The coefficient of $x^{10}$ in the expansion of $(1+2 x)^{-7}$ is

$2^{10}\left(\begin{array}{c}16 \\ 9\end{array}\right)$

$\left(\begin{array}{c}16 \\ 6\end{array}\right)$

If the expansion in powers of $x$ of the function $\frac{1}{(1-a x)(1-b x)}$ is $a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots$, then $a_n$ is

$\frac{b^{\mathrm{n}}-a^n}{b-a}$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 331-360 of 468 questions

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331 Medium

331 of 468

Application +4 | -1

The term independent of

x

in the expansion of

\left[\left(t^{-1}-1\right) x+\left(t^{-1}+1\right)^{-1} x^{-1}\right]^8

View answer and explanation

Correct option: C

70\left(\frac{1-t}{1+t}\right)^4

Option C (

70\left(\frac{{1-t}}{{1+t}}\right)^4

) is correct because the term independent of x is found by setting the power of x to zero and simplifying the resulting expression.

332 Medium

332 of 468

Application +4 | -1

In how many ways 12 identical pens can be distributed among 5 students?

View answer and explanation

Correct option: B

\left(\begin{array}{c}16 \\ 4\end{array}\right)

Option B

\left(\begin{array}{c}16 \\ 4\end{array}\right)

is correct because it represents the number of ways to distribute 12 identical pens among 5 students.

333 Medium

333 of 468

Application +4 | -1

n > 1

then value of the expression is

C_0-2 C_1+3 C_2-4 C_3+\ldots+(-1)^n(n+1) C_n

View answer and explanation

Correct option: B

Option B (0) is correct because differentiating the binomial expansion and substituting x=-1 results in the given expression equaling zero.

334 Medium

334 of 468

Application +4 | -1

The integral part of

(\sqrt{2}+1)^6

View answer and explanation

Correct option: C

197

Option C (197) is correct because it is the integral part of the expression

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

calculated using binomial expansion and conjugate properties.

335 Medium

335 of 468

Application +4 | -1

In the binomial

\left(2^{1 / 3}+3^{-1 / 3}\right)^n

, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is

1 / 6

then

\mathrm{n}=

View answer and explanation

Correct option: B

Option B (9) is correct because equating the ratio of the seventh terms from the beginning and end leads to n=9.

336 Medium

336 of 468

Application +4 | -1

The coefficient of

y

in the expansion of

\left(y^2+c / y\right)^5

View answer and explanation

Correct option: A

10 c^3

Option A (

10 c^3

) is correct because it is the coefficient of

y

in the expansion of the given binomial.

337 Medium

337 of 468

Application +4 | -1

The approximate value of

{\left( {7.995} \right)^{1/3}}

correct to four decimal places is

View answer and explanation

Correct option: B

1.9996

Option B (1.9996) is correct. Using binomial approximation, the value of (7.995)^(1/3) is approximately 1.9996.

338 Medium

338 of 468

Application +4 | -1

{\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_n}{x^n}

, then the value of

{C_0} + 2{C_1} + 3{C_2} + ... + \left( {n + 1} \right){C_n}

will be

View answer and explanation

Correct option: A

\left( {n + 2} \right){2^{n - 1}}

Option A (

\left( {n + 2} \right){2^{n - 1}}

) is correct because differentiating the binomial expansion and substituting x=1 yields the given sum, which simplifies to

\left( {n + 2} \right){2^{n - 1}}

339 Medium

339 of 468

Application +4 | -1

The value of

\sum\limits_{k = 1}^\infty {\sum\limits_{r = 0}^k {\frac{1}{{{3^k}}}{(^k}{C_r})} }

View answer and explanation

Correct option: C

Option C (2) is correct because the double summation simplifies to a geometric series with a sum of 2.

340 Medium

340 of 468

Analyzing +4 | -1

The value of

\sum\limits_{{r=0}}^n(-1)^r \cdot r^2 \cdot{ }^n C_r

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the value of the summation simplifies to 0.

