$\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + . \ldots {\rm{is\;equal\;to}}$

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

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

Back to BINOMIAL THEOREM

Your Progress

0 / 434 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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 91-120 of 434 questions

Preparing practice and saved progress…

91 Easy

91 of 434

Analyzing +4 | -1

\alpha \,\,\mathrm{and}\,\,\beta

be the coefficients of

{x^4}\,\,\mathrm{and}\,\,{x^2}

respectively in the expansion of

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

then

View answer and explanation

Correct option: B

\alpha - \beta = - 132

Option B (

\alpha - \beta = - 132

) is correct because it is the result of calculating the coefficients of x⁴ and x² in the given expansion and then subtracting them.

92 Easy

92 of 434

Understanding +4 | -1

The expansion of

\frac{1}{(4-3 x)^{1 / 2}}

, binomial theorem will be valid, if

View answer and explanation

Correct option: D

None of these

None of these is correct because the binomial expansion is valid when

|x| < \frac{4}{3}

93 Easy

93 of 434

Application +4 | -1

Let

P_n

denote the product of all the coefficients in the expansion of

(1+x)^n

. If

(20) ! P_{n+1}=21^{20} P_n

, then

n

is equal to

View answer and explanation

Correct option: B

Option B (20) is correct. By comparing the product of coefficients in consecutive binomial expansions, the value of n is determined.

94 Easy

94 of 434

Understanding +4 | -1

m

and

n

are positive integers, then value of

{ }^m C_0{ }^n C_k+{ }^m C_1{ }^n C_{k-1}+\ldots+{ }^m C_k{ }^n C_0

View answer and explanation

Correct option: A

{ }^{m+n} C_k

Option D (None of these) is correct because the value of

\arg(z) + \arg(\bar{z})

depends on whether z is a negative real number or not. If z is a negative real number, the sum is

2\pi

, otherwise it is 0.

95 Easy

95 of 434

Application +4 | -1

If the coefficient of

rth

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

and

\left( {r + 2} \right)th

terms in the binomial expansion of

{(1 + y)^m}

are in AP, then

m

and

r

satisfy the equation

View answer and explanation

Correct option: B

{m^2} - m\left( {4r + 1} \right) + 4{r^2} - 2 = 0

Option B (

{m^2} - m\left( {4r + 1} \right) + 4{r^2} - 2 = 0

) is correct because it is derived from the condition that the coefficients of the given terms are in Arithmetic Progression (AP).

96 Easy

96 of 434

Application +4 | -1

n

is even, then the middle term in the expansion of

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

is 924

{x^6}

, then

n

is equal to

View answer and explanation

Correct option: B

Option B (12) is correct because the middle term in the expansion of

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

is given as 924

{x^6}

, and solving for n yields n=12.

97 Easy

97 of 434

Application +4 | -1

The ninth term of the expansion

{\left( {3x - \frac{1}{{2x}}} \right)^8}

View answer and explanation

Correct option: D

\frac{1}{{256{x^8}}}

Option D (

\frac{1}{{256{x^8}}}

) is correct because substituting r=8 into the general term of the binomial expansion and simplifying leads to this result.

98 Easy

98 of 434

Analyzing +4 | -1

In the expansion of

2 \log _e x-\log _e(x+1)-\log _e(x-1)

, the coefficient of

x^{-4}

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (

\frac{{1}}{{2}}

) is correct because the coefficient of

x^{-4}

in the expansion is found by using logarithmic properties and series expansion, which leads to the coefficient being

\frac{{1}}{{2}}

99 Easy

99 of 434

Application +4 | -1

If the coefficient of 4 th term in the expansion of

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

is 20 , then the respective values of

\alpha

and

n

are

View answer and explanation

Correct option: D

2,6

The correct answer is D (2,6) because by equating the power of x to zero and comparing the constant term, we can find the values of alpha and n.

