Back to BINOMIAL THEOREM

Your Progress

0 / 951 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

Practice format and chapter quiz

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.

Take a chapter quiz →

Showing 391-420 of 951 questions

Preparing practice and saved progress…

391 Medium

391 of 951

Application +4 | -1

If the coefficient of

x^7

\left(a x-\frac{1}{b x^2}\right)^{13}

and the coefficient of

x^{-5}

\left(a x+\frac{1}{b x^2}\right)^{13}

are equal, then

a^4 b^4

is equal to:

View answer and explanation

Correct option: C

Option C (22) is correct because after equating the coefficients of x⁷ and x^-5 and simplifying, we find that a⁴b⁴ equals 22.

392 Easy

392 of 951

Application +4 | -1

If in the binomial expansion of

(1-x)^m(1+x)^n

, the coefficients of

x

and

x^2

are respectively

3

and

-4,

then the ratio

m: n

is equal to:

View answer and explanation

Correct option: D

7: 10

Option D (

7: 10

) is correct because by setting up equations based on the coefficients of

x

and

x^2

, we can solve for

m

and

n

and find their ratio.

393 Medium

393 of 951

Application +4 | -1

The coefficient of

x^{11}

in the expansion of

\left(x^3-\frac{2}{x^2}\right)^{12}

View answer and explanation

Correct option: A

-25344

Option A (-25344) is correct because it is the coefficient of x¹¹ obtained after applying the binomial theorem and solving for the appropriate term.

394 Medium

394 of 951

Application +4 | -1

x > 0

, then the number of positive terms in the expansion of

(1+i x)^{4 n}

View answer and explanation

Correct option: A

n+1

Option A (n+1) is correct because the positive terms occur when the power of i is a multiple of 4, resulting in n+1 positive terms.

395 Easy

395 of 951

Understanding +4 | -1

\alpha

and

\beta

are the coefficients of

x^r

and

x^{n-r}

respectively in the expansion of

(1+x)^n

, then

View answer and explanation

Correct option: A

\alpha=\beta

Option A (

a_p=a_q

) is correct because

a_r

represents the coefficient of

x^r

in the expansion of

(1+x)^{p+q}

, and

a_p = { }^{p+q}C_p

and

a_q = { }^{p+q}C_q

. Since

{ }^{n}C_r = { }^{n}C_{n-r}

, and

{ }^{p+q}C_p = { }^{p+q}C_{(p+q)-p} = { }^{p+q}C_q

, therefore

a_p = a_q

396 Medium

396 of 951

Analyzing +4 | -1

If the constant term in the expansion of

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

2^k \cdot I

, where

I

is an odd integer, then the value of

k

is equal to

View answer and explanation

Correct option: D

The correct answer is D (9) because the value of k is determined by finding the constant term in the multinomial expansion and extracting the power of 2.

397 Easy

397 of 951

Analyzing +4 | -1

\sum\limits_{p=1}^n \sum\limits_{m=p}^n\left(\begin{array}{l}n \\ m\end{array}\right)\left(\begin{array}{l}m \\ p\end{array}\right)

is equal to

View answer and explanation

Correct option: D

3^n-2^n

Option D (

3^n-2^n

) is correct because the double summation can be simplified using combinatorial identities to this expression.

398 Easy

398 of 951

Application +4 | -1

The co-efficient of correlation between two variables

x

and

y

is 0.8 while the regression coefficient of

y

x

is 0.2 , Then the regression co-efficient of

x

y

View answer and explanation

Correct option: B

3.2

Option B (3.2) is correct because it is the regression coefficient of x on y, calculated using the given correlation coefficient and regression coefficient of y on x.

399 Medium

399 of 951

Application +4 | -1

If the digits at ten's and hundred's places in (11)

{ }^{2016}

are

\mathrm{x}

and

\mathrm{y}

respectively, then the ordered pair

(x, y)

is equal to :

View answer and explanation

Correct option: C

(6,1)

Option C ((6,1)) is correct because it represents the digits at the ten's and hundred's places in the expansion of (11)

^{2016}

, which are 6 and 1 respectively.