341 Medium

341 of 468

Application +4 | -1

The coefficient of

x^{10}

in the expansion of

(1+2 x)^{-7}

View answer and explanation

Correct option: B

2^{10}\left(\begin{array}{l}16 \\ 10\end{array}\right)

Option A (

2^{10}\left(\begin{array}{l}17 \\ 10\end{array}\right)

) is incorrect. Option B (

2^{10}\left(\begin{array}{l}16 \\ 10\end{array}\right)

) is the correct answer because it is derived from the general term in the binomial expansion of

(1+2x)^{-7}

342 Medium

342 of 468

Analyzing +4 | -1

(1+x)^n=C_0+C_1 x+\ldots+C_n x^n

, then the value of

\sum\limits_{0 \le r} {\sum\limits_{ < s \le n} {{C_r}{C_r}} }

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\left[2^{2 n}-{ }^{2 n} C_n\right]

Option A (

\frac{1}{2}\left[2^{2 n}-{ }^{2 n} C_n\right]

) is the correct answer because it is derived from the binomial identity and combinatorial arguments.

343 Medium

343 of 468

Understanding +4 | -1

The sum of the series

\sum \limits_{r = 0}^{10} {\;^{20}}{C_r}

, is

View answer and explanation

Correct option: C

{2^{19}} + \frac{1}{2}{\;^{20}}{C_{10}}

Option C (

{2^{19}} + \frac{1}{2}{\;^{20}}{C_{10}}

) is correct because the sum of the series can be simplified using the properties of binomial coefficients and the binomial theorem.

344 Medium

344 of 468

Understanding +4 | -1

\left(1+2 x+x^2\right)^n=\sum\limits_{{r=0}}^{2 n} a_r x^r

, then

a_r

is equal to

View answer and explanation

Correct option: C

{ }^{2n} C_r

Option C (

{ }^{2n} C_r

) is correct because the given expression simplifies to

(1+x)^{2n}

, and

a_r

is the coefficient of

x^r

in its binomial expansion.

345 Medium

345 of 468

Application +4 | -1

{\left( {1 + 2x + 3{x^2}} \right)^{10}} = {a_0} + {a_1}\;x + {a_2}\;{x^2} + \ldots + {a_{20}}\;{x^{20}},

then

{a_1}

equals

View answer and explanation

Correct option: B

Option B (20) is correct because the coefficient of x in the expansion of

{\left( {1 + 2x + 3{x^2}} \right)^{10}}

is 20.

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346 Medium

346 of 468

Analyzing +4 | -1

n

is positive integer and

(3 \sqrt{3}+5)^{2 n+1}=\alpha+\beta

, where

\alpha

is an integer and

0 < \beta < 1

, then

View answer and explanation

Correct option: A

\alpha

is an even integer

Option A is correct because when we consider the conjugate expression and add it to the original, the irrational parts cancel out, resulting in an even integer.

347 Medium

347 of 468

Application +4 | -1

In the expansion of

(1+x)^{18}

, if the coefficients of

(2 r+4)^{\mathrm{th}}

and

(r-2)^{\mathrm{th}}

terms are equal, then the value of

r

is :

View answer and explanation

Correct option: D

Option D (6) is correct because by equating the coefficients using the property of binomial coefficients, we find that r = 6.

348 Medium

348 of 468

Analyzing +4 | -1

Coefficient of

x^{2009}

\left(1+x+x^2+x^3+x^4\right)^{1001}(1-x)^{1002}

View answer and explanation

Correct option: A

Option A (0) is correct because the coefficient of x²⁰⁰⁹ in the simplified expression is zero.

349 Medium

349 of 468

Application +4 | -1

If the expansion in powers of

x

of the function

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

a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots

, then

a_n

View answer and explanation

Correct option: C

\frac{b^{n+1}-a^{n+1}}{b-a}

Option C (

\frac{{b^{n+1}-a^{n+1}}}{{b-a}}

) is correct because it represents the coefficient of

x^n

in the expansion of the given function using partial fractions or series expansion.

350 Medium

350 of 468

Application +4 | -1

\mathrm{C}_{\mathrm{r}}=\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right)

, then

\sum_{\mathrm{r}=1}^{\mathrm{n}}(-1)^{\mathrm{r}-1} \frac{\mathrm{C}_{\mathrm{r}}}{\mathrm{r}}=

View answer and explanation

Correct option: C

\sum_{r=1}^n \frac{1}{r}

Option C (

\sum_{{r=1}}^n \frac{{1}}{{r}}

) is correct because it is the result of integrating the given binomial series.

