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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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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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Showing 61-90 of 434 questions

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61 Easy

61 of 434

Application +4 | -1

\,\left| x \right| < 1,\;\mathrm{then\;}\,1 + n\left( {\frac{{2x}}{{1 + x}}} \right) + \frac{{n\left( {n + 1} \right)}}{{2!}}{\left( {\frac{{2x}}{{1 + x}}} \right)^2} +

…is equal to

View answer and explanation

Correct option: D

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

The correct answer is D (

{(\frac{{1 + x}}{{1 - x}})^n}

) because it is the simplified form of the given infinite series using the binomial theorem for negative exponents.

62 Easy

62 of 434

Analyzing +4 | -1

n

is the number of irrational terms in the expansion of

{\left( {{3^{\frac{1}{4}}} + {5^{\frac{1}{8}}}} \right)^{60}}

, then

(n - 1)

is divisible by:

View answer and explanation

Correct option: B

Option B (26) is correct because after calculating the number of irrational terms, subtracting 1 gives a number divisible by 26.

63 Easy

63 of 434

Application +4 | -1

The greatest term in the expansion of

(3+5 x)^{15}

, when

x=1 / 5

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the greatest term is

{ }^{15} C_4(3)^{11}

, which is not present in the given options.

64 Easy

64 of 434

Analyzing +4 | -1

\{ p\}

denotes the fractional part of the number p, then

\left\{ {{{\frac{3}{8}}^{200}}} \right\}

is equal to

View answer and explanation

Correct option: D

\frac{1}{8}

Option D is correct because by using the binomial theorem and properties of fractional parts, the fractional part of the given expression is

\frac{1}{8}

65 Easy

65 of 434

Application +4 | -1

If in the expansion of

{\left( {1 + ax} \right)^n},n \in N,

the coefficients of

x

and

{x^2}

are 8 and 24 respectively, then

View answer and explanation

Correct option: A

a = 2,n = 4

Option A (

a = 2, n = 4

) is correct because the coefficients of x and x² in the binomial expansion of

(1 + ax)^n

are given as 8 and 24 respectively, leading to the solution a = 2 and n = 4.

66 Easy

66 of 434

Application +4 | -1

The remainder when

{3^{2022}}

is divided by 5 is:

View answer and explanation

Correct option: D

Option D (4) is correct because when

{3^{2022}}

is divided by 5, the remainder is 4, as determined by analyzing the pattern of powers of 3 modulo 5.

67 Easy

67 of 434

Application +4 | -1

If the absolute term in the expansion of

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

is 405 , then

k

is equal to

View answer and explanation

Correct option: B

\pm 3

Option B (

\pm

3) is correct because these values of k make the absolute term in the expansion equal to 405.

68 Easy

68 of 434

Analyzing +4 | -1

\sum\limits_{r=0}^n(-1)^r{ }^n C_r\left[\frac{1}{2^r}+\frac{3^r}{2^{2 r}}+\frac{7^r}{2^{3 r}}+\ldots\right.

upto

m

terms

]

=f(n)\left(1-\frac{1}{2^{m n}}\right)

\int_{-3}^3 f\left(x^3 \ln x\right) d\left(x^3 \ln x\right)

is equal to

View answer and explanation

Correct option: D

Cannot be determined

Option D (Cannot be determined) is correct because the integral cannot be calculated due to the presence of

\ln x

with a lower limit of integration at -3, which is outside the domain of the natural logarithm.

69 Easy

69 of 434

Analyzing +4 | -1

The term independent of

x

in the expansion of

{\left( {x - \frac{1}{x}} \right)^4}{\left( {x + \frac{1}{x}} \right)^3}\mathrm{\;}

View answer and explanation

Correct option: B

Option B (0) is correct because when the expression is expanded, there is no term independent of x.