400 Easy

400 of 951

Application +4 | -1

Sum of the last 30 coefficients in the expansion of

{\left( {1 + x} \right)^{59}},

when expanded in ascending powers of

x

View answer and explanation

Correct option: B

{2^{58}}

The correct answer is B (

{2^{58}}

) because the sum of the last 30 coefficients is half of the total sum of all coefficients in the binomial expansion.

401 Hard

401 of 951

Application +4 | -1

Find the coefficient of

x^5

in the expansion of

\left(1+x+x^3\right)^9

View answer and explanation

Correct option: A

378

Option A (378) is the correct answer because it is the calculated coefficient of

x^5

in the expansion of the given expression.

402 Easy

402 of 951

Application +4 | -1

n > 1,

then

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

is divisible by

View answer and explanation

Correct option: B

{x^2}

Option B (

{x^2}

) is correct because the expansion of

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

contains terms with powers of

x

starting from

{x^2}

, making it divisible by

{x^2}

403 Medium

403 of 951

Application +4 | -1

If the second term in the expansion

{\left[ {\sqrt[{13}]{a} + \frac{a}{{\sqrt {{a^{ - 1}}} }}} \right]^n}

is,

14{a^{5/2}}

then the value of

\frac{{{n_{{C_3}}}}}{{{n_{{C_2}}}}}

View answer and explanation

Correct option: A

Option A (4) is correct because by comparing the second term in the expansion with the given value, we find

n = 14

, and then

\frac{{{n_{{C_3}}}}}{{{n_{{C_2}}}}} = 4

404 Easy

404 of 951

Application +4 | -1

{49^n} + 16n - 1\;

is divisible by

View answer and explanation

Correct option: D

Option D (64) is correct because

{49^n} + 16n - 1

can be rewritten as

{(1 + 48)^n} + 16n - 1

, and using the binomial theorem, it can be shown that the expression is divisible by 64.

405 Medium

405 of 951

Understanding +4 | -1

The number of distinct terms in the expansion of

\left(x_1+x_2+\cdots+x_n\right)^3

View answer and explanation

Correct option: B

{ }^{n+2} C_3

Option B (

{ }^{n+2} C_3

) is correct because the number of distinct terms corresponds to the number of non-negative integer solutions to the equation

\lambda_1 + \lambda_2 + ... + \lambda_n = 3

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

406 Easy

406 of 951

Analyzing +4 | -1

If the fractional part of the number

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

\frac{k}{{15}},

then

k

is equal to

View answer and explanation

Correct option: D

Option D (8) is correct because when 2⁴⁰³ is divided by 15, the remainder is 8, making the fractional part 8/15.

407 Easy

407 of 951

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

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

View answer and explanation

Correct option: B

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

Option B is correct. The coefficient of x^n in the expansion is found by summing the coefficients of x^n and x^(n-1) in (1-x)^n.

408 Medium

408 of 951

Analyzing +4 | -1

The value of

\sum\limits_{1 \le i < } {\sum\limits_{j \le n - 1} {{{(i.j)}^n}{C_i}{.^n}{C_j}} }

is equal to

View answer and explanation

Correct option: A

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

Option A is correct because the summation can be simplified using properties of binomial coefficients and results in

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

409 Medium

409 of 951

Application +4 | -1

If the coefficients of three consecutive terms of

(1+x)^{n+5}

are in the ratio 5:10:14, then

n

is equal to

View answer and explanation

Correct option: B

Option B (6) is correct because the given ratio of consecutive coefficients leads to a system of equations that can be solved for n.

410 Hard

410 of 951

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

{\left( {1 + {x^2}} \right)^5}{\left( {1 + x} \right)^4}

View answer and explanation

Correct option: B

Option B (60) is correct because by expanding the given expression and finding the coefficient of

x^5

, we get 60.

