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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 1-30 of 951 questions

1 Medium

1 of 951

Application +4 | -1

\left({ }^{10} C_0\right)^2-\left({ }^{10} C_1\right)^2+\ldots-\left({ }^{10} C_9\right)^2+\left({ }^{10} C_{10}\right)^2

equals

View answer and explanation

Correct option: B

-{ }^{10} C_5

Option B (-

{ }^{10} C_5

) is correct because the given expression is the coefficient of

x^{10}

in the expansion of

(1-x^2)^{10}

, which can be found using the binomial theorem.

2 Easy

2 of 951

Analyzing +4 | -1

a

and

d

are two complex numbers, then the sum to

\left( {n + 1} \right)

terms of the following series

a{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - ... + ...

View answer and explanation

Correct option: C

Option C (0) is correct because the given series can be expressed as a combination of binomial coefficients with alternating signs, which sums to zero.

3 Medium

3 of 951

Analyzing +4 | -1

The number of different terms in the expansion of

(1-x)^{201}\left(1+x+x^2\right)^{200}

View answer and explanation

Correct option: D

402

Option D (402) is correct because the expression simplifies to (1-x)(1-x³)²⁰⁰, which has terms from x⁰ to x⁶⁰¹, and considering the (1-x) factor, the number of terms is 2*201 = 402.

4 Medium

4 of 951

Application +4 | -1

The sum of the coefficient of the binomial expansion of

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

is equal to 6561 . The constant term in the expansion is

View answer and explanation

Correct option: B

16 \cdot{ }^8 C_4

Option B (

16 \cdot{ }^8 C_4

) is correct because it calculates the constant term in the binomial expansion after finding the value of n.

5 Easy

5 of 951

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

\frac{{1 + {x^2}}}{{1 + x}},\left| x \right| < 1,

View answer and explanation

Correct option: D

- 2

Option D (-2) is correct because the coefficient of x⁵ in the expansion of (1+x²)/(1+x) is found by multiplying the series expansion of (1+x²) and (1+x)^-1 and identifying the x⁵ term.

6 Medium

6 of 951

Application +4 | -1

\left(\begin{array}{c}2 n+1 \\ 0\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 3\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 6\end{array}\right)+\ldots=170

, then

n

equals

View answer and explanation

Correct option: B

Option B (4) is correct because substituting n=4 into the given equation yields 170, satisfying the condition.

7 Easy

7 of 951

Analyzing +4 | -1

Let

{\left( {1 + x} \right)^n} = \sum \limits_{r = 0}^n {C_r}\;{x^r}

and

\frac{{{C_1}}}{{{C_0}}} + 2\frac{{{C_2}}}{{{C_1}}} + 3\frac{{{C_3}}}{{{C_2}}} + \ldots + n\frac{{{C_n}}}{{{C_{n - 1}}}} = \frac{1}{k}n\left( {n + 1} \right),

then the value of

k

, is

View answer and explanation

Correct option: B

Option B (2) is correct because by simplifying the summation and comparing it to the given expression, we find that k = 2.

8 Medium

8 of 951

Analyzing +4 | -1

S=\sum\limits_{{n=1}}^{\infty}\left(\frac{{ }^n C_0+{ }^n C_1+{ }^n C_2+\ldots+{ }^n C_n}{{ }^n P_n}\right)

, then

S

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the sum evaluates to

e^2 - 1

, which is not among the given options.

9 Easy

9 of 951

Application +4 | -1

The sum of the coefficients in the expansion of

{\left( {1 + 2x + 3{x^2} + ... + n{x^n}} \right)^2}

View answer and explanation

Correct option: D

\sum {n^3}

Option D (

\sum {n^3}

) is correct because substituting x=1 into the given expression directly yields the sum of the coefficients, which simplifies to

(\sum n)^2 = \sum n^3

10 Hard

10 of 951

Application +4 | -1

For natural numbers

m,n

{\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n} = 1 + {a_1}{y^2} +

….and

{a_1} = {a_2} = 10,\;

then

\left( {m,n} \right)

View answer and explanation

Correct option: C

\left( {35,45} \right)

Option C ((35,45)) is correct because these values satisfy the equations derived from the given conditions on the binomial expansion.

