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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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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 121-150 of 434 questions

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121 Easy

121 of 434

Application +4 | -1

The greatest term in the expansion of

{\left( {1 + 3x} \right)^{54}},

where

x = \frac{1}{3}

View answer and explanation

Correct option: A

{T_{28}}

Option A (

{T_{28}}

) is correct because it is the greatest term in the expansion, determined by the given condition and the binomial theorem.

122 Easy

122 of 434

Analyzing +4 | -1

If the fourth term of

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

is equal to 200 and

x>1, x

is equal to

View answer and explanation

Correct option: B

10

Option B (10) is correct because substituting the given condition into the binomial expansion and solving the resulting logarithmic equation yields x = 10.

123 Easy

123 of 434

Analyzing +4 | -1

If the second, third and fourth term in the expansion of

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

are 240,720 and 1080 respectively, then the value of

n

View answer and explanation

Correct option: D

The correct answer is D (5) because by setting up ratios of consecutive terms in the binomial expansion, we can solve for n.

124 Easy

124 of 434

Application +4 | -1

The two values of the rational exponent

m

such that in the binomial expansion of

(1-x)^m

, the coefficient of

x^2

is 3 , are

View answer and explanation

Correct option: C

3,-2

Option C (3, -2) is correct because these are the two values of m that satisfy the given condition for the coefficient of x² in the binomial expansion.

125 Easy

125 of 434

Application +4 | -1

The coefficient of

{a^3}{b^4}c

in the expansion of

{\left( {1 + a - b + c} \right)^9}

is equal to

View answer and explanation

Correct option: D

\frac{{9!}}{{3!4!}}

Option D (9!/(3!4!)) is correct because the coefficient of a³b⁴c in the multinomial expansion of (1+a-b+c)⁹ is calculated using the multinomial coefficient formula.

126 Easy

126 of 434

Analyzing +4 | -1

n

is an integer greater than 1, then

a - {\;^n}{C_1}\left( {a - 1} \right) + {\;^n}{C_2}\left( {a - 2} \right) + ... + {( - 1)^n}\left( {a - n} \right)

is equal to

View answer and explanation

Correct option: B

Option B (0) is correct because the given expression simplifies to 0 using binomial theorem properties.

127 Easy

127 of 434

Application +4 | -1

Last four digits of the number

N=7^{100}-3^{100}

View answer and explanation

Correct option: D

8000

Option D (8000) is correct because the last four digits of the expression

7^{100}-3^{100}

can be found using binomial expansion and modular arithmetic.

128 Easy

128 of 434

Application +4 | -1

If the ratio of the coefficient of third and fourth term in the expansion of

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

1:2

, then the value of

n

will be

View answer and explanation

Correct option: D

- 10

The correct answer is D (-10). The value of n is found by equating the ratio of the coefficients of the third and fourth terms to 1:2 and solving for n.

129 Easy

129 of 434

Understanding +4 | -1

The value of

r

for which

{}^{20}{C_r}\,\,{}^{20}{C_0} + {}^{20}{C_{r - 1}}\,\,{}^{20}{C_1} + {}^{20}{C_{r - 2}}\,\,{}^{20}{C_2} + .... + {}^{20}{C_0}{}^{20}{C_r}

is maximum, is

View answer and explanation

Correct option: D

Option D (20) is correct because the expression represents the coefficient of

x^r

(1+x)^{40}

, which is maximum when r = 20.

130 Easy

130 of 434

Application +4 | -1

The value of

{ }^{15} C_0^2-{ }^{15} C_1^2+{ }^{15} C_2^2-\ldots-{ }^{15} C_{15}^2

View answer and explanation

Correct option: C

Option C (0) is correct because the alternating sum of squares of binomial coefficients is zero when n is odd.

131 Easy

131 of 434

Application +4 | -1

The last digit of the number

(32)^{32}

View answer and explanation

Correct option: B

Option B (6) is correct because the last digit of (32)³² is determined by the last digit of 2³², which repeats in a cycle of 4, and 32 is a multiple of 4, so the last digit is 6.

132 Easy

132 of 434

Application +4 | -1

The coefficient of

x^7

in the expansion o

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

, is

View answer and explanation

Correct option: B

-144

Option B (-144) is correct because the coefficient of

x^7

in the expansion of

(1-x-x^2+x^3)^6

can be found by factoring and then applying the binomial theorem.

133 Easy

133 of 434

Application +4 | -1

For all integers

n \ge 1,

which of the following is divisible by 9?

View answer and explanation

Correct option: B

{4^n} - 3n - 1

Option B (

{4^n} - 3n - 1

) is correct because when expanding

4^n

using the binomial theorem, the expression becomes divisible by 9.

134 Easy

134 of 434

Application +4 | -1

{\;^{20}}{C_4} + 2.{\;^{20}}{C_3} + {\;^{20}}{C_2} - {\;^{22}}{C_{18}}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct. Using the properties of binomial coefficients, the expression simplifies to zero.

135 Easy

135 of 434

Application +4 | -1

The term independent of

x

in the expansion of

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

View answer and explanation

Correct option: D

none of these

Option D (none of these) is correct because the value of r is not a whole number, indicating there's no term independent of x.