411 Medium

411 of 951

Application +4 | -1

7^{103}

is divided by 25 , then the remainder is

View answer and explanation

Correct option: C

Option C (18) is correct because when

7^{{103}}

is divided by 25, the remainder is 18, which is found using modular arithmetic.

412 Easy

412 of 951

Application +4 | -1

The positive value of a so that the coefficients of

{x^5}

and

{x^{15}}

are equal in the expansion of

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

View answer and explanation

Correct option: A

\frac{1}{{2\sqrt 3 }}

Option A (

\frac{1}{{2\sqrt 3 }}

) is correct because it is the positive value of 'a' that makes the coefficients of x⁵ and x¹⁵ equal in the given binomial expansion.

413 Easy

413 of 951

Application +4 | -1

For

n \in \mathbf{N}, x \in \mathbf{R}

, the sum of the series

C_1 x(1-x)^{n-1}+2 C_2 x^2(1-x)^{n-2} +3 C_3 x^3(1-x)^{n-3}+\cdots+n C_n x^n

View answer and explanation

Correct option: A

n x

Option A (nx) is correct because the series can be recognized as the derivative of a binomial expansion.

414 Easy

414 of 951

Application +4 | -1

The sum

{\;^{40}}{C_0} + {\;^{40}}{C_1} + {\;^{40}}{C_2} + \ldots + {\;^{40}}{C_{20}}

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given sum can be simplified to

{2^{39}} + \frac{1}{2} \times \frac{{40!}}{{{{\left( {20!} \right)}^2}}}

, which does not match any of the provided options.

415 Medium

415 of 951

Analyzing +4 | -1

The value of

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

is equal to

View answer and explanation

Correct option: D

n^2.{ }^{2 n-2} C_n

Option D (

n^2.{ }^{2 n-2} C_n

) is correct because the given summation can be simplified using combinatorial identities to match this expression.

416 Medium

416 of 951

Analyzing +4 | -1

Coefficient of

x^{50}

in the expansion of

(1+x)^{41}\left(1-x+x^2\right)^{40}

View answer and explanation

Correct option: A

Option A (0) is correct because the expression can be simplified to

(1+x)(1-x^3)^{40}

, and the coefficient of

x^{50}

in this expression is 0.

417 Easy

417 of 951

Analyzing +4 | -1

Let

(5+2 \sqrt{6})^n=I+f

, where

n, I \in N

and

0 < f < 1

, then the value of

f^2-f+I \cdot f-I

, is

View answer and explanation

Correct option: B

a negative integer

Option B (a negative integer) is correct because the expression simplifies to -1, which is a negative integer.

418 Hard

418 of 951

Application +4 | -1

k

and

n

are positive integer and

s_k=1^k+2^k+\ldots .+n^k

, then

\sum\limits_{{r=1}}^m{ }^{m+1} C_r s_r

is equal to

View answer and explanation

Correct option: A

(n+1)^{m+1}-(n+1)

Option A (

(n+1)^{m+1}-(n+1)

) is correct because the given summation simplifies to this expression using binomial theorem and telescoping series.

419 Easy

419 of 951

Application +4 | -1

In the expansion of

{\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}},

the coefficient of

{x^{39}},

View answer and explanation

Correct option: D

- 455

Option D (-455) is correct because it is the coefficient of x³⁹ in the binomial expansion of the given expression, calculated using the general term formula.

420 Medium

420 of 951

Application +4 | -1

\frac{1}{1 ! 10 !}+\frac{1}{3 ! 8 !}+\frac{1}{5 ! 6 !}+\ldots+\frac{1}{11 ! 1 !}=\frac{2^k}{11 !}

then

\frac{k}{2}

is equal to

View answer and explanation

Correct option: A

Option A (5) is correct because the value of k is 10, and k/2 equals 5, after simplifying the given expression using binomial coefficients.

Revision notes and key takeaways

Review BINOMIAL THEOREM in your textbook or class notes, then use this set to check your understanding.

Showing 391-420 of 951 questions

Your Progress

Binomial theorem practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice