Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 11

HARD

JEE Main

IMPORTANT

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The coefficient of $x^{6}$ in $((1 + x) {(1 + x^{2})}^{2} {(1 + x^{3})}^{3} \dots {(1 + x^{n})}^{n})$ , where $\{n \in N, n > 6\}$ , is equal to

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Important Questions on Binomial Theorem

MEDIUM

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 12

In the expansion of

{(\frac{x}{\cos θ} + \frac{1}{x \sin θ})}^{16}

, if

l_{1}

is the least value of the term independent of

x

when

\frac{π}{8} \leq θ \leq \frac{π}{4}

and

l_{2}

is the least value of the term independent of

x

when

\frac{π}{16} \leq θ \leq \frac{π}{8}

, then the value of

\frac{l_{2}}{l_{1}}

is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 13

Given ${(1 - x^{3})}^{n} = \sum_{k = 0}^{n} a_{k} x^{k} {(1 - x)}^{3 n - 2 k} .$ Then, the value of $3 a_{k - 1} + a_{k}$ is

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JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 14

If

P_{n}

denotes the product of all the coefficients of

{(1 + x)}^{n}

and

9! P_{n + 1} = 10 \cdot^{9} P_{n}

, then

n

is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 15

If

\sum_{r = 0}^{n} {\{\frac{{}^{n}C_{r - 1}}{{}^{n}C_{r} +^{n} C_{r - 1}}\}}^{3} = \frac{25}{24}

, then

n

is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 16

If the expansion in powers of

x

of the function

\frac{1}{(1 - a x) (1 - b x)}

is

a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots,

then

a_{n}

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 17

The sum of coefficients of integral powers of

x

in the binomial expansion of

{(1 - 2 \sqrt{x})}^{50}

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 18

The coefficient of

p^{n} q^{n}

in the expansion of

{[(1 + p) (1 + q) (p + q)]}^{n}

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 3 - Binomial Theorem>Level 2>Q 19

The sum

\frac{1}{1! 2004!} + \frac{1}{3! 2002!} + \frac{1}{5! 2000!} + \dots + \frac{1}{2003! 2!} = (\frac{2^{a} - 1}{b!})

where

a

and

b

are positive integers. Then
find

b - a .