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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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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.

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Showing 811-840 of 951 questions

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811 Hard

811 of 951

Application +4 | -1

If the ninth term in the expansion of

{\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{ - 1/8{{\log }_3}\left( {{5^{x - 1}} + 1} \right)}}} \right\}^{10}}

is equal to 180 and

x > 1,

then

x

equals

View answer and explanation

Correct option: B

{\log _5}15

Option B (

{\log _5}15

) is correct because it is the solution to the equation derived from the given binomial expansion and condition.

812 Medium

812 of 951

Application +4 | -1

(1+x)^n=c_0+c_1 x+c_2 x^2+\ldots+c_n x^n

, then

c_0-\frac{1}{2} c_1+\frac{1}{3} c_2-\ldots+(-1)^n \frac{1}{n+1} c_n=

View answer and explanation

Correct option: D

\frac{1}{n+1}

Option D is correct because the given expression can be simplified using integration and the binomial theorem.

813 Medium

813 of 951

Application +4 | -1

The digit at the unit place in the number

{19^{2005}} + {11^{2005}} - {9^{2005}}

View answer and explanation

Correct option: B

Option B (1) is correct because analyzing the unit digits of each term reveals that the unit digit of the entire expression is 1.

814 Hard

814 of 951

Application +4 | -1

\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\cdots+a_{2 n} x^{2 n}

, then the value of

a_0+a_3+a_6+\cdots

View answer and explanation

Correct option: A

a_1+a_4+a_7+\ldots

Option A (

a_1+a_4+a_7+\ldots

) is correct because by substituting complex roots of unity into the given expression, we can show the equality of the sums.

815 Medium

815 of 951

Application +4 | -1

If coefficients of

2 \mathrm{nd}, 3 \mathrm{rd}

and 4 th terms in the expansion of

(1+x)^{2 n}

are in AP, then

View answer and explanation

Correct option: B

2 n^2-9 n+7=0

Option B (

2 n^2-9 n+7=0

) is correct because the coefficients being in AP leads to the given quadratic equation after simplification.

816 Easy

816 of 951

Application +4 | -1

x^2-x+1=0

, then

\sum\limits_{{n=1}}^5\left(x^n+\frac{1}{x^n}\right)=

View answer and explanation

Correct option: A

Option A (8) is correct because it represents the sum of the given expression for the roots of the quadratic equation.

817 Easy

817 of 951

Application +4 | -1

A positive value of

{m}

for which the coefficient of

x^2

in the expansion

(1+x)^m

is 6, is

View answer and explanation

Correct option: A

Option A (4) is correct because the coefficient of

x^2

in the expansion of

(1+x)^m

being 6 leads to the equation

\frac{m(m-1)}{2} = 6

, which gives

m=4

818 Medium

818 of 951

Application +4 | -1

The coefficient of

x^{10}

in the expansion of

\left(3 x^2-\frac{1}{x^2}\right)^{15}

View answer and explanation

Correct option: B

-\frac{15 !}{10 ! 5 !} 3^{10}

Option B (

-\frac{15 !}{10 ! 5 !} 3^{10}

) is correct because it represents the coefficient of

x^{10}

in the binomial expansion of

\left(3 x^2-\frac{1}{x^2}\right)^{15}

, calculated using the binomial theorem.

819 Easy

819 of 951

Application +4 | -1

The coefficient of

\frac{1}{x}

in the expansion of

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

View answer and explanation

Correct option: B

{\;^{2n}}{C_{n - 1}}\;

Option B (

\;{\;^{2n}}{C_{n - 1}}\;

) is correct because simplifying the expression and finding the coefficient of 1/x leads to this result.

820 Easy

820 of 951

Application +4 | -1

If the ratio of the coefficient of

x^{10}

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

and term independent of

x

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

\frac{p}{q}

(where

{p}

and

{q}

are relative prime numbers), then

\frac{{q}}{8}-{p}

View answer and explanation

Correct option: A

Option A (3) is correct because the ratio of the coefficients simplifies to 1/32, leading to p=1, q=32, and (q/8)-p = 3.

821 Medium

821 of 951

Application +4 | -1

There are at least 10 books of each of the subjects Mathematics, Physics and Chemistry. In how many ways 5 books can be chosen from these books?

View answer and explanation

Correct option: A

The correct answer is 21 because it represents the number of ways to choose 5 books from the three subjects.

822 Medium

822 of 951

Application +4 | -1

1+x^4+x^5=\sum\limits_{{i=0}}^5 a_i(1+x)^i

, for all

x

{R}

, then

a_2

is:

View answer and explanation

Correct option: A

-4

Option A (-4) is correct because by substituting

x

with

x-1

and comparing coefficients, we can find the value of

a_2

823 Medium

823 of 951

Analyzing +4 | -1

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

, then

\sum\limits_{0 \le i} {\sum\limits_{ < j \le n} {{{({C_i}{ + C_j})}^2}} }

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

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

824 Medium

824 of 951

Application +4 | -1

The coefficient of

x^{7}

in the expansion of

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

View answer and explanation

Correct option: B

-144

Option B (-144) is correct because after factoring and expanding, the coefficient of x⁷ is -144.

825 Medium

825 of 951

Application +4 | -1

The term independent of

x

{\left\{ {\sqrt {\frac{x}{3}} + \frac{3}{{2\;{x^2}}}} \right\}^{10}},

View answer and explanation

Correct option: C

\frac{5}{4}

Option C (

\frac{5}{4}

) is the correct answer because it represents the term independent of x after expanding the binomial and setting the power of x to zero.

