If ${\left( {1 + 2\;x + {x^2}} \right)^n} = \mathop \sum \limits_{r = 0}^{2\;n} a,\;{x^r},$ Then ${a_r} = $

${\left( {{\;^n}{C_r}} \right)^2}$

${\;^n}{C_r}.{\;^n}{C_{r + 1}}$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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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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451 Medium

451 of 951

Application +4 | -1

The ratio of the coefficient of

x^{10}

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

and the term independent of

x

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

View answer and explanation

Correct option: B

1: 32

Option B (1: 32) is correct because the ratio of the coefficient of x¹⁰ in (1+x²)¹⁰ to the term independent of x in (x+2/x)¹⁰ is 1:32.

452 Medium

452 of 951

Application +4 | -1

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

Then

{a_r} =

View answer and explanation

Correct option: C

{\;^{2n}}{C_r}

Option C (

\;^{2n}{C_r}

) is correct because simplifying the expression and applying the binomial theorem leads to this result.

453 Easy

453 of 951

Application +4 | -1

The remainder when

(11)^{1011}+(1011)^{11}

is divided by 9 is

View answer and explanation

Correct option: D

Option D (8) is correct because when

(11)^{1011}+(1011)^{11}

is divided by 9, the remainder is equivalent to the remainder of

2^{1011}+3^{11}

divided by 9, which simplifies to 8.

454 Medium

454 of 951

Analyzing +4 | -1

The greatest integer less than

(\sqrt{2}+1)^6

is equal to

View answer and explanation

Correct option: B

197

The correct answer is B (197) because it is the greatest integer less than

(\sqrt{2}+1)^6

, calculated by considering

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

and the fact that

(\sqrt{2}-1)^6

is between 0 and 1.

455 Medium

455 of 951

Application +4 | -1

The value of

\sum\limits_{{i=0}}^{n-r}{ }^{n-i} C_r, r \in[1, n]

is equal to

View answer and explanation

Correct option: D

{ }^{n+1} C_{r+1}

Option D (

{ }^{n+1} C_{r+1}

) is correct because the summation represents a hockey stick identity, which simplifies to

{ }^{n+1} C_{r+1}

456 Easy

456 of 951

Application +4 | -1

The sum of all values of '

p

' so that the term independent of

\mathrm{x}

in the expansion of

\left(\mathrm{x}+\frac{\mathrm{p}^2-2}{\mathrm{x}^4}\right)^{10}

is 45 is

View answer and explanation

Correct option: A

Option A (0) is correct because after finding the values of p, which are

\pm 1

and

\pm \sqrt{3}

, their sum is 0.

457 Easy

457 of 951

Analyzing +4 | -1

C_r={ }^n C_r

, then value of

2\left(\frac{C_1}{C_0}+2 \frac{C_2}{C_1}+3 \frac{C_3}{C_2}+\cdots n \frac{C_n}{C_{n-1}}\right)

View answer and explanation

Correct option: B

n(n+1)

Option B (

n(n+1)

) is correct because the given expression simplifies to

2 \sum_{r=1}^n(n-r+1)

, which equals

n(n+1)

458 Medium

458 of 951

Analyzing +4 | -1

p

is prime number and

n < p < 2 n

, if

N={ }^{2 n} C_n

, then

View answer and explanation

Correct option: A

p

divides

N

Option A is correct because since p is a prime number between n and 2n, it will divide the numerator of the binomial coefficient but not the denominator, thus dividing N.

459 Medium

459 of 951

Application +4 | -1

There are numerous balls of red, white, black and green colour. In how many ways 11 balls can be chosen from these?

View answer and explanation

Correct option: B

120

The correct answer is 120 because it represents the number of ways to choose 11 balls from the four colors.

460 Easy

460 of 951

Analyzing +4 | -1

{\left( {1 + x + {x^2}} \right)^n} = \sum \limits_{r = 0}^{2n} {a_r}{x^r},\;\mathrm{then\;}{a_1} - 2{a_2} + 3{a_3} \ldots - 2n{a_{2n}}

is equal to

View answer and explanation

Correct option: B

- n

Option B (-n) is correct because it is derived by differentiating the given expression and substituting x = -1.

