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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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241 Medium

241 of 468

Application +4 | -1

The coefficient of

x^3

\left(\sqrt{x^5}+\frac{3}{\sqrt{x^3}}\right)^6

View answer and explanation

Correct option: D

540

Option D (540) is the correct answer because it is the coefficient of

x^3

after expanding and simplifying the given expression.

242 Medium

242 of 468

Application +4 | -1

r

and

n

are positive integers

r>1, n>2

and coefficient of

(r+2)^{\text {th }}

term and

3 r^{\text {th }}

term in the expansion of

(1+x)^{2 n}

are equal, then

n

equals)

View answer and explanation

Correct option: C

2 r

Option C (2r) is correct because equating the coefficients and solving for n yields n = 2r.

243 Medium

243 of 468

Application +4 | -1

\left(1+x-2 x^2\right)^{20}=a_0+a_1 x+a_2 x^2+a_3 x^3

+\ldots+a_{40} x^{40}

, then

a_1+a_3+a_5+\ldots+a_{39}

is equal to

View answer and explanation

Correct option: A

-2^{19}

Option A (

-2^{19}

) is correct because substituting x=1 and x=-1 into the polynomial and subtracting the results allows us to isolate the sum of odd-powered coefficients.

244 Medium

244 of 468

Application +4 | -1

If the term free from

x

in the expansion of

\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}

is 405 , then the value of

k

View answer and explanation

Correct option: B

\pm 3

Option B (

\pm 3

) is the correct value of k because it satisfies the condition that the term free from x in the given expansion is 405.

245 Medium

245 of 468

Understanding +4 | -1

The value of

\sum\limits_{0 \le i} {\sum\limits_{ < j < } {\sum\limits_{k < l} {\sum\limits_{ \le n} 2 } } }

is equal to

View answer and explanation

Correct option: B

2^{n+1} C_4

Option B (

2^{n+1} C_4

) is correct because the quadruple summation can be expressed as a combination multiplied by 2.

246 Medium

246 of 468

Application +4 | -1

\sum\limits_{{k=1}}^{n-r} { }^{{n}-{k}} {C}_{{r}}={ }^{{x}} {C}_{{y}}

then

View answer and explanation

Correct option: B

x=n, y=r+1

Option B (

x=n, y=r+1

) is correct because the sum of the series can be simplified to a single binomial coefficient using the identity

\sum\limits_{{k=1}}^{n-r} { }^{{n}-{k}} {C}_{{r}}={ }^{{n}} {C}_{{r}+1}

247 Medium

247 of 468

Application +4 | -1

If in the expansion of

{\left( {3x - \frac{2}{{{x^2}}}} \right)^{15}}

r\mathrm{th}

term is independent of

x

,then value of

r

View answer and explanation

Correct option: A

Option A (6) is correct because it is the value of r that makes the term in the binomial expansion independent of x.

248 Medium

248 of 468

Application +4 | -1

The sum of binomial coefficient in the expansion of

\left(2-\frac{3}{2} x+\frac{1}{2 x^2}\right)^{2001}

View answer and explanation

Correct option: C

Option C (1) is correct because substituting

x=1

into the expression gives

(2 - \frac{3}{2} + \frac{1}{2})^{2001} = 1^{2001} = 1

, which is the sum of the binomial coefficients.

249 Medium

249 of 468

Application +4 | -1

If three successive coefficient in the expansion of

(1+x)^n

are in A.P., then

(n+2)

View answer and explanation

Correct option: C

a perfect square

Option C (a perfect square) is correct because the condition that three successive coefficients are in A.P. leads to an expression that can be written as a perfect square.

250 Medium

250 of 468

Analyzing +4 | -1

a_{r}

is the coefficient of

x^{10-r}

in the Binomial expansion of

(1+x)^{10}

, then

\sum\limits_{{r=1}}^{10} r^{3}\left(\frac{a_{r}}{a_{r-1}}\right)^{2}

is equal to

View answer and explanation

Correct option: A

1210

Option A (1210) is correct because the summation simplifies to a telescoping series that can be evaluated using standard summation formulas.