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826 Easy

826 of 951

Application +4 | -1

\mathrm{\;}{x^{2r}}

occurs in

\mathrm{\;}{\left( {x + \frac{2}{{{x^2}}}} \right)^n}

, then

\mathrm{\;}n - 2r

must be of the form

View answer and explanation

Correct option: B

3k

Option B (3k) is correct because the general term of the binomial expansion leads to the conclusion that n-2r must be a multiple of 3.

827 Easy

827 of 951

Application +4 | -1

The sum of the coefficient in the expansion of

{(1 + x - 3{x^2})^{3148}}\;

View answer and explanation

Correct option: C

Option C (1) is correct because the sum of the coefficients is found by substituting x = 1 into the given expression.

828 Medium

828 of 951

Analyzing +4 | -1

The coefficient of

\lambda^n \mu^n

in the expansion of

(1+\lambda)^n(1+\mu)^n(\lambda+\mu)^n

View answer and explanation

Correct option: D

\sum\limits_{{r=0}}^n\left({ }^n C_r\right)^3

Option D (

\sum\limits_{{r=0}}^n\left({ }^n C_r\right)^3

) is correct because the coefficient of

\lambda^n \mu^n

in the expansion is the sum of the cubes of the binomial coefficients.

829 Medium

829 of 951

Application +4 | -1

|x| < 1

, then the coefficient of

x^n

in expansion of

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

View answer and explanation

Correct option: D

n+1

Option D (

n+1

) is correct because the coefficient of

x^n

in the expansion of

(1-x)^{-2}

n+1

830 Easy

830 of 951

Application +4 | -1

The reminder of

3^{2003}

when divided by 28

View answer and explanation

Correct option: A

Option A (19) is correct because after simplifying the expression using modular arithmetic, the remainder is found to be 19.

831 Easy

831 of 951

Application +4 | -1

If the coefficient of

(3 r+4)

th term and

(2 r-2)

th term in the expansion of

(1+x)^{20}

are equal then

r

is equal to:

View answer and explanation

Correct option: A

Option A (4) is correct because if the coefficients of the (3r+4)th and (2r-2)th terms in the expansion of (1+x)²⁰ are equal, then r = 4.

832 Easy

832 of 951

Analyzing +4 | -1

For

r=0,1,2, \ldots, 10

, let

A_r, B_r

and

C_r

denote respectively, the coefficients of

x^r

in the expansion of

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

and

(1+x)^{30}, \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 by substituting the binomial coefficients and simplifying the summation, the expression reduces to

C_{10}-B_{10}

833 Hard

833 of 951

Application +4 | -1

\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2 \ldots \ldots \mathrm{C}_{\mathrm{n}}

are binomial coefficients in the expansion of

(1+\mathrm{x})^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}

, then

\frac{C_1}{2}+\frac{C_3}{4}+\frac{C_5}{6}+\ldots \ldots+\frac{C_b}{n+1}

is equal to -

View answer and explanation

Correct option: C

\frac{2^{\mathrm{n}}-1}{\mathrm{n}+1}

Option C (

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

) is correct because integrating the binomial expansion and then using specific values for x allows us to isolate the desired sum.

834 Medium

834 of 951

Application +4 | -1

If sum of middle terms is

S

in the expansion of

\left(2 a-\frac{a^2}{4}\right)^9

, then the value(s) of

S

is/are

View answer and explanation

Correct option: D

\left(\frac{63}{32}\right) a^{13}(8-a)

Option D is correct because the sum of the middle terms in the binomial expansion is calculated by finding T5 and T6, then simplifying the expression.

835 Easy

835 of 951

Application +4 | -1

In the expansion of

{\left( {1 + x} \right)^{50}}

, the sum of the coefficients of odd powers of

x

View answer and explanation

Correct option: B

{2^{49}}

Option B (

{2^{49}}

) is correct because the sum of the coefficients of odd powers of x in the expansion of

{\left( {1 + x} \right)^{50}}

is equal to

{2^{50 - 1}} = {2^{49}}

836 Medium

836 of 951

Application +4 | -1

The coefficient of

{x^4}

in the expansion of

{\left( {1 + x + {x^3} + {x^4}} \right)^{10}},

View answer and explanation

Correct option: D

310

Option D (310) is correct because the coefficient of x⁴ is found by multiplying the relevant terms in the expansion of (1+x)¹⁰ and (1+x³)¹⁰.

837 Easy

837 of 951

Application +4 | -1

Coefficient of

{x^{ - 4}}

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

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the calculated coefficient of x^-4 does not match any of the given options. The correct coefficient is

\frac{{3^{12} \times 5}}{{{{2^8}}}}

838 Medium

838 of 951

Application +4 | -1

If the coefficients of

{x^3}

and

{x^4}

in the expansion of

(1 + ax + b{x^2}){(1 - 2x)^{18}}

in powers of

x

are both zero, then

(a,\,b)

is equal to

View answer and explanation

Correct option: D

\left( {16,\,\frac{{272}}{3}} \right)

Option D

\left( {16,\frac{{272}}{3}} \right)

is correct because solving the equations formed by setting the coefficients of

x^3

and

x^4

to zero yields these values for a and b.

839 Easy

839 of 951

Application +4 | -1

In what direction a line be drawn through the point

(1,2)

so that it point of intersection with the line

x+y=4

is at a distance

\frac{\sqrt{6}}{3}

from the given point?

View answer and explanation

Correct option: D

75^{\circ}

Option D (75°) is correct, but the provided explanation is insufficient and lacks proper steps. A more detailed solution is needed to confirm this answer.

840 Medium

840 of 951

Application +4 | -1

Coefficient of

x^8

in the expansion of

\left(1+2 x-x^3\right)^6

View answer and explanation

Correct option: A

528

Option A (528) is correct because it is the sum of the coefficients of x⁸ obtained from the multinomial expansion of the given expression.

Revision notes and key takeaways

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

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