351 Medium

351 of 468

Application +4 | -1

The sum of the rational terms in the binomial expansion of

\left(2^{\frac{1}{2}}+3^{\frac{1}{5}}\right)^{10}

is:

View answer and explanation

Correct option: D

Option D (41) is correct because only the first and last terms of the expansion are rational, and their sum is 41.

352 Medium

352 of 468

Analyzing +4 | -1

A={ }^{2 n} C_0{ }^{2 n} C_1+{ }^{2 n} C_1{ }^{2 n-1} C_1+{ }^{2 n} C_2{ }^{2 n-2} C_1+\ldots

, then

A

View answer and explanation

Correct option: C

n \cdot 2^{2 n}

Option C is correct because the expression can be interpreted as the coefficient of x in a binomial expansion, which simplifies to

n \cdot 2^{2 n}

353 Medium

353 of 468

Application +4 | -1

{\left( {1 + x - 3{x^2}} \right)^{10}} = 1 + {a_1}x + {a_2}{x^2} + ... + {a_{20}}{x^{20}},

then

{a_2} + {a_4} + {a_6} + ... + {a_{20}}

is equal to

View answer and explanation

Correct option: C

\frac{{{3^{10}} - 1}}{2}

Option C (

\frac{{{3^{10}} - 1}}{2}

) is correct because it is the value obtained by substituting x = 1 and x = -1 into the given polynomial and solving for the sum of even-powered coefficients.

354 Medium

354 of 468

Application +4 | -1

3^{51}

when divided by 8 leaves the remainder,

View answer and explanation

Correct option: D

Option D (3) is correct because 3⁽⁵¹⁾ can be written as 3 * (3²)²⁵ = 3 * (9)²⁵ = 3 * (8+1)²⁵, and when divided by 8, the remainder is 3.

355 Medium

355 of 468

Application +4 | -1

The value of

\sum\limits_{{r=0}}^7 \frac{{ }^7 C_r}{{ }^{r+2} C_2}

is equal to

View answer and explanation

Correct option: B

\frac{251}{18}

Option B,

\frac{251}{18}

, is correct because it is the result of simplifying the summation using binomial coefficient identities.

356 Medium

356 of 468

Application +4 | -1

The coefficient of

{a^5}{b^6}{c^7}

in the expansion of

{\left( {bc + ca + ab} \right)^9}

View answer and explanation

Correct option: D

1260

Option D (1260) is correct because it is the coefficient of

{a^5}{b^6}{c^7}

obtained by expanding the given expression using binomial theorem properties.

357 Medium

357 of 468

Analyzing +4 | -1

For

r=0,1, \ldots, 10

, let

A_r, B_r

and

C_r

denote, respectively the coefficient of

x^r

in the expansions of

(1+x)^{10},(1+x)^{20}

and

\left(1+x^{30}\right)

Then,

\sum\limits_{{r=1}}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)

is equal to

View answer and explanation

Correct option: D

C_{10}=B_{10}

Option D (

C_{10}=B_{10}

) is correct because simplifying the summation using properties of binomial coefficients leads to the conclusion that

C_{10} - B_{10} = 0

, thus

C_{10} = B_{10}

358 Medium

358 of 468

Analyzing +4 | -1

Statement I:The integral part of

(8+3 \sqrt{7})^{20}

is an even integer.

Statement II:

(8+3 \sqrt{7})^{20}+(8-3 \sqrt{7})^{20}

is an even integer.

View answer and explanation

Correct option: D

Statement-I is false, statement-II is true

Option D (Statement-I is false, statement-II is true) is correct because the integral part of the expression is odd, and the given sum is an even integer.

359 Medium

359 of 468

Analyzing +4 | -1

The value of

\sum\limits_{{r=0}}^{2 n}(-1)^r \cdot\left({ }^{2 n} C_r\right)^2

is equal to

View answer and explanation

Correct option: C

(-1)^n \cdot{ }^{2 n} C_n

Option C (

(-1)^n \cdot{ }^{2 n} C_n

) is correct because it is obtained by recognizing the sum as the coefficient of a term in the product of two binomial expansions.

360 Medium

360 of 468

Analyzing +4 | -1

(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots+C_n x^n

, then

\sum\limits_{{r=0}}^n \sum\limits_{{s=0}}^n(r+s) C_r C_s

is equal to

View answer and explanation

Correct option: B

n \cdot 2^{2 n}

Option B (

n \cdot 2^{2 n}

) is correct because the summation can be separated and simplified using properties of binomial coefficients.

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