251 Medium

251 of 468

Application +4 | -1

\frac{\mathrm{C}_0}{1}+\frac{\mathrm{C}_2}{3}+\frac{\mathrm{C}_4}{5}+\frac{\mathrm{C}_6}{7}+\ldots \ldots . .=

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the given series can be expressed as a binomial coefficient sum, which simplifies to

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

252 Medium

252 of 468

Application +4 | -1

If the coefficient of

r

th,

(r+1)

th and

(r+2)

th terms in the expansion of

(1+x)^n

are in A.P., then

n

is a root of the equation

View answer and explanation

Correct option: D

x^2-x(4 r+1)-4 r^2-2=0

Option D (

x^2-x(4r+1)-4r^2-2=0

) is correct because the condition that the coefficients are in A.P. leads to a quadratic equation in n.

253 Medium

253 of 468

Application +4 | -1

\frac{T_2}{T_3}

in the expansion of

(a+b)^n

and

\frac{T_3}{T_4}

in the expansion of

(a+b)^{n+3}

are equal, then

n

is equal to

View answer and explanation

Correct option: C

Option C (5) is correct because by equating the ratios of consecutive terms in the binomial expansions, we can solve for n.

254 Medium

254 of 468

Application +4 | -1

Let

{\left( {1 + x} \right)^n} = \mathop {\sum \,}\limits_{r = 0}^n {a_r}\;{x^r}.

Then,

\left( {1 + \frac{{{a_1}}}{{{a_0}}}} \right)\left( {1 + \frac{{{a_2}}}{{{a_1}}}} \right) \ldots \left( {1 + \frac{{{a_n}}}{{{a_{n - 1}}}}} \right)\;

is equal to

View answer and explanation

Correct option: B

\frac{{{{\left( {n + 1} \right)}^n}}}{{n\;!}}

Option B is correct because the given expression simplifies to

\frac{{{{\left( {n + 1} \right)}^n}}}{{n\;!}}

after applying the binomial coefficient properties and simplifying the product.

255 Medium

255 of 468

Analyzing +4 | -1

The value of

1+\left(\frac{1}{2}+\frac{1}{3}\right) \frac{1}{4}+\left(\frac{1}{4}+\frac{1}{5}\right) \frac{1}{4^2}+\left(\frac{1}{6}+\frac{1}{7}\right) \frac{1}{4^3}+\ldots

View answer and explanation

Correct option: A

\log _e \sqrt{12}

Option A (

\log _e \sqrt{{12}}

) is correct because the given infinite series can be simplified and expressed in terms of logarithms, ultimately resulting in

\log _e \sqrt{{12}}

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256 Medium

256 of 468

Application +4 | -1

The greatest term in the expansion of

\sqrt 3 {\left( {1 + \frac{1}{{\sqrt 3 }}} \right)^{20}}

View answer and explanation

Correct option: C

\frac{{25840}}{9}

Option C (

\frac{{25840}}{9}

) is correct because it is the greatest term calculated by finding the value of r for which T_{r+1} >= T_r in the binomial expansion.

257 Medium

257 of 468

Application +4 | -1

Coeffiecient of

t^{24}

in the expansion of

\left(1+t^2\right)^{12}\left(1+t^{12}\right)\left(1+t^{24}\right)

View answer and explanation

Correct option: A

{ }^{12} C_6+2

Option A (

{ }^{12} C_6+2

) is correct because the coefficient of t²⁴ is found by considering the different ways to obtain t²⁴ from the product of the three terms.

258 Medium

258 of 468

Application +4 | -1

The sum of coefficients of integral powers of

x

in the binomial expansion

{(1 - 2\sqrt x )^{50}}

View answer and explanation

Correct option: A

\frac{1}{2}({3^{50}} + 1)

Option A (

\frac{1}{2}({3^{50}} + 1)

) is correct because, by using the binomial theorem and considering only the integral powers of x, the sum of the coefficients simplifies to the given expression.

259 Medium

259 of 468

Application +4 | -1

The first integral term in the expansion of

(\sqrt{3}+\sqrt[3]{2})^9

, is the

View answer and explanation

Correct option: C

4^{\text {th}}

term

Option C,

4^{\text {th}}

term, is correct because the first integral term occurs when r=3 in the binomial expansion.

