Back to BINOMIAL THEOREM

Your Progress

0 / 951 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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 751-780 of 951 questions

Preparing practice and saved progress…

751 Medium

751 of 951

Analyzing +4 | -1

{ }^{n+1} C_{r+1}:{ }^n C_r:{ }^{n-1} C_{r-1}=11: 6: 3

, then

n r

is equal to

View answer and explanation

Correct option: D

Option D (50) is correct because solving the given ratio equations yields n=10 and r=5, thus nr = 50.

752 Medium

752 of 951

Analyzing +4 | -1

The expression

{\left[ {x + {{\left( {{x^3} - 1} \right)}^{1/2}}} \right]^5} + {\left[ {x - {{\left( {{x^3} - 1} \right)}^{1/2}}} \right]^5}

is a polynomial of degree

View answer and explanation

Correct option: C

Option C (7) is correct because after expanding and simplifying, the highest power of x is 7.

753 Medium

753 of 951

Analyzing +4 | -1

Number of different terms in the expression

(1+x)^{300}+\left(1+x^2\right)^{200}+\left(1+x^3\right)^{150}

are

View answer and explanation

Correct option: A

385

Option A (385) is correct because it is the result of applying the principle of inclusion-exclusion to find the total number of distinct terms.

754 Easy

754 of 951

Application +4 | -1

If the coefficient of

{x^7}

in the expansion of

{\left( {a{x^2} + {b^{ - 1}}{x^{ - 1}}} \right)^{11}}

is equal to the coefficient of

{x^{ - 7}}

{\left( {ax - {b^{ - 1}}{x^{ - 2}}} \right)^{11}},

, then

ab =

View answer and explanation

Correct option: A

Option A (1) is correct because equating the coefficients of

{x^7}

and

{x^{ - 7}}

leads to the relationship

ab = 1

755 Easy

755 of 951

Understanding +4 | -1

The coefficients of

x^{p}

and

x^{q}

in the expansion of

(1+x)^{p+q}

are

View answer and explanation

Correct option: A

equal

Option A is correct because the coefficients of x^p and x^q in the expansion of (1+x)^(p+q) are equal due to the property of binomial coefficients: nCr = nC(n-r).

756 Easy

756 of 951

Analyzing +4 | -1

s_n=\sum\limits_{{r=0}}^n \frac{1}{{ }^n C_r}

and

t_n=\sum\limits_{{r=0}}^n \frac{r}{{ }^n C_r}

, then

\frac{t_n}{s_n}

is equal to

View answer and explanation

Correct option: A

\frac{n}{2}

Option A (

\frac{n}{2}

) is correct because by using the property

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

and manipulating the summations, we find that

\frac{t_n}{s_n} = \frac{n}{2}

757 Medium

757 of 951

Application +4 | -1

The largest real value of

x

, such that

\sum\limits_{r=0}^4\left(\frac{5^{4-r}}{(4-r) !}\right)\left(\frac{x^r}{r !}\right)=\frac{8}{3}

View answer and explanation

Correct option: A

2 \sqrt{2}-5

Option A (

2 \sqrt{2}-5

) is correct because the largest real value of

x

satisfying the given summation can be found by recognizing the summation as a binomial expansion and solving the resulting equation.

758 Medium

758 of 951

Application +4 | -1

The value of

\sum\limits_{{r=1}}^{10} r \cdot \frac{{ }^n C_r}{{ }^n C_{r-1}}

is equal to

View answer and explanation

Correct option: A

5(2 n-9)

Option A,

5(2n-9)

, is correct because it is the simplified form of the summation after applying the formula for combinations and simplifying the expression.

759 Medium

759 of 951

Application +4 | -1

The coefficient of

a^8 b^6 c^4

in the expansion of

(a+b+c)^{18}

View answer and explanation

Correct option: B

13063680

Option B (13063680) is correct because it is the coefficient of x³y⁴z² in the expansion of (2x-3y+4z)⁹, calculated using the multinomial theorem.