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

136 Easy

136 of 434

Application +4 | -1

If the binomial expansion of

{\left( {a + b\;x} \right)^{ - 2}}

\frac{1}{4} - 3\;x + \ldots ,

then

\left( {a,b} \right) =

View answer and explanation

Correct option: A

\left( {2,\;12} \right)

Option A

\left( {2,\;12} \right)

is correct because it satisfies the given binomial expansion condition by comparing coefficients.

137 Easy

137 of 434

Application +4 | -1

The coefficient of

{x^5}

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

View answer and explanation

Correct option: C

Option C (60) is correct because the coefficient of x⁵ is obtained by multiplying the appropriate terms in each expansion and summing the results.

138 Easy

138 of 434

Application +4 | -1

The last two digits of the number

19^{9^4}

View answer and explanation

Correct option: A

Option A (19) is correct because by using the binomial theorem and focusing on the last two digits, we find the remainder when divided by 100 is 19.

139 Easy

139 of 434

Analyzing +4 | -1

The largest integer

\lambda

such that

2^\lambda

divides

3^{2^n}-1, n \in N

View answer and explanation

Correct option: D

n+2

Option D (

n+2

) is correct because the largest integer

\lambda

such that

2^\lambda

divides

3^{2^n}-1

n+2

140 Easy

140 of 434

Application +4 | -1

The ratio of the coefficient of

{x^{15}}

to the term independent of

x

{\left( {{x^2} + \frac{2}{x}} \right)^{15}},

View answer and explanation

Correct option: C

\frac{1}{32}

Option C (1/32) is correct because it represents the ratio of the coefficient of x¹⁵ to the term independent of x after expanding the given binomial expression.

141 Easy

141 of 434

Analyzing +4 | -1

{7^9} + {9^7}

is divided by

View answer and explanation

Correct option: C

Option C (64) is correct because by expressing

{7^9}

and

{9^7}

(8-1)^9

and

(8+1)^7

respectively and using the binomial theorem, we can show that their sum is divisible by 64.

142 Easy

142 of 434

Application +4 | -1

The number of terms in the expansion of

{\left( {1 + 5\sqrt 2 \;x} \right)^9} + {\left( {1 - 5\sqrt 2 \;x} \right)^9},

View answer and explanation

Correct option: A

Option A (5) is correct because after expanding and simplifying the expression, only the even power terms remain, resulting in 5 terms.

143 Easy

143 of 434

Analyzing +4 | -1

If the last term in the binomial expansion of

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

{\left( {\frac{1}{{{3^{5/3}}}}} \right)^{{{\log }_3}8}}

, then the 5th term from the beginning is

View answer and explanation

Correct option: A

210

Option A (210) is correct because after finding n=10, the 5th term can be calculated using the binomial theorem.

144 Easy

144 of 434

Application +4 | -1

\sum \limits_{k = 1}^\infty k{\left( {1 + \frac{1}{n}} \right)^{k - 1}} =

View answer and explanation

Correct option: C

{n^2}

The correct answer is C (

{n^2}

) because the infinite sum can be simplified to the derivative of a geometric series and evaluated.

145 Easy

145 of 434

Application +4 | -1

The coefficient of

{x^4}

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

View answer and explanation

Correct option: A

\frac{{405}}{{256}}

Option A (405/256) is correct because by finding the general term in the binomial expansion and setting the power of x to 4, we can determine the coefficient of x⁴.

146 Easy

146 of 434

Application +4 | -1

The value of

2 C_0+\frac{2^2}{2} C_1+\frac{2^3}{3} C_2+\ldots+\frac{2^{11}}{11} C_{10}

, is

View answer and explanation

Correct option: B

\frac{1}{11}\left(3^{11}-1\right)

Option B (

\frac{1}{11}(3^{11}-1)

) is correct because the given expression can be represented as an integral of a binomial expansion.

147 Easy

147 of 434

Application +4 | -1

The number of non-zero terms in the expansion of

{\left( {1 + 3\sqrt 2 \;x} \right)^9} + {\left( {1 - 3\sqrt 2 \;x} \right)^9},

View answer and explanation

Correct option: C

Option C (5) is correct because when expanding the given expression, only the even powers of

x

remain, resulting in 5 non-zero terms.

148 Easy

148 of 434

Analyzing +4 | -1

C_r={ }^n C_r

, then

C_0-\frac{1}{3} C_1+\frac{1}{5} C_2 \cdots

upto

(n+1)

terms equal

View answer and explanation

Correct option: B

\int\limits_{0}^1\left(1-x^2\right)^n d x

Option B (

\int_0^1\left(1-x^2\right)^n d x

) is correct because the given series can be represented as an integral of

(1-x^2)^n

from 0 to 1.

149 Easy

149 of 434

Analyzing +4 | -1

The sum of the series

1 + \frac{1}{5} + \frac{{1.3}}{{5.10}} + \frac{{1.3.5}}{{5.10.15}} + . \ldots

is equal to

View answer and explanation

Correct option: D

\sqrt {\frac{5}{3}}

Option D (

\sqrt{\frac{5}{3}}

) is correct because the sum of the series can be found by recognizing it as a binomial expansion.

150 Easy

150 of 434

Application +4 | -1

If the sum of the coefficient in the expansion of

{({a^2}{x^2} - 6ax + 11)^{10}}

, where

a

is constant is 1024, then the value of

a

View answer and explanation

Correct option: D

Option D (3) is correct because substituting x=1 into the polynomial gives the sum of the coefficients, which is then solved for a.

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