461 Medium

461 of 951

Application +4 | -1

\sum\limits_{{r=1}}^n r(r+1) \frac{C_r}{C_{r-1}}=77

, then

n

equals

View answer and explanation

Correct option: B

Option B (6) is correct because substituting n=6 into the simplified summation formula yields 77, satisfying the given relation.

462 Easy

462 of 951

Application +4 | -1

The

\left( {r + 1} \right)th

term in the expansion of

{\left( {1 - x} \right)^{ - 4}}

will be

View answer and explanation

Correct option: B

\frac{{\left( {r + 1} \right)\left( {r + 2} \right)\left( {r + 3} \right)}}{6}{x^r}

Option B is correct because the (r+1)th term in the expansion of (1-x)^-4 is derived using the binomial theorem for negative exponents.

463 Easy

463 of 951

Analyzing +4 | -1

\frac{1}{n !}+\frac{1}{2 !(n-2) !}+\frac{1}{4 !(n-4) !}+\ldots

is equal to

View answer and explanation

Correct option: A

\frac{2^{n-1}}{n !}

Option A (

\frac{2^{n-1}}{n !}

) is correct because the given expression simplifies to this value using binomial coefficients.

464 Easy

464 of 951

Analyzing +4 | -1

Sum of the series

\sum\limits_{k = 0}^n {^n{C_k}{{\left( { - 1} \right)}^k}\frac{1}{{{a_k}}}}

where

a_k=\sum\limits_{{i=0}}^k{ }^k C_i b_i

where

b_i=\sum\limits_{{j=0}}^i{ }^i C_j\left(\frac{-2}{3}\right)^j

View answer and explanation

Correct option: C

\frac{1}{4^n}

Option C,

\frac{{1}}{{4^n}}

, is correct because after simplifying the series, it becomes a binomial expansion equal to

(\frac{{1}}{{4}})^n

465 Medium

465 of 951

Analyzing +4 | -1

The value of

{ }^n C_0,{ }^{2 n} C_n-{ }^n C_1,{ }^{2 n-2} C_n +{ }^n C_2 \cdot{ }^{2 n-4} C_n-\ldots \text { is }

View answer and explanation

Correct option: C

2^n

Option C (

2^n

) is correct because the expression simplifies to the coefficient of

x^n

(2x+x^2)^n

, which is

2^n

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466 Medium

466 of 951

Understanding +4 | -1

The expansion of

\frac{1}{\sqrt[5]{7-3 x}}

in exponents of

x

is possible, if

View answer and explanation

Correct option: A

|x| < \frac{7}{3}

Option A (

|x| < \frac{7}{3}

) is correct because the binomial expansion is valid only when

|\frac{3x}{7}| < 1

467 Easy

467 of 951

Application +4 | -1

The coefficient of

{x^2}

term in the binomial expansion of

{\left( {\frac{1}{3}{x^{1/2}} + {x^{ - 1/4}}} \right)^{10}}

View answer and explanation

Correct option: A

\frac{{70}}{{243}}

Option A (70/243) is correct because it is the coefficient of the x² term in the binomial expansion of the given expression, calculated by finding the appropriate term in the expansion.

468 Easy

468 of 951

Analyzing +4 | -1

\left( {^m{C_0}{ + ^m}{C_1}{ - ^m}{C_2}{ - ^m}{C_3}} \right) + \left( {^m{C_4}{ + ^m}{C_5}{ - ^m}{C_6}{ - ^m}{C_7}} \right) + \ldots = 0

if and only if for some positive integer

k, m

is equal to

View answer and explanation

Correct option: C

4 k-1

Option C (

4 k-1

) is correct because it satisfies the condition for the given sum to be zero, derived from De Moivre's theorem.