11 Easy

11 of 951

Analyzing +4 | -1

The coefficient of the term independent of

x

in the expansion of

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

View answer and explanation

Correct option: A

210

Option A (210) is correct because the coefficient of the term independent of x is found by simplifying the expression and using binomial theorem.

12 Easy

12 of 951

Application +4 | -1

If term independent of

x

in the expansion of

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

\lambda 3^5

, then find the value of

\lambda

View answer and explanation

Correct option: A

Option A (5) is correct because the term independent of x is found by setting the power of x to zero and solving for the binomial coefficient.

13 Easy

13 of 951

Application +4 | -1

a C_0+(a+b) C_1+(a+2 b) C_2+\ldots+(a+n b) C_n

is equal to

View answer and explanation

Correct option: B

(2 a+n b) 2^{n-1}

Option B (

(2 a+n b) 2^{n-1}

) is correct because the summation can be split into two parts, one involving 'a' and the other involving 'b', and then simplified using binomial coefficient identities.

14 Medium

14 of 951

Analyzing +4 | -1

The coefficient of

x^r\{0 \leq r \leq(n-1)\}

in the expansion of

(x+2)^{n-1}+(x+2)^{n-2}(x+1)+(x+2)^{n-3}

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

is equal to

View answer and explanation

Correct option: C

\left(2^{n-r}-1\right)^n C_r

Option C (

\left(2^{{n-r}}-1\right)^n C_r

) is correct because the given series can be simplified to

(x+2)^n - (x+1)^n

, and the coefficient of

x^r

in this expression is

\left(2^{{n-r}}-1\right)^n C_r

15 Medium

15 of 951

Application +4 | -1

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

, then

\frac{\left(C_0+C_1\right)\left(C_1+C_2\right) \ldots\left(C_{n-1}+C_n\right)}{C_1 C_2 \ldots C_n}

equals-

View answer and explanation

Correct option: B

\frac{(n+1)^n}{n !}

Option B (

\frac{{(n+1)^n}}{{n!}}

) is correct because simplifying the expression by using the binomial coefficient formula leads to this result.

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16 Easy

16 of 951

Application +4 | -1

For a positive integer

n

, if the mean of the binomial coefficients in the expansion of

(a+b)^{2 n-3}

is 16, then

n

is equal to:

View answer and explanation

Correct option: A

Option A (5) is correct because the mean of the binomial coefficients is given as 16, and solving the equation derived from this information yields

n=5

17 Medium

17 of 951

Application +4 | -1

The greatest integer less than or equal to

(\sqrt{2}+1)^6

View answer and explanation

Correct option: B

197

Option B (197) is correct because the greatest integer less than or equal to

(\sqrt{2}+1)^6

is found by considering its conjugate and using binomial expansion.

18 Easy

18 of 951

Application +4 | -1

Coefficient of

x^{\prime \prime}

in the expression

\left(1-2 x+3 x^2-4 x^3+\cdots\right)^{-n}

equals

View answer and explanation

Correct option: A

{ }^{2 n} C_n

Option A (

{ }^{2 n} C_n

) is correct because the given expression simplifies to (1+x)⁽²ⁿ⁾, and the coefficient of x^n is

{ }^{2 n} C_n

19 Medium

19 of 951

Analyzing +4 | -1

If the coefficient of

x^{15}

in the expansion of

\left(a x^3+\frac{1}{b x^{1 / 3}}\right)^{15}

is equal to the coefficient of

x^{-15}

in the expansion of

\left(a x^{1 / 3}-\frac{1}{b x^3}\right)^{15}

, where

a

and

b

are positive real numbers, then for each such ordered pair

(a, b)

View answer and explanation

Correct option: A

a b=1

The correct answer is ab=1. By equating the coefficients of x¹⁵ and x^-15 in the given expansions and simplifying, we arrive at the relationship ab=1.

