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If the coefficient of $y^{8}$ in $(1 + y^{12}) (1 + y^{2})^{12} (1 + y^{24})$ is $λ$ then sum of digits of $λ$ is

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Important Questions on Binomial Theorem

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

For

x \in R, x \neq - 1,

{(1 + x)}^{2016} + x {(1 + x)}^{2015} + x^{2} {(1 + x)}^{2014} + \dots + x^{2016} = \sum_{i = 0}^{2016} a_{i} x_{i}

, then

a_{17}

is equal to

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The sum of the co-efficient of all odd degree terms in the expansion of

{(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}, (x > 1)

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

If the coefficients of

x^{- 2}

and

x^{- 4}

, in the expansion of

{(x^{\frac{1}{3}} + \frac{1}{2 x^{\frac{1}{3}}})}^{18}, (x > 0)

, are

m

and

n

respectively, then

\frac{m}{n}

is equal to

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The smallest natural number

n

, such that the coefficient of

x

in the expansion of

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

{}^{n}C_{23}

, is

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The total number of irrational terms in the binomial expansion of

{(7^{\frac{1}{5}} - 3^{\frac{1}{10}})}^{60}

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

If the fourth term in the binomial expansion of

{(\sqrt{x^{\frac{1}{1 + {l o g}_{10} x}}} + x^{\frac{1}{12}})}^{6}

is equal to

200,

and

x > 1,

then the value of

x

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

Coefficient of

x^{11}

in the expansion of

{(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

If the fourth term in the Binomial expansion of

{(\frac{2}{x} + x^{{l o g}_{8} x})}^{6}, (x > 0)

20 \times 8^{7},

then a value of

x

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The term independent of $x$ in the expansion of

{(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The term independent of

x

in the binomial expansion of

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

If the coefficients of

x^{2}

and

x^{3}

, are both zero, in the expansion of the expression

(1 + a x + b x^{2}) {(1 - 3 x)}^{15}

, in powers of

x

, then the ordered pair

(a, b)

is equal to

EASY

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

A ratio of the

5^{t h}

term from the beginning to the

5^{t h}

term from the end in the binomial expansion of

{(2^{\frac{1}{3}} + \frac{1}{2 {(3)}^{\frac{1}{3}}})}^{10}

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The coefficient of

x^{18}

in the product

(1 + x) {(1 - x)}^{10} {(1 + x + x^{2})}^{9}

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The sum of the co-efficient of all even degree terms in

x

in the expansion of

{(x + \sqrt{x^{3} - 1})}^{6} {+ (x - \sqrt{x^{3} - 1})}^{6}, (x > 1)

is equal to

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The term independent of

x

in the expansion of

(\frac{1}{60} - \frac{x^{8}}{81}) . {(2 x^{2} - \frac{3}{x^{2}})}^{6}

is equal to

EASY

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

α

and

β

be the coefficients of

x^{4}

and

x^{2}

, respectively in the expansion of

{(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}

, then

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

In the expansion of

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

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 ratio

l_{2} : l_{1}

is equal to:

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The coefficient of

x^{- 5}

in the binomial expansion of

{(\frac{x + 1}{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1} - \frac{x - 1}{x - x^{\frac{1}{2}}})}^{10}

where

x \neq 0,1

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

The positive value of

λ

for which the co-efficient of

x^{2}

in the expansion

x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}

720,

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms in Binomial Expansion

If the third term in the binomial expansion of

{(1 + x^{\log_{2} x})}^{5}

equals

2560,

then a possible value of

x

If the coefficient of y8 in (1 + y12) (1 + y2)12 (1 + y24) is λ then sum of digits of λ is

Important Questions on Binomial Theorem

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If the coefficient of $y^{8}$ in $(1 + y^{12}) (1 + y^{2})^{12} (1 + y^{24})$ is $λ$ then sum of digits of $λ$ is