760 Medium

760 of 951

Analyzing +4 | -1

The coefficient of the term independent of

x

in the expansion of

\left( {1 + x + 2{x^3}} \right){\left( {\frac{3}{2}{x^2} - \frac{1}{{3x}}} \right)^9}

View answer and explanation

Correct option: C

\frac{{17}}{{54}}

Option C (

\frac{{17}}{{54}}

) is correct because it is the sum of the coefficients of the constant terms obtained by multiplying the terms in the first expression with the appropriate terms in the binomial expansion.

761 Easy

761 of 951

Application +4 | -1

Which term in the expansion of the binomial

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

contains

a

and

b

to one and the same power?

View answer and explanation

Correct option: C

10 th

Option B (6) is correct because the area of the quadrilateral OABC is 6 times the area of the parallelogram formed by vectors OA and OC.

762 Easy

762 of 951

Application +4 | -1

The coefficient of

{x^7}

in the expansion of

{\left( {x - 2\;{x^2}} \right)^{ - 3}},

View answer and explanation

Correct option: C

67584

Option C is correct. The coefficient of x⁷ is found by identifying the correct term in the expansion and calculating its coefficient.

763 Easy

763 of 951

Application +4 | -1

In the expansion of

\left(\sqrt{\frac{q}{p}}+\sqrt[10]{\frac{p^7}{q^3}}\right)^n

, there is a term similar to

p q

, then that term is equal to

View answer and explanation

Correct option: D

252 p q

Option D (

252 p q

) is correct because when the binomial expansion is simplified and the powers of p and q are set to 1, the resulting coefficient is 252.

764 Easy

764 of 951

Understanding +4 | -1

The largest term in the expansion of

\left(\frac{b}{2}+\frac{b}{2}\right)^{100}

View answer and explanation

Correct option: C

{ }^{100} C_{50}\left(\frac{b}{2}\right)^{100}

Option C (

{ }^{100} C_{50}(\frac{b}{2})^{100}

) is correct because the largest term in a binomial expansion occurs at the middle term when the coefficients are equal.

765 Easy

765 of 951

Application +4 | -1

5^{99}

is divided by 13 , the remainder is

View answer and explanation

Correct option: D

Option D (8) is correct because when

5^{99}

is divided by 13, the remainder can be found using modular arithmetic and binomial theorem.

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

766 Medium

766 of 951

Analyzing +4 | -1

In the expansion of

\left(x^{\frac{4}{5}}+x^{-\frac{1}{5}}\right)^n

, the coeficient of the

8^{\text {th }}

and

19^{\text {th }}

terms are equal. The term independent of

x

is given by

View answer and explanation

Correct option: B

{ }^{25} C_{20}

The correct answer is B (

{ }^{25} C_{20}

) because it represents the term independent of x in the binomial expansion, found by setting the power of x to zero and using the binomial coefficient.

767 Medium

767 of 951

Application +4 | -1

Coefficient of

x^{25}

in expansion of expression

\sum\limits_{{r=0}}^{50}{ }^{50} C_r(2 x-3)^r(2-x)^{50-r}

View answer and explanation

Correct option: D

-{ }^{50} \mathrm{C}_{25}

Option D (-

{ }^{50} \mathrm{C}_{25}

) is correct because after simplification, the expression becomes (x-1)⁵⁰, and the coefficient of x²⁵ is -

{ }^{50} \mathrm{C}_{25}

768 Hard

768 of 951

Application +4 | -1

{(1 + 2x + 3{x^2})^{10}} = {a_0} + {a_1}x + {a_2}{x^2} + ... + {a_{20}}{x^{20}},

then

{a_1}

equals

View answer and explanation

Correct option: B

Option B (20) is correct because the coefficient of x in the expansion can be found by considering the term with x or by differentiating the given equation and substituting x=0.

769 Easy

769 of 951

Analyzing +4 | -1

The term independent of

x

in expansion of

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

View answer and explanation

Correct option: C

210

Option C (210) is correct because after simplifying the expression inside the parentheses and applying the binomial theorem, the term independent of x is found to be 210.

770 Medium

770 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 \le r} {\sum\limits_{ < s \le n} {(r + s)} } ({C_r} + {C_s} + {C_r}{C_s})

View answer and explanation

Correct option: B

n^2 \cdot 2^n+\frac{n}{2}\left[2^{2 n}-{ }^{2 n} C_n\right]

Option B (

n^2 \cdot 2^n+\frac{n}{2}[2^{2 n}-{ }^{2 n} C_n]

) is correct because it combines the results from previous questions to find the final answer.