100 Easy

100 of 434

Application +4 | -1

The value of

\left\{\frac{3^{2003}}{28}\right\}

, where

\{ .\}

denotes the fractional part function is

View answer and explanation

Correct option: B

19 / 28

Option B (19/28) is correct because the fractional part of

\frac{3^{2003}}{28}

can be found using modular arithmetic and binomial theorem.

101 Easy

101 of 434

Application +4 | -1

The coefficient of

x^m

(1+x)^m+(1+x)^{m+1}+\ldots+(1+x)^n, m \leq n

, is

View answer and explanation

Correct option: A

{ }^{n+1} C_{m+1}

Option A (

{ }^{n+1} C_{m+1}

) is correct because the sum of binomial coefficients can be simplified to a single binomial coefficient using the hockey-stick identity.

102 Easy

102 of 434

Application +4 | -1

The value of the sum

\sum\limits_{{j=0}}^8 \frac{1}{(j+1)(j+2)}\left(\begin{array}{l} 8 \\ j \end{array}\right)

View answer and explanation

Correct option: B

\frac{1013}{90}

Option B (

\frac{1013}{90}

) is correct because by rewriting the sum using binomial coefficient identities and then evaluating, we find the value to be

\frac{1013}{90}

103 Easy

103 of 434

Application +4 | -1

The number of irrational terms in the expansion of

\left(2^{1 / 5}+3^{1 / 10}\right)^{55}

View answer and explanation

Correct option: C

Option C (50) is correct because the number of irrational terms is found by subtracting the number of rational terms from the total number of terms.

104 Easy

104 of 434

Application +4 | -1

The number of distinct terms in the expansion of

(x+y-z)^{16}

View answer and explanation

Correct option: B

153

Option B (153) is correct because the number of distinct terms in the expansion is the sum of integers from 1 to n+1, where n is the power.

105 Easy

105 of 434

Application +4 | -1

In the expansion of

{(2 - 3{x^3})^{20}},

if the ratio of 10th term to 11th term is

\frac{{45}}{{22}}

, then

x

is equal to

View answer and explanation

Correct option: A

- \frac{2}{3}

Option A (

- \frac{2}{3}

) is correct because setting up the ratio of the 10th and 11th terms and solving for x gives

x = - \frac{2}{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

106 Easy

106 of 434

Application +4 | -1

Coefficient of

x^4

in the expansion of

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

View answer and explanation

Correct option: B

\frac{405}{256}

Option B (405/256) is correct. The coefficient of x⁴ is found by determining the correct term in the binomial expansion.

107 Easy

107 of 434

Application +4 | -1

The total number of terms in the expansion of

{\left( {x + a} \right)^{100}} + {\left( {x - a} \right)^{100}}

after simplification will be

View answer and explanation

Correct option: B

Option B (51) is correct because only the even power terms remain after simplification, resulting in half the number of terms plus one.

108 Easy

108 of 434

Analyzing +4 | -1

The value of

\frac{1}{{{{81}^n}}} - \frac{{10}}{{{{81}^n}}}{\;^{2n}}{C_1}\;\frac{{{{10}^2}}}{{{{81}^n}}}{\;^{2n}}{C_2} - \frac{{{{10}^3}}}{{{{81}^n}}}2{\;^{2n}}{C_3} + ... + \frac{{{{10}^{2n}}}}{{{{81}^n}}}\;\mathrm{is}

View answer and explanation

Correct option: D

The correct answer is D (1) because the expression can be recognized as a binomial expansion, which simplifies to 1.

109 Easy

109 of 434

Understanding +4 | -1

The middle term in the expansion of

{\left( {x - a} \right)^8}

View answer and explanation

Correct option: B

{\;^8}{C_4}{x^4}{a^4}

Option B (

{\;^8}{C_4}{x^4}{a^4}

) is correct because the middle term in the expansion of

{\left( {x - a} \right)^8}

is the 5th term, which is

{\;^8}{C_4}{x^4}{\left( { - a} \right)^4} = {\;^8}{C_4}{x^4}{a^4}

110 Easy

110 of 434

Application +4 | -1

The 8 th term of

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

, when expanded in ascending power of

x

, is

View answer and explanation

Correct option: A

\frac{228096}{x^3}

Option A (

\frac{{228096}}{{x^3}}

) is correct because it is the 8th term in the expansion.