70 Easy

70 of 434

Application +4 | -1

The lowest integer which is greater than

{\left( {1 + \frac{1}{{{{10}^{100}}}}} \right)^{{{10}^{100}}}}

View answer and explanation

Correct option: A

Option A (3) is correct because the expression approaches 'e' as the exponent increases, and since 'e' is between 2 and 3, the lowest integer greater than the expression is 3.

71 Easy

71 of 434

Analyzing +4 | -1

If the second term in the expansion of

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

14 a^{5 / 2}

, the value of

\frac{{ }^n C_3}{{ }^n C_2}

View answer and explanation

Correct option: A

Option A (4) is correct because solving for n in the given condition and then calculating the ratio of binomial coefficients results in 4.

72 Easy

72 of 434

Analyzing +4 | -1

The sum

\sum \limits_{i = 0}^m \left( {\begin{array}{cccccccccccccccccccc} {10}\\ i \end{array}} \right)\left( {\begin{array}{cccccccccccccccccccc} {20}\\ {m - i} \end{array}} \right),\;\;\left( {\mathrm{where},\;\left( {\begin{array}{cccccccccccccccccccc} \mathrm{p}\\ q \end{array}} \right) = 0\;\mathrm{if}\;p < q} \right)

is maximum, when

m

View answer and explanation

Correct option: C

Option C (15) is correct because the sum is the coefficient of

x^m

(1+x)^{30}

, which is maximum when

m = 30/2 = 15

73 Easy

73 of 434

Application +4 | -1

The value of

{\left( {1.002} \right)^{12}}

upto fourth place of decimal is

View answer and explanation

Correct option: A

1.0242

Option A (1.0242) is correct because it is the value of (1.002)¹² calculated up to the fourth decimal place using the binomial theorem.

74 Easy

74 of 434

Application +4 | -1

The number of rational terms in the expansion of

\left(5^{1 / 6}+7^{1 / 9}\right)^{1824}

View answer and explanation

Correct option: B

102

Option B (102) is correct because it counts the number of terms in the binomial expansion that result in rational numbers, which occurs when the powers of 5^(1/6) and 7^(1/9) are integers.

75 Easy

75 of 434

Application +4 | -1

If the number of terms in

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

is 401 , then

n

is greater than

View answer and explanation

Correct option: C

199

Option C (199) is correct because the number of terms in the expansion is related to n, and solving for n shows it's greater than 199.

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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

76 Easy

76 of 434

Application +4 | -1

\frac{{{\;^8}{C_0}}}{6} = {\;^8}{C_1} + {\;^8}{C_2}.6 - {\;^8}{C_3}{.6^2} + \ldots + {\;^8}{C_8}{.6^7}\;\mathrm{is\;equal\;to}

View answer and explanation

Correct option: D

\frac{{{5^8}}}{6}

Option D (5⁸/6) is correct because the given expression can be simplified using the binomial theorem.

77 Easy

77 of 434

Understanding +4 | -1

The coefficient of the middle term in the expansion of

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

, is

View answer and explanation

Correct option: A

\frac{{1.3.5 \ldots \left( {2n - 1} \right)}}{{n\;!}}{2^n}

Option A is correct because the coefficient of the middle term in the expansion of

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

is derived using binomial theorem and simplification.

78 Easy

78 of 434

Analyzing +4 | -1

If the coefficients of

{x^2}

and

{x^3}

are both zero, in the expansion of the expression

\left( {1 + ax + b{x^2}} \right){(1 - 3x)^{15}}

in powers of

x,

then the ordered pair

(a,b)

is equal to

View answer and explanation

Correct option: A

\left( {28,315} \right)

Option A (

\left( {28,315} \right)

) is correct because by setting the coefficients of

x^2

and

x^3

to zero, we can solve the resulting system of equations to find

a = 28

and

b = 315

79 Easy

79 of 434

Application +4 | -1

a, b, c

are odd positive integers, then number of integral solutions of

a+b+c=13

, is

View answer and explanation

Correct option: A

Option A (21) is correct because it represents the number of non-negative integer solutions to the transformed equation.