771 Easy

771 of 951

Analyzing +4 | -1

a=99^{50}+100^{50}

and

b=101^{50}

, then

View answer and explanation

Correct option: A

a < b

Option A (

a < b

) is correct because

101^{50}-99^{50}

can be expanded using the binomial theorem, showing that

b

is greater than

a

772 Medium

772 of 951

Application +4 | -1

\,\,\, f(x)=1-x+x^2-x^3+\ldots-x^{15}+x^{16}-x^{17}

. Then, the coefficient of

x^2

f(x-1)

View answer and explanation

Correct option: B

816

Option B (816) is correct because the coefficient of x² in f(x-1) is equal to the coefficient of x³ in 1-(x-1)¹⁸, which is calculated as

{ }^{18} C_3=816

773 Medium

773 of 951

Analyzing +4 | -1

{a_r}

is the coefficient of

{x^{r - 1}}

{\left( {1 + x} \right)^n} + {\left( {1 + x} \right)^{n + 1}} + ... + {\left( {1 + x} \right)^{n + k}}\;\left( {n < r - 1 \le n + k} \right)

, then

\sum \limits_{r = 0}^{n + k + 1} {\left( { - 1} \right)^r}{a_r}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the alternating sum of coefficients in the expansion equals zero.

774 Easy

774 of 951

Application +4 | -1

In the expansion of

{\left( {1 + x} \right)^{50}},

the sum of the coefficient of add power of

x

View answer and explanation

Correct option: B

{2^{49}}

Option B (

{2^{49}}

) is correct because the sum of the coefficients of odd powers of x in the expansion of

{\left( {1 + x} \right)^{50}}

is equal to

{2^{50 - 1}}

775 Easy

775 of 951

Application +4 | -1

5^{40}

is divided by 11 , then remainder is

\alpha

and when

2^{2003}

is divided by 17 , then remainder is

\beta

, then the value of

\beta-\alpha

View answer and explanation

Correct option: C

Option C (7) is correct because after calculating the remainders α and β, their difference is 7.

776 Easy

776 of 951

Application +4 | -1

The number of distinct terms in the expansion of

(x+2 y-3 z+5 w-7 u)^n

View answer and explanation

Correct option: D

None of these

Option D is correct because the number of distinct terms in the expansion is calculated using combinations, and the result does not match the other options.

777 Easy

777 of 951

Analyzing +4 | -1

The remainder, if

1+2+2^2+2^3+\ldots+2^{1999}

is divided by 5 , is

View answer and explanation

Correct option: A

Option A (0) is correct because the sum is a geometric progression that simplifies to a form divisible by 5, leaving no remainder.

778 Medium

778 of 951

Analyzing +4 | -1

Let

x=(8 \sqrt{3}+13)^{13}

and

y=(7 \sqrt{2}+9)^9

. If [f] denotes the greatest integer

\leq t

, then

View answer and explanation

Correct option: B

[x]+[y]

is even

The correct answer is

[x]+[y]

is even because both

[x]

and

[y]

are even, which is shown by considering the conjugate terms and their integer parts.

779 Medium

779 of 951

Application +4 | -1

T_{r+1}

is the term independent of

x

\left(\sqrt{a} x^2+\frac{1}{2 x^3}\right)^{10}

then

r=

View answer and explanation

Correct option: A

Option A (2) is correct because there are two solutions for θ in the given interval that satisfy the trigonometric equation.

780 Easy

780 of 951

Application +4 | -1

If the coefficient of

r th

and

\left( {r + 1} \right)

th terms in the expansion of

{(3 + 7x)^{29}}

are equal, then

r

equals

View answer and explanation

Correct option: B

Option B (21) is correct because it is the value of

r

that makes the coefficients of the

r

th and

(r+1)

th terms equal in the expansion of

{(3 + 7x)^{29}}

Revision notes and key takeaways

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

Showing 751-780 of 951 questions

Your Progress

Binomial theorem practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice