Mathematical Induction And Binomial Theorem, Solved Paper Question: Jee 1 year MATH, Math

The sum of the real values of x for which the middle term in the binomial expansion of

{(\frac{x^{3}}{3} + \frac{3}{x})}^{8}

equals 5670 is:

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JEE Mains 2019

If the third term in the binomial expansion of

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

equals 2560, then a possible value of x is:

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JEE Mains 2019

A ratio of the 5^th term from the beginning to the 5^th term from the end in the binomial expansion of

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

is:

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JEE Mains 2019

The coefficient of t⁴ in the expansion of

{(\frac{1 - t^{6}}{1 - t})}^{3}

is:

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JEE Mains 2019

If the fractional part of the number

\frac{2^{403}}{15}

is

\frac{k}{15},

then k is equal to:

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JEE Mains 2019

The positive value of λ for which the co-efficient of x² in the expression

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

is 720, is :

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JEE Mains 2019

The total number of irrational terms in the binomial expansion of

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

is :

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JEE Mains 2019

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

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

is :

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JEE Mains 2018

Let

X = {({}^{10}C_{1})}^{2} + 2 {({}^{10}C_{2})}^{2} + 3 {({}^{10}C_{3})}^{2} + . . . + 10 {({}^{10}C_{10})}^{2},

where ¹⁰C_r, r ∈ {1, 2,⋯, 10} denote binomial coefficients. Then, the value of

\frac{1}{1430} X

is _____ .

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JEE Advanced 2018

If the number of terms in the expansion of

{(1 - \frac{2}{x} + \frac{4}{x^{2}})}^{n}, x \neq 0,

is 28, then the sum of the coefficients of the terms in this expansion, is :

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JEE Mains 2016

Let m be the smallest positive integer such that the coefficient of x² in the expansion of (1 + x)² + (1 + x)³ + ... + (1 + x)⁴⁹ + (1 + mx)⁵⁰ is (3n + 1) ⁵¹C₃ for some positive integer n. Then the value of n is

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JEE Advanced 2016

The coefficient of x⁹ in the expansion of (1+x) (1+x²) (1+x³) .... (1+x¹⁰⁰) is

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JEE Advanced 2015

The sum of coefficient of integral power of x in the binomial expansion of

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

is :

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JEE Mains 2015

If the coefficients of x³ and x⁴ in the expansion of (1 + ax + bx²) (1 − 2x)¹⁸ in powers of x are both zero, then (a, b) is equal to :

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JEE Mains 2014

If X = {4ⁿ − 3n − 1 : n ∈ N} and Y = {9(n − 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to:

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JEE Mains 2014

Coefficient of x¹¹ in the expansion of (1 + x²)⁴ (1 + x³)⁷ (1 + x⁴)¹² is

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JEE Advanced 2014

The coefficients of three consecutive terms of (1 + x)ⁿ^{+ 5} are in the ratio 5 : 10 : 14. Then n =

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JEE Advanced 2013

The term independent of x in expansion of

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JEE Mains 2013

For r = 0,1,…..10, let A_r, B_r and C_r denote, respectively, the coefficient of x^r in the expansions of , and Then is equal to

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JEE 2010

Prove that

2^{k} (\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} n \\ k \end{matrix}) - 2^{k - 1} (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} n - 1 \\ k - 1 \end{matrix}) . . . . . . . . . {(- 1)}^{k} (\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} n - k \\ 0 \end{matrix}) = (\begin{matrix} n \\ k \end{matrix}), where (\begin{matrix} n \\ m \end{matrix}) =^{n} C_{m} .

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JEE 2003

Use mathematical induction to show that (25)ⁿ⁺¹ – 24n + 5735 is divisible by (24)² for all n = 1, 2, 3, ....

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JEE 2002