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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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Showing 871-900 of 951 questions

871 Easy

871 of 951

Application +4 | -1

^{n-1} C_r=\left(k^2-3\right) \cdot{ }^n C_{r+1}

then

k

belongs to

View answer and explanation

Correct option: D

(\sqrt{3}, 2]

Option D ((

\sqrt{3}, 2]

) is correct because the given condition leads to

3 < k^2 \leq 4

, which means

k

belongs to the interval

(\sqrt{3}, 2]

872 Easy

872 of 951

Application +4 | -1

The sum of the coefficients of all even degree terms is

x

in the expansion of

{(x + \sqrt {{x^3} - 1} )^6}\, + \,{(x - \sqrt {{x^3} - 1} )^6}\,,

(x > 1)

is equal to

View answer and explanation

Correct option: D

Option D (24) is correct because it accurately calculates the sum of the coefficients of all even degree terms in the given expansion.

873 Easy

873 of 951

Application +4 | -1

The number of terms in the expansion of

(\sqrt{5}+\sqrt[4]{11})^{124}

which are integers, is equal to

View answer and explanation

Correct option: D

Option D (32) is correct because there are 32 integer terms in the expansion, determined by the conditions on r.

874 Medium

874 of 951

Application +4 | -1

The sum of the coefficients

x^{2 r}, r=1,2,3, \ldots

, in the expansion of

(1+x)^n

View answer and explanation

Correct option: B

2^{w-1}-1

Option B (

2^{w-1}-1

) is correct because it represents the sum of coefficients of even powers of x in the binomial expansion.

875 Medium

875 of 951

Application +4 | -1

The number of solutions of the equation

x_1+x_2+x_3+x_4=31, x_1, x_2, x_3, x_4 \in \mathrm{W}

View answer and explanation

Correct option: B

\left(\begin{array}{c}34 \\ 3\end{array}\right)

Option C, 120, is correct because it represents the number of non-negative integer solutions to the equation with the given constraints.

876 Easy

876 of 951

Application +4 | -1

The coefficient of

{x^m}

{\left( {1 + x} \right)^p} + {\left( {1 + x} \right)^{p + 1}} + \ldots + {\left( {1 + x} \right)^n},p \le m \le n

View answer and explanation

Correct option: A

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

The correct answer is

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

because the coefficient of

{x^m}

can be found by manipulating the sum into a single term and then finding the coefficient of

{x^{m+1}}

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

877 Easy

877 of 951

Application +4 | -1

If the ratio of the 7th term from the beginning to the seventh term from the end in the expansion of

{\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^x}

\frac{1}{6}

then

x

, is

View answer and explanation

Correct option: A

Option A (9) is correct because it satisfies the given ratio condition of the 7th terms from the beginning and end of the expansion.

878 Medium

878 of 951

Analyzing +4 | -1

\sum\limits_{0 \le r} {\sum\limits_{ < s \le n} {{(^n}{C_r}{ + ^n}Cs)} }

View answer and explanation

Correct option: C

n 2^n

Option C (

n 2^n

) is correct because the summation can be simplified using binomial theorem properties and combinatorial arguments.

879 Medium

879 of 951

Application +4 | -1

The sum of the coefficients in the polynomial expansion of

\left(1+x-3 x^2\right)^{2163}

View answer and explanation

Correct option: B

-1

Option B (-1) is correct because substituting x=1 into the polynomial results in (1+1-3)²¹⁶³ = (-1)²¹⁶³ = -1.

880 Medium

880 of 951

Application +4 | -1

n

is odd, then

C_0^2 - C_1^2 + C_2^2 - C_3^2 + ... + {\left( { - 1} \right)^n}C_n^2

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because when n is odd, the given expression evaluates to zero due to the properties of binomial coefficients.

