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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Medium

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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 468 questions

1 Medium

1 of 468

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 Medium

2 of 468

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.

3 Medium

3 of 468

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.

4 Medium

4 of 468

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.

5 Medium

5 of 468

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.

6 Medium

6 of 468

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

7 Medium

7 of 468

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.

8 Medium

8 of 468

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.

9 Medium

9 of 468

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.

10 Medium

10 of 468

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.

11 Medium

11 of 468

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.

12 Medium

12 of 468

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.

13 Medium

13 of 468

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.

14 Medium

14 of 468

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.

15 Medium

15 of 468

Application +4 | -1

The term independent of

x

in the expansion of

{\left( {{x^3} + \frac{2}{{{x^2}}}} \right)^{15}}\mathrm{is}

View answer and explanation

Correct option: D

{T_{10}}

Option D (T_{10}) is correct. The term independent of x in the expansion of (x³ + 2/x²)¹⁵ is T_{10}, obtained by setting the power of x to zero.

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16 Medium

16 of 468

Analyzing +4 | -1

Sum of the series

S=n^2\left(\frac{C_0}{C_1}\right)+(n-1)^2\left(\frac{C_1}{C_2}\right)+\cdots+1^2\left(\frac{C_{n-1}}{C_n}\right)

equals

View answer and explanation

Correct option: C

\frac{1}{6} n(n+1)(n+2)

Option C (

\frac{1}{6} n(n+1)(n+2)

) is correct because the given series can be simplified using the identity

\frac{C_k}{C_{k+1}}=\frac{k+1}{n-k}

and then evaluated to

\frac{1}{6} n(n+1)(n+2)

17 Medium

17 of 468

Analyzing +4 | -1

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

, then the value of

\sum\limits_{{0 \leq r}} \sum\limits_{{< s \leq n}}(r+s)\left(C_r+C_s\right)

View answer and explanation

Correct option: A

n^2 \cdot 2^n

Option A (

n^2 \cdot 2^n

) is correct because it uses the properties of binomial coefficients and summation to arrive at the simplified result.

18 Medium

18 of 468

Application +4 | -1

The sum of the series

2.{}^{20}{C_0} + 5.{}^{20}{C_1} + 8.{}^{20}{C_2} + 11.{}^{20}{C_3} + .... + 62.{}^{20}{C_{20}}\,

is equal to

View answer and explanation

Correct option: B

{2^{25}}

Option B is correct because it accurately calculates the sum of the series using properties of binomial coefficients.

19 Medium

19 of 468

Analyzing +4 | -1

Let

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

and

g(x)=b_0+b_1 x+b_2 x^2 +\ldots+b_{n-1} x^{n-1} +x^n+x^{n+1}+\ldots+x^{2 n}

f(x)=g(x+1)

, then

a_n

in terms of

n

is equal to

View answer and explanation

Correct option: A

{ }^{2 n+1} C_{n+1}

Option B (

{ }^{2 n+1} C_{n+1}

) is correct because it represents the coefficient of

x^n

after transforming the given polynomial expression using binomial theorem.

20 Medium

20 of 468

Analyzing +4 | -1

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

when written in the ascending power of

x

, then the highest exponent of

x

\lambda(1010)

, where

\lambda

equals___.

View answer and explanation

Correct option: A

Option A (5) is correct because the highest exponent is the sum of the exponents from 1 to 100, which is 5050, and 5050 divided by 1010 is 5.

21 Medium

21 of 468

Analyzing +4 | -1

The sum

1 \cdot{ }^{20} C_1-2 \cdot{ }^{20} C_2+3 \cdot{ }^{20} C_3-\ldots-20 \cdot{ }^{20} C_{20}

is equal to

View answer and explanation

Correct option: B

Option B (0) is correct because the sum is equal to zero.

22 Medium

22 of 468

Application +4 | -1

The sum of the coefficients in the expansion of

(5 x-4 y)^n

, where

n

is a positive integer, is

View answer and explanation

Correct option: C

Option C (1) is correct because the sum of the coefficients is obtained by setting x=1 and y=1 in the expression.

23 Medium

23 of 468

Application +4 | -1

\sum_{\mathrm{r}=0}^{\mathrm{n}}\left(\frac{\mathrm{r}+2}{\mathrm{r}+1}\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=\frac{2^8-1}{6}

; then

\mathrm{n}

is -

View answer and explanation

Correct option: D

Option D (5) is correct because solving the summation and equating it to the given value leads to n=5.

24 Medium

24 of 468

Application +4 | -1

f(x)=x^n

, then the value of

f(1)+\frac{f^1(1)}{1 !}+\frac{f^2(1)}{2 !}

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

, where

f^r(x)

denotes the

r

th order derivative of

f(x)

with respect to

x

, is

View answer and explanation

Correct option: B

2^n

Option B (

2^n

) is correct because the given expression is equivalent to the binomial expansion of

(1+1)^n

25 Medium

25 of 468

Application +4 | -1

(5+2 \sqrt{6})^{\mathrm{n}}=\mathrm{I}+\mathrm{f} ; \mathrm{n}, \mathrm{I} \in \mathrm{N}

; and

0 \leq \mathrm{f}<1

, then

\mathrm{I}

equals-

View answer and explanation

Correct option: D

\frac{1}{1-\mathrm{f}}-\mathrm{f}

Option D (

\frac{1}{1-f}-f

) is correct because it is derived from the properties of conjugate surds and their integer part.

26 Medium

26 of 468

Analyzing +4 | -1

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

, then the value of

{\sum\limits_{0 \le r} {\sum\limits_{ < s \le n} {({C_r} \pm {C_s})} } ^2}

View answer and explanation

Correct option: D

(n \mp 1)^{2 n} C_n \pm 2^{2 n}

Option D is correct because the expression simplifies to the given result using combinatorial identities and properties of binomial coefficients.

27 Medium

27 of 468

Application +4 | -1

If the term independent of

x

in the expansion of

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

\lambda

, then

\frac{\lambda}{81}

is equal to

View answer and explanation

Correct option: C

7920

Option C (7920) is correct because it is the value of

\frac{\lambda}{81}

calculated from the term independent of

x

in the binomial expansion.

28 Medium

28 of 468

Application +4 | -1

The two successive terms in the expansion of

(1+x)^{24}

whose coefficients are in the ratio

4: 1

are

View answer and explanation

Correct option: C

5^{\text {th }}

and

6^{\text {th }}

Option C (

5^{\text {th }}

and

6^{\text {th }}

) is correct because by setting up the ratio of consecutive binomial coefficients, we can solve for the term number.

29 Medium

29 of 468

Application +4 | -1

\sum\limits_{{r=0}}^n r\left(\begin{array}{l}n \\ r\end{array}\right)\left(\frac{2}{7}\right)^r\left(\frac{5}{7}\right)^{n-r}=

View answer and explanation

Correct option: C

\frac{2 n}{7}

Option C (

\frac{{2 n}}{{7}}

) is correct because it is derived from differentiating the binomial expansion and substituting the given values.

30 Medium

30 of 468

Application +4 | -1

The sum of the coefficients of all odd degree terms in the expansion of

{\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5},\left( {x > 1} \right)

View answer and explanation

Correct option: D

Option D (

\frac{9}{2}

) is correct because it satisfies the condition for orthogonal intersection of the two curves.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 468 single-correct-answer MCQs on BINOMIAL THEOREM.
✓ Try each question before opening its answer. Read the explanation where one is available.
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