469 Easy

469 of 951

Application +4 | -1

If the coefficient of 7th and 13th term in the expansion of

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

are equal, then

n

is equal to

View answer and explanation

Correct option: C

Option C (18) is correct because if the coefficients of the 7th and 13th terms are equal, then

n

must be

6 + 12 = 18

470 Easy

470 of 951

Application +4 | -1

If the sum of the coefficient in the expansion of

{\left( {x + y} \right)^n}

is 1024, then the value of the greatest coefficient in the expansion is

View answer and explanation

Correct option: B

252

Option B (252) is correct because after finding n=10 from the sum of coefficients, the greatest coefficient in the expansion of (x+y)¹⁰ is calculated as

^{10}C_5

, which equals 252.

471 Easy

471 of 951

Application +4 | -1

Cube root of 217 is

View answer and explanation

Correct option: A

6.01

Option A (6.01) is correct because it is the approximate cube root of 217 calculated using the binomial theorem.

472 Medium

472 of 951

Application +4 | -1

In the binomial expansion of

{\left( {a - b} \right)^n},\;n \ge 5,

the sum of 5th and 6th terms is zero, then

\frac{a}{b}

equals

View answer and explanation

Correct option: D

\frac{{n - 4}}{5}

Option D (

\frac{{n - 4}}{5}

) is correct because setting the sum of the 5th and 6th terms of the binomial expansion to zero and solving for a/b yields this result.

473 Easy

473 of 951

Application +4 | -1

If the fourth term in the binomial expansion of

{\left( {\sqrt {{x^{\left( {\frac{1}{{1 + \log {{10}^x}}}} \right)}}} + {x^{\frac{1}{{12}}}}} \right)^\mathrm{6}}

is equal to 200, and

x > 1,

then the value of

x

View answer and explanation

Correct option: C

Option C (10) is correct because after simplifying the binomial expansion and solving the resulting equation, the value of x that satisfies the condition x > 1 is 10.

474 Medium

474 of 951

Application +4 | -1

The coefficient of

x^3

\left(\sqrt{x^5}+\frac{3}{\sqrt{x^3}}\right)^6

View answer and explanation

Correct option: D

540

Option D (540) is correct because after finding the correct term in the expansion, its coefficient is calculated to be 540.

475 Easy

475 of 951

Application +4 | -1

If the sum of the coefficients in the expansion of

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

a

and if the sum of the coefficients in the expansion of

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

b

, then

View answer and explanation

Correct option: B

a = {b^3}

Option B (

a = {b^3}

) is correct because substituting x=1 in the first expression yields a = 8^n and substituting x=1 in the second expression yields b = 2^n, thus a = b³.

476 Medium

476 of 951

Application +4 | -1

A set contains

(2 n+1)

elements. If the number of subsets of this set which contain at most

\mathrm{n}

elements is 4096 , then the value of

\mathrm{n}

View answer and explanation

Correct option: A

Option A (6) is correct because by using the properties of binomial coefficients and the given condition, we can derive that n = 6.

477 Easy

477 of 951

Application +4 | -1

The number of distinct terms in the expansion of

\left(1+3 x+3 x^2+x^3\right)^7

View answer and explanation

Correct option: D

Option D (22) is correct because the expression simplifies to (1+x)²¹, which has 22 terms.

478 Medium

478 of 951

Application +4 | -1

Coefficient of constant term in the expansion of

\left(x^3+5(3)^{-\log \sqrt{3}} \sqrt{x^3}\right)^4

View answer and explanation

Correct option: D

150

Option D (150) is correct. The constant term is found by determining the term independent of x in the binomial expansion.

479 Hard

479 of 951

Analyzing +4 | -1

The coefficient of

x^{10}

in the expansion of

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

\left(1+x^{3}\right)^{4}

is equal to

View answer and explanation

Correct option: A

The correct answer is A (52) because by expanding the given expression and identifying the combinations that result in x¹⁰, the sum of their coefficients is 52.

480 Hard

480 of 951

Application +4 | -1

C_0, C_1, C_2, \cdots, C_n

are binomial coefficients, then

\lim \limits_{{n \rightarrow \infty}}\left[C_n-\left(\frac{2}{3}\right) C_{n-1}+\left(\frac{2}{3}\right)^2 C_{n-2}+\cdots+(-1)^n\left(\frac{2}{3}\right)^n C_0\right]

View answer and explanation

Correct option: A

Option A (0) is the correct answer because the limit of the given expression as n approaches infinity is 0.

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