260 Medium

260 of 468

Analyzing +4 | -1

In the expansion of

(x+y+z)^{25}

View answer and explanation

Correct option: A

every term is of the form

{ }^{25} C_r \cdot{ }^r C_k \cdot x^{25-r} \cdot y^{r k} \cdot z^k

Option A is correct because it accurately represents the general form of terms in the multinomial expansion of (x+y+z)²⁵.

261 Medium

261 of 468

Analyzing +4 | -1

The coefficient of

{x^4}

in the expansion of

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

View answer and explanation

Correct option: D

{\;^n}{C_4} + {\;^n}{C_2} + {\;^n}{C_1}.{\;^n}{C_2}

The correct answer is D (

{\;^n}{C_4} + {\;^n}{C_2} + {\;^n}{C_1}.{\;^n}{C_2}

) because it is found by expanding the given expression and identifying the coefficient of

{x^4}

262 Medium

262 of 468

Analyzing +4 | -1

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

, then

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

=k \cdot C_1 C_2 C_3 \ldots C_n

, where

k

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the given expression can be simplified using binomial coefficient identities.

263 Medium

263 of 468

Application +4 | -1

If the coefficients of

x^{7}

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

and

x^{-7}

\left(a x-\frac{1}{3 b x^{2}}\right)^{11}

are equal, then

View answer and explanation

Correct option: A

729 a b=32

Option A (

729 a b=32

) is correct because equating the coefficients of

x^7

and

x^{-7}

and simplifying leads to the relationship

729ab = 32

264 Medium

264 of 468

Analyzing +4 | -1

The expansion

\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 expression, the highest power of

x

is 7.

265 Medium

265 of 468

Understanding +4 | -1

The number of terms in

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

View answer and explanation

Correct option: D

201

Option D is correct because the expansion results in terms from x^-300 to x³⁰⁰, with a difference of 3 between each power, leading to 201 terms.

266 Medium

266 of 468

Analyzing +4 | -1

n \in N

and

n

is not a multiple of three and

\left(1+x+x^2\right)^n=\sum\limits_{{r=0}}^{2 n} a_r x^r

, then the value of

\sum\limits_{{r=0}}^n(-1)^r \cdot a_r \cdot{ }^n C_r

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 0, which is not listed.

267 Medium

267 of 468

Application +4 | -1

The coefficient of

x^4

in the expansion of

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

View answer and explanation

Correct option: D

{ }^n C_4+{ }^n C_2+{ }^n C_1 \cdot{ }^n C_2

Option D (

{ }^n C_4+{ }^n C_2+{ }^n C_1 \cdot{ }^n C_2

) is correct because expanding (1+x+x²+x³)^n as (1+x)^n(1+x²)^n and finding the x⁴ term gives the correct coefficient.

268 Medium

268 of 468

Application +4 | -1

The coefficient of

x^{-2}

in the expansion of

(1+x)^{40}\left(1+\frac{1}{x}\right)^{40}

View answer and explanation

Correct option: D

\left(\begin{array}{l}80 \\ 38\end{array}\right)

Option D

\left(\begin{array}{l}80 \\ 38\end{array}\right)

is correct because the coefficient of

x^{-2}

is equivalent to the coefficient of

x^{38}

in the expansion of

(1+x)^{80}

269 Medium

269 of 468

Application +4 | -1

Let

m

and

n

be the coefficient of

7^{\text {th}}

and

13^{\text {th}}

term in expansion of

\left(\frac{1}{3} x^{1 / 3}+\frac{1}{2 x^{2 / 5}}\right)^{18}

, then

\left(\frac{m}{n}\right)^{1 / 3}

View answer and explanation

Correct option: D

\frac{4}{9}

Option D (4/9) is correct because after calculating the coefficients m and n of the 7th and 13th terms in the binomial expansion, the expression (m/n)^(1/3) simplifies to 4/9.

270 Medium

270 of 468

Application +4 | -1

If for positive integers

r > 1,\;n > 2

, the coefficient of the

\left( {3r} \right)

th and

\left( {r + 2} \right)

th powers of

x

in the expansion of

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

are equal, then

View answer and explanation

Correct option: C

n = 2r + 1

Option C (

n = 2r + 1

) is correct because it satisfies the condition for equality of binomial coefficients in the expansion of

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

when

r > 1

and

n > 2

Revision notes and key takeaways

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