881 Easy

881 of 951

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

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

View answer and explanation

Correct option: B

{\left( { - 1} \right)^n}

Option B (

{\left( { - 1} \right)^n}

) is correct because the given expression simplifies to

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

, and the coefficient of

{x^n}

is then

{\left( { - 1} \right)^n}

882 Medium

882 of 951

Application +4 | -1

The term independent of

x

in the expansion of

\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x > 0

, is

\alpha

times the corresponding binomial coefficient. Then,

\alpha

is___.

View answer and explanation

Correct option: A

Option A (1) is correct because after finding the value of r for the term independent of x, alpha is found to be 1.

883 Medium

883 of 951

Application +4 | -1

The greatest term (numerically) in the expansion of

(2+3 x)^9

, when

x=\frac{3}{2}

, is

View answer and explanation

Correct option: C

\frac{7 \times 3^{13}}{2}

Option C (

\frac{{7 \times 3^{{13}}}}{{2}}

) is correct because it is the greatest term calculated using the binomial theorem with the given x value.

884 Medium

884 of 951

Analyzing +4 | -1

The value of

\sum\limits_{{r=0}}^{2 n} r \cdot\left({ }^{2 n} C_r\right) \cdot \frac{1}{(r+2)}

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because evaluating the summation leads to this expression.

885 Medium

885 of 951

Analyzing +4 | -1

The value of

\left(\begin{array}{l}30 \\ 0\end{array}\right)\left(\begin{array}{l}30 \\ 10\end{array}\right)-\left(\begin{array}{l}30 \\ 1\end{array}\right)\left(\begin{array}{l}30 \\ 11\end{array}\right)

+\left(\begin{array}{l}30 \\ 2\end{array}\right)\left(\begin{array}{l}30 \\ 12\end{array}\right)+\ldots+\left(\begin{array}{l}30 \\ 20\end{array}\right)\left(\begin{array}{l}30 \\ 30\end{array}\right)

is equal to

View answer and explanation

Correct option: B

{ }^{30} C_{10}

Option B (

{ }^{30} C_{10}

) is correct because the given expression is the coefficient of

x^{20}

in the product of

(1-x)^{30}

and

(x+1)^{30}

, which simplifies to

(1-x^2)^{30}

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886 Hard

886 of 951

Application +4 | -1

\sum_{r=0}^m{ }^{n+r} \mathrm{C}_n

is equal to :

View answer and explanation

Correct option: A

{ }^{n+m+1} \mathrm{C}_{n+1}

Option A (

{ }^{n+m+1} \mathrm{C}_{n+1}

) is correct because by repeatedly applying the identity

{ }^n \mathrm{C}_r+{ }^n \mathrm{C}_{r+1}={ }^{n+1} \mathrm{C}_{r+1}

, the summation simplifies to the given expression.

887 Medium

887 of 951

Understanding +4 | -1

Let

n

be a positive integer such that

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

, then

\sum\limits_{{r=0}}^{2 n} a_r

View answer and explanation

Correct option: A

3^n

Option A (

3^n

) is correct because substituting x=1 into the given polynomial expression directly yields the sum of the coefficients.

888 Medium

888 of 951

Analyzing +4 | -1

The coefficient of

x^{-5}

in the binomial expansion of

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

, where

x \neq 0,1

View answer and explanation

Correct option: A

Option A (1) is correct because after simplifying the expression and finding the general term, the coefficient of

x^{-5}

is found to be 1 when r = 10.

889 Easy

889 of 951

Application +4 | -1

Find the coefficient of

x^{13}

in the expansion of

(1-x)^5\left(1+x+x^2+x^3\right)^4

View answer and explanation

Correct option: A

Option A (4) is correct because it is the coefficient of

x^{13}

after simplifying the expression

(1-x)^5(1+x+x^2+x^3)^4

890 Medium

890 of 951

Application +4 | -1

The coefficient of

t^{4}

in the expansion of

\left(\frac{1-t^{6}}{1-t}\right)^{3}

View answer and explanation

Correct option: B

Option B (15) is correct because it is the coefficient of t⁴ in the expansion after simplifying the expression.