80 Easy

80 of 434

Application +4 | -1

\cot 7 \frac{1^{\circ}}{2}=\sqrt{n_1}+\sqrt{n_2}+\sqrt{n_3}+\sqrt{n_4}

(where

n_1, n_2

n_3, n_4

are natural numbers) then the value of

\sum_{i=1}^4 n_i

View answer and explanation

Correct option: A

Option D (8) is correct because the area of the quadrilateral formed by the foci of the two ellipses is 8 square units.

81 Easy

81 of 434

Application +4 | -1

If in the expansion of

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

, the coefficient of

r\,th

and

\left( {r + 2} \right)

th term be equal, then

r

is equal to

View answer and explanation

Correct option: C

\frac{n}{2}

Option C (

\frac{n}{2}

) is correct because if the coefficients of the r-th and (r+2)-th terms are equal, then

{\;^n}{C_{r - 1}} = {\;^n}{C_{r + 1}}

, leading to

r = \frac{n}{2}

82 Easy

82 of 434

Analyzing +4 | -1

If the coefficients of three consecutive terms in the expansion of

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

are in the ratio

1:7:42

, then the value of

n

View answer and explanation

Correct option: C

Option D (None of these) is correct because the value of n is 55, which is not among the provided options.

83 Easy

83 of 434

Application +4 | -1

If the sum of the coefficients in the expansion of

\left(\alpha x^2-2 x+1\right)^{35}

is equal to the sum of the coefficients in the expansion of

(x-\alpha y)^{35}

, then

\alpha

is equal to

View answer and explanation

Correct option: C

Option C (1) is correct because equating the sum of coefficients and solving the resulting equation leads to alpha = 1.

84 Easy

84 of 434

Application +4 | -1

The sum of coefficient in the expansion of

(1-x)^{10}

View answer and explanation

Correct option: B

Option B (0) is correct because when x=1, (1-x)¹⁰ becomes (1-1)¹⁰ = 0¹⁰ = 0, which is the sum of the coefficients.

85 Easy

85 of 434

Application +4 | -1

If the sum of the coefficient in the expansion of

{\left( {x - 2y + 3z} \right)^n}

is 128, then the greatest coefficient in the expansion of

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

View answer and explanation

Correct option: A

Option A (35) is correct because the sum of coefficients gives n=7, and the greatest coefficient in (1+x)⁷ is 7C3 or 7C4, which equals 35.

86 Easy

86 of 434

Analyzing +4 | -1

Let

f(x)=|x-a|, a \neq 0

then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because none of the given equations hold true for a general

a \neq 0

when

f(x) = |x-a|

87 Easy

87 of 434

Application +4 | -1

5^{81}+\lambda

is divisible by 26, then minimum positive value of

\frac{\lambda}{3}

must be

View answer and explanation

Correct option: A

Option A (7) is correct because by using the binomial theorem and divisibility rules, the minimum positive value of lambda/3 is found to be 7.

88 Easy

88 of 434

Application +4 | -1

The coefficient of

{x^6}\;{a^{ - 2}}

in the expansion of

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

View answer and explanation

Correct option: C

Option C (0) is correct because the powers of x and a in the general term of the expansion cannot simultaneously satisfy the conditions for x⁶ and a^-2.

89 Easy

89 of 434

Understanding +4 | -1

The expansion of

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

by binomial theorem 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 when

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

, which means

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

. None of the provided options match this condition.

90 Easy

90 of 434

Application +4 | -1

If the number of integral terms in the expansion of

{\left( {{3^{\frac{1}{2}}} + {5^{\frac{1}{8}}}} \right)^n}

is exactly 33, then the least value of

n

View answer and explanation

Correct option: C

256

Option C (256) is correct because it is the least value of n that results in exactly 33 integral terms in the binomial expansion.

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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