1>Q 39

If the number of integral terms in the expansion of

{(3^{\frac{1}{2}} + 5^{\frac{1}{8}})}^{n}

is exactly

33

, then the least value of

n

is