891 Easy

891 of 951

Analyzing +4 | -1

(1+a x)^n=1+10 x+40 x^2+\ldots

, then value of

\frac{a+n}{a-n}

View answer and explanation

Correct option: D

-7 / 3

Option D (-7 / 3) is correct because by comparing coefficients and solving the resulting equations, we find a=2 and n=5, leading to the correct value of (a+n)/(a-n).

892 Medium

892 of 951

Analyzing +4 | -1

Let

n

be a positive integer such that

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

, then

(n-r) a_r+(2 n-r+1) a_{r-1}, 0 < r < 2 n

View answer and explanation

Correct option: A

(r+1) a_{r+1}

Option A (

(r+1) a_{r+1}

) is correct because it correctly relates the coefficients using differentiation and equating coefficients.

893 Medium

893 of 951

Application +4 | -1

For the natural numbers

m,\,\,n,\,\,\mathrm{if}\,\,{\left( {1 - y} \right)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + ...... + {a_{m + n}}{y^{m + n}}\mathrm{)}

and

{a_1} = {a_2} = 10

, then the value of

(m + n)

is equal to

View answer and explanation

Correct option: D

Option D (80) is correct because by expanding the given expression and equating coefficients, we can solve for m and n, and find m + n = 80.

894 Easy

894 of 951

Application +4 | -1

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

. Then value of

a_0+a_3+a_6 \ldots \ldots

. is equal to

View answer and explanation

Correct option: C

3^{n-1}

Option C (

3^{n-1}

) is correct because by substituting

x=1, \omega, \omega^2

and adding them, we can isolate the desired sum.

895 Easy

895 of 951

Application +4 | -1

If the

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

term in the expansion of

{\left( {\frac{{{a^{1/3}}}}{{{b^{1/6}}}} + \frac{{{b^{1/2}}}}{{{a^{1/6}}}}} \right)^{21}}

has equal exponents of both

a

and

b

, then value of

r

View answer and explanation

Correct option: B

Option 9 is correct. By equating the exponents of a and b in the (r+1)th term, we find r=9.

896 Medium

896 of 951

Application +4 | -1

The coefficient of

x^4

in the expansion of

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

View answer and explanation

Correct option: B

-91

Option B (-91) is correct because the coefficient of x⁴ is calculated by expanding the expression using the binomial theorem and summing the relevant terms.

897 Easy

897 of 951

Analyzing +4 | -1

X = \left\{ {{4^n} - 3n - 1:n \in N} \right\}

and

Y = \left\{ {9\left( {n - 1} \right):n \in N} \right\}

then

X \cup Y

is equal to:

View answer and explanation

Correct option: B

Option B (Y) is correct because

X \subseteq Y

, therefore

X \cup Y = Y

898 Easy

898 of 951

Analyzing +4 | -1

The sum to infinity of the series

1+\frac{2}{3}+\frac{6}{3^2}+\frac{10^3}{3^3}+\frac{14}{3^4}+\ldots .

. , is

View answer and explanation

Correct option: B

Option B (3) is correct because the sum to infinity of the series is 3.

899 Medium

899 of 951

Application +4 | -1

The coefficient of

a^{10} b^7 c^3

in the expansion of

(b c+c a+a b)^{10}

View answer and explanation

Correct option: C

120

Option C, 120, is correct because it is the multinomial coefficient calculated from the given expansion with the correct powers of a, b, and c.

900 Hard

900 of 951

Analyzing +4 | -1

Let

S\left( k \right) = 1 + 3 + 5 + ... + \left( {2k - 1} \right) = 3 + {k^2}

. Then, which of the following is true?

View answer and explanation

Correct option: B

S\left( k \right) \Rightarrow S\left( {k + 1} \right)

Option B (

S(k) \Rightarrow S(k + 1)

) is correct because if we assume

S(k)

is true, then

S(k+1)

is also true, showing the implication holds.

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