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If a³ + b⁶ = 2, then the maximum value of the term independent of x in the expansion of ${(a x^{1 / 3} + b x^{- 1 / 6})}^{9}$ $(a > 0, b > 0)$ is

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Important Questions on Binomial Theorem

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

Coefficient of

x^{11}

in the expansion of

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

f (x) = x^{n}

, then the value of

f (1) - \frac{f^{'} (1)}{1!} + \frac{f^{''} (1)}{2!} - \frac{f^{'''} (1)}{3!} + \dots + \frac{{(- 1)}^{n} f^{n} (1)}{n!}

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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)

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

If the third term in the binomial expansion of

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

equals

2560,

then a possible value of

x

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

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

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

For each positive integer n, let

A_{n} = \max \{(\frac{n}{r}) | 0 \leq r \leq n\}

. Then the numbers of elements in

\{1, 2, 3, . . . . . . . ., 19, 20\}

for which

1.9 \leq \frac{A_{n}}{A_{n - 1}} \leq 2

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

EASY

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

The coefficient of

x

in the expansion of

(1 - 3 x + 7 x^{2}) {(1 - x)}^{16}

EASY

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

The coefficient of

x^{5}

in the expansion of

{(x + 3)}^{8}

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

The total number of irrational terms in the binomial expansion of

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

EASY

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

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

HARD

Mathematics>Algebra>Binomial Theorem>General and Middle Terms

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

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,

If a3 + b6 = 2, then the maximum value of the term independent of x in the expansion of a x 1 / 3 + b x - 1 / 6 9 (a>0, b>0) is

Important Questions on Binomial Theorem

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If a³ + b⁶ = 2, then the maximum value of the term independent of x in the expansion of ${(a x^{1 / 3} + b x^{- 1 / 6})}^{9}$ $(a > 0, b > 0)$ is