20 Easy

20 of 951

Analyzing +4 | -1

The sum

\sum\limits_{i=0}^m\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}20 \\ m-i\end{array}\right)

, (where

\frac{p}{q}=0

, if

p < q)

is maximum when

m

View answer and explanation

Correct option: C

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

x^m

(1+x)^{30}

, which is maximum when

m = 30/2 = 15

21 Medium

21 of 951

Application +4 | -1

The number of irrational terms in the expansion of

{({5^{1/6}} + {2^{1/8}})^{100}}

View answer and explanation

Correct option: B

Option B (97) is correct because there are 4 rational terms in the expansion, and thus 101 - 4 = 97 irrational terms.

22 Hard

22 of 951

Understanding +4 | -1

{ }^{\mathrm{n}} \mathrm{C}_0-\frac{1}{2}{ }^{\mathrm{n}} \mathrm{C}_1+\frac{1}{3}{ }^{\mathrm{n}} \mathrm{C}_2-\ldots \ldots .+(-1)^{\mathrm{n}} \frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}}{\mathrm{n}+1}=

View answer and explanation

Correct option: C

\frac{1}{n+1}

Option C (

\frac{{1}}{{n+1}}

) is correct because the given expression is the result of integrating the binomial expansion of

(1+x)^n

and then substituting

x = -1

23 Easy

23 of 951

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

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

View answer and explanation

Correct option: B

Option B (60) is correct because the coefficient of

{x^5}

can be found by multiplying the relevant terms in the expansions of

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

and

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

24 Medium

24 of 951

Application +4 | -1

The remainder when

(2021)^{2022}+(2022)^{2021}

is divided by 7 is

View answer and explanation

Correct option: A

Option A is correct because when we divide

(2021)^{2022}+(2022)^{2021}

by 7, the remainder is 0.

25 Medium

25 of 951

Analyzing +4 | -1

The value of

\frac{1+\frac{2^2}{2 !}+\frac{2^4}{3 !}+\frac{2^6}{4 !}+\ldots}{1+\frac{1}{2 !}+\frac{2}{3 !}+\frac{2^2}{4 !}+\ldots}

View answer and explanation

Correct option: C

e^2-1

Option C (

e^2-1

) is correct because the numerator and denominator can be expressed in terms of exponential series.

26 Medium

26 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

a_0+a_2+a_4+\cdots+a_{2 n}

is equal to

View answer and explanation

Correct option: A

\frac{3^n+1}{2}

Option A (

\frac{3^n+1}{2}

) is the correct answer because it is the calculated sum of the even-indexed coefficients.

27 Easy

27 of 951

Analyzing +4 | -1

The maximum value of the term independent of

't'

in the expansion of

{\left( {t{x^{1/5}} + \frac{{{{\left( {1 - x} \right)}^{1/10}}}}{t}} \right)^{10}}

, where

x \in \left( {0,\,\,1} \right)

View answer and explanation

Correct option: B

\frac{{2.10!}}{{3\sqrt 3 {{\left( {5!} \right)}^2}}}

Option B is correct. The maximum value of the term independent of

't'

is found by using the binomial theorem and maximizing the resulting expression using calculus.

28 Easy

28 of 951

Application +4 | -1

The sum of coefficients of the expansion

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

in 6561. The coefficient of term independent of

x

View answer and explanation

Correct option: A

16\;{\;^8}{C_4}

Option A (

16{\;^8}{C_4}

) is correct because it represents the coefficient of the term independent of x in the expansion, calculated by finding the value of r for which the power of x is zero.

29 Medium

29 of 951

Application +4 | -1

The greatest term in the expansion of

(10+3 x)^{12}

, when

x=4

is _____ and its value is _____

View answer and explanation

Correct option: A

\mathrm{T}_8,\left(\begin{array}{c}12 \\ 5\end{array}\right)(120)^5 \times 144

Option A is correct because calculating the ratio of consecutive terms and finding where it becomes less than 1 determines the greatest term, which is T_8.

30 Easy

30 of 951

Application +4 | -1

x>0, y>0

then

\sum\limits_{{r=1}}^{\infty}\left(\frac{x}{2 x+3 y}\right)^{r-1}=

View answer and explanation

Correct option: B

\frac{2 x+3 y}{x+3 y}

Option B (

\frac{{2 x+3 y}}{{x+3 y}}

) is correct because it is derived from summing the infinite geometric series.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 951 single-correct-answer MCQs on BINOMIAL THEOREM.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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