111 Easy

111 of 434

Application +4 | -1

Let

{T_n}

denotes the number of triangles which can be formed using the vertices of a regular polygon of

n

sides. If

{T_{n + 1}} - {T_n} = 21

then

n

equals

View answer and explanation

Correct option: B

The correct answer is B (7) because it satisfies the given condition

{T_{n + 1}} - {T_n} = 21

where

{T_n}

represents the number of triangles formed using vertices of a regular polygon.

112 Easy

112 of 434

Application +4 | -1

t_n=n \cdot n

! then

\sum_{n=1}^{30} t_n=

View answer and explanation

Correct option: B

31 !-1

Option B (

31 !-1

) is correct because the summation telescopes, resulting in

31! - 1! = 31! - 1

113 Easy

113 of 434

Application +4 | -1

a > 0, b > 0

and

a^2+b=2

, then the maximum value of the term independent of

x

in the expansion of

\left(a x^{\frac{1}{6}}+b x^{-\frac{1}{3}}\right)^9

View answer and explanation

Correct option: B

Option B (84) is correct because the maximum value of the term independent of x is found by using the binomial theorem and AM >= GM inequality.

114 Easy

114 of 434

Application +4 | -1

The coefficient of

x

in the expansion of

\left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right) \ldots \ldots \left( {1 + 100x} \right)

View answer and explanation

Correct option: A

5050

Option A (5050) is correct because the coefficient of x is the sum of the integers from 1 to 100, which is calculated as

\frac{{100(100 + 1)}}{2} = 5050

115 Easy

115 of 434

Application +4 | -1

The remainder when

7^{2022}+3^{2022}

is divided by 5 is:

View answer and explanation

Correct option: C

The correct answer is C (3) because when

7^{2022}+3^{2022}

is divided by 5, the remainder is 3.

116 Easy

116 of 434

Application +4 | -1

The coefficient of

{x^{ - 7}}

in the expansion of

{\left[ {ax - \frac{1}{{b{x^2}}}} \right]^{11}}

will be

View answer and explanation

Correct option: B

\frac{{462{a^5}}}{{{b^6}}}

Option B is correct because the coefficient of

{x^{-7}}

is found by determining the appropriate term in the binomial expansion and calculating its value.

117 Easy

117 of 434

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

\frac{{\left( {a - bx} \right)}}{{{e^x}}}

View answer and explanation

Correct option: A

\frac{{{{\left( { - 1} \right)}^n}}}{{n!}}\left( {a + bn} \right)

Option A is correct because it represents the coefficient of x^n in the expansion of the given function using the Taylor series expansion of e^(-x).

118 Easy

118 of 434

Application +4 | -1

The expansion of

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

by Only One Correct Option will be valid, if

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the expansion is valid only when

|\frac{3}{4}x| < 1

, which means

-\frac{4}{3} < x < \frac{4}{3}

, and none of the provided options match this range.

119 Easy

119 of 434

Application +4 | -1

x

is so small that

\left(\frac{1+3 x}{1-2 x}\right)^2=1+k x

, then

k=

View answer and explanation

Correct option: C

Option C (10) is correct because it is the coefficient of x in the expansion of the given expression, considering x is very small.

120 Easy

120 of 434

Application +4 | -1

\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + . \ldots \mathrm{is\;equal\;to}

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the given series can be expressed as an integral, and the result simplifies to

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

Revision notes and key takeaways

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

Showing 91-120 of 434 questions

Your Progress

Binomial theorem practice · Easy

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice