Back to BINOMIAL THEOREM

Your Progress

0 / 951 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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.

Take a chapter quiz →

Showing 31-60 of 951 questions

31 Easy

31 of 951

Application +4 | -1

If the

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

term in the expansion of

{\left\{ {\sqrt[3]{{\frac{a}{{\sqrt b }}}} + \sqrt {\frac{b}{{\sqrt[3]{a}}}} } \right\}^{21}}

contains

a

and

b

to one and the same power, then the value of

r

, is

View answer and explanation

Correct option: A

Option A (9) is correct because when the powers of a and b are equal in the (r+1)th term, solving the equation yields r=9.

32 Medium

32 of 951

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.

33 Easy

33 of 951

Application +4 | -1

If the third term in the binomial expansion of

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

- \frac{1}{8}{x^2},

then the rational value of

m

, is

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (1/2) is correct because by setting the third term of the binomial expansion equal to -1/8 x², we can solve for m.

34 Medium

34 of 951

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)

35 Medium

35 of 951

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.

36 Medium

36 of 951

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.

37 Easy

37 of 951

Analyzing +4 | -1

Coefficient of

x^{11}

in the expansion of

\left(1+x^2\right)^4\left(1+x^3\right)^7\left(1+x^4\right)^{12}

View answer and explanation

Correct option: C

1113

Option C (1113) is correct because it is the sum of the products of coefficients that result in

x^{11}

when expanding the given expression.

38 Medium

38 of 951

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.

39 Medium

39 of 951

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.

40 Easy

40 of 951

Understanding +4 | -1

Assertion: General term of the expansion

(x+2 y)^9

{ }^9 {C}_{{r}} \cdot 2^{{r}} \cdot {x}^{9-{r}} \cdot {y}^{{r}}

Reason: General term of the expansion

(x+a)^n

is given by

{T}_{{r}+1}={ }^{{n}} {C}_{{r}} \, {x}^{{n}-{r}} \cdot {a}^{{r}}

View answer and explanation

Correct option: A

Assertion is correct, reason is correct; reason is a correct explanation for assertion

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the general term formula is correctly applied and stated.

41 Medium

41 of 951

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.

42 Medium

42 of 951

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.

43 Medium

43 of 951

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.

44 Medium

44 of 951

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

45 Medium

45 of 951

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.

Explore Other Chapters in MATHEMATICS

View All →

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

46 Easy

46 of 951

Application +4 | -1

If the coefficient of

{r^{\mathrm{th}}},{\left( {r + 1} \right)^{\mathrm{th}}}

and

{\left( {r + 2} \right)^{\mathrm{th}}}

terms in the expansion of

{\left( {1 + x} \right)^{14}}

are in A.P., then the value of

r

, is

View answer and explanation

Correct option: A

5,9

Option A (

5,9

) is correct because these values of

r

satisfy the given condition that the coefficients are in A.P.

47 Easy

47 of 951

Understanding +4 | -1

The coefficient of

x^p

and

x^q

(

p

and

q

are positive integers) in the expansion of

(1+x)^{p+q}

are

View answer and explanation

Correct option: A

equal

Option A is correct because the binomial coefficients

{ }^{p+q}C_p

and

{ }^{p+q}C_q

are equal since

{ }^{n}C_r = { }^{n}C_{n-r}

48 Easy

48 of 951

Analyzing +4 | -1

The larger of

{99^{50}} + {100^{50}}

and

{101^{50}}

View answer and explanation

Correct option: C

{101^{50}}

Option C (

{101^{50}}

) is correct because

{(101)^{50}}

is greater than the sum of

{99^{50}}

and

{100^{50}}

after expansion and comparison.

49 Easy

49 of 951

Application +4 | -1

The interval in which

x

must lie so that the greatest term in the expansion of

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

has the greatest coefficient, is

View answer and explanation

Correct option: B

\left( {\frac{n}{{n + 1}},\frac{{n + 1}}{n}} \right)

Option B ((n/(n+1)),((n+1)/n)) is correct because it defines the interval for x such that the term with the greatest coefficient in the expansion of (1+x)⁽²ⁿ⁾ is also the greatest term.

50 Medium

50 of 951

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.

51 Medium

51 of 951

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.

52 Easy

52 of 951

Application +4 | -1

If in the expansion of

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

, the sum of the coefficients of

x^5

and

x^{10}

is 0, then the coefficient of the third term is:

View answer and explanation

Correct option: B

105

Option B (105) is correct because by finding the value of

n

using the given condition and then calculating the third term's coefficient, we get 105.

53 Medium

53 of 951

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.

54 Medium

54 of 951

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.

55 Easy

55 of 951

Understanding +4 | -1

The greatest coefficient in the expansion of

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

View answer and explanation

Correct option: A

{\;^{2n}}{C_n}

Option A (

{\;^{2n}}{C_n}

) is correct because the middle term in the binomial expansion of

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

has the greatest coefficient.

56 Medium

56 of 951

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.

57 Easy

57 of 951

Application +4 | -1

The coefficient of

x^{3}

in the expansion of

3^{x}

View answer and explanation

Correct option: D

\frac{(\log 3)^{3}}{6}

The correct answer is D (

\frac{(\log 3)^{3}}{6}

). The coefficient of

x^3

in the expansion of

3^x

is found using the Taylor series expansion of

e^{x \log 3}

58 Hard

58 of 951

Application +4 | -1

Coefficient of

x^{10}

in the expansion of

\left(1+x^2-x^3\right)^8

View answer and explanation

Correct option: A

476

Option A (476) is correct because the coefficient of

x^{10}

is found by expanding the given expression using the binomial theorem and identifying the relevant terms.

59 Medium

59 of 951

Application +4 | -1

In how many ways 4 coins can be chosen out of 20 coins of ₹1, 15 coins of ₹2 and 10 coins of ₹5?

View answer and explanation

Correct option: D

The correct answer is 15 because it represents the number of ways to choose 4 coins from the given set of coins.

60 Easy

60 of 951

Application +4 | -1

If the coefficients of the second, third and fourth terms in the expansion of

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

are in AP, then

n

is equal to

View answer and explanation

Correct option: A

Option A (7) is correct because when the coefficients of the second, third, and fourth terms are in AP, n can be either 2 or 7, but n=2 is not acceptable, so n=7.

Revision notes and key takeaways

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

Showing 31-60 of 951 questions

Continue Your Practice

📚 Switch Set (ALL)

Chapter Sets

ALL 951 MCQs EASY 434 MCQs HARD 49 MCQs MEDIUM 468 MCQs

0 /951

0✓ 0✗

0% Accuracy