The coefficient of $x^{60}$ in $(1+x)^{51}\left(1-x+x^2\right)^{50}$ is

$-\left({ }^{50} C_{20}\right)$

$-\left({ }^{51} C_{20}\right)$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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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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211 Easy

211 of 434

Application +4 | -1

The

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

term in the expansion of

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

will be

View answer and explanation

Correct option: B

\frac{{\left( {r + 1} \right)\left( {r + 2} \right)\left( {r + 3} \right)}}{6}{x^r}

Option B is correct because the (r+1)th term in the expansion of (1-x)^-4 is derived using the binomial theorem for negative exponents.

212 Easy

212 of 434

Analyzing +4 | -1

\frac{1}{n !}+\frac{1}{2 !(n-2) !}+\frac{1}{4 !(n-4) !}+\ldots

is equal to

View answer and explanation

Correct option: A

\frac{2^{n-1}}{n !}

Option A (

\frac{2^{n-1}}{n !}

) is correct because the given expression simplifies to this value using binomial coefficients.

213 Easy

213 of 434

Analyzing +4 | -1

Sum of the series

\sum\limits_{k = 0}^n {^n{C_k}{{\left( { - 1} \right)}^k}\frac{1}{{{a_k}}}}

where

a_k=\sum\limits_{{i=0}}^k{ }^k C_i b_i

where

b_i=\sum\limits_{{j=0}}^i{ }^i C_j\left(\frac{-2}{3}\right)^j

View answer and explanation

Correct option: C

\frac{1}{4^n}

Option C,

\frac{{1}}{{4^n}}

, is correct because after simplifying the series, it becomes a binomial expansion equal to

(\frac{{1}}{{4}})^n

214 Easy

214 of 434

Application +4 | -1

The coefficient of

{x^2}

term in the binomial expansion of

{\left( {\frac{1}{3}{x^{1/2}} + {x^{ - 1/4}}} \right)^{10}}

View answer and explanation

Correct option: A

\frac{{70}}{{243}}

Option A (70/243) is correct because it is the coefficient of the x² term in the binomial expansion of the given expression, calculated by finding the appropriate term in the expansion.

215 Easy

215 of 434

Analyzing +4 | -1

\left( {^m{C_0}{ + ^m}{C_1}{ - ^m}{C_2}{ - ^m}{C_3}} \right) + \left( {^m{C_4}{ + ^m}{C_5}{ - ^m}{C_6}{ - ^m}{C_7}} \right) + \ldots = 0

if and only if for some positive integer

k, m

is equal to

View answer and explanation

Correct option: C

4 k-1

Option C (

4 k-1

) is correct because it satisfies the condition for the given sum to be zero, derived from De Moivre's theorem.

216 Easy

216 of 434

Application +4 | -1

If the coefficient of 7th and 13th term in the expansion of

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

are equal, then

n

is equal to

View answer and explanation

Correct option: C

Option C (18) is correct because if the coefficients of the 7th and 13th terms are equal, then

n

must be

6 + 12 = 18

217 Easy

217 of 434

Application +4 | -1

If the sum of the coefficient in the expansion of

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

is 1024, then the value of the greatest coefficient in the expansion is

View answer and explanation

Correct option: B

252

Option B (252) is correct because after finding n=10 from the sum of coefficients, the greatest coefficient in the expansion of (x+y)¹⁰ is calculated as

^{10}C_5

, which equals 252.

218 Easy

218 of 434

Application +4 | -1

Cube root of 217 is

View answer and explanation

Correct option: A

6.01

Option A (6.01) is correct because it is the approximate cube root of 217 calculated using the binomial theorem.

219 Easy

219 of 434

Application +4 | -1

If the fourth term in the binomial expansion of

{\left( {\sqrt {{x^{\left( {\frac{1}{{1 + \log {{10}^x}}}} \right)}}} + {x^{\frac{1}{{12}}}}} \right)^\mathrm{6}}

is equal to 200, and

x > 1,

then the value of

x

View answer and explanation

Correct option: C

Option C (10) is correct because after simplifying the binomial expansion and solving the resulting equation, the value of x that satisfies the condition x > 1 is 10.

220 Easy

220 of 434

Application +4 | -1

If the sum of the coefficients in the expansion of

{\left( {1 - 3\;x + 10\;{x^2}} \right)^n}

a

and if the sum of the coefficients in the expansion of

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

b

, then

View answer and explanation

Correct option: B

a = {b^3}

Option B (

a = {b^3}

) is correct because substituting x=1 in the first expression yields a = 8^n and substituting x=1 in the second expression yields b = 2^n, thus a = b³.

221 Easy

221 of 434

Application +4 | -1

The number of distinct terms in the expansion of

\left(1+3 x+3 x^2+x^3\right)^7

View answer and explanation

Correct option: D

Option D (22) is correct because the expression simplifies to (1+x)²¹, which has 22 terms.

222 Easy

222 of 434

Application +4 | -1

The coefficient of

{x^r}

in the expansion of

{(1 - x)^{ - 2}}

View answer and explanation

Correct option: B

r + 1

Option B (r + 1) is correct because the coefficient of x^r in the expansion of (1-x)^(-2) is (r+1).

223 Easy

223 of 434

Application +4 | -1

If middle term in the expansion to

\left(x^{1 / 2}+x^k\right)^{16}

is also independent term

View answer and explanation

Correct option: C

16 x^7

The correct answer is C (

16 x^7

) because it is obtained by finding the value of k that makes the middle term independent of x, and then finding the term independent of x.

224 Easy

224 of 434

Analyzing +4 | -1

1 + \frac{{2.1}}{{3.2}} + \frac{{2.5}}{{3.6}}{\left( {\frac{1}{2}} \right)^2} + \frac{{2.5.8}}{{3.6.9}}.{\left( {\frac{1}{2}} \right)^3} + ...

is equal to

View answer and explanation

Correct option: C

{4^{1/3}}

Option C (

{4^{1/3}}

) is correct because the given series can be recognized as a binomial expansion of the form (1-x)^(-n), which simplifies to

{4^{1/3}}

225 Easy

225 of 434

Application +4 | -1

If the coefficients of

pth

\left( {p + 1} \right)th

and

\left( {p + 2} \right)th

terms in the expansion of

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

are in AP, then

View answer and explanation

Correct option: B

{n^2} - n\left( {4p + 1} \right) + 4{p^2} - 2 = 0

Option B (

{n^2} - n\left( {4p + 1} \right) + 4{p^2} - 2 = 0

) is correct because it is the relationship between n and p when the coefficients of the pth, (p+1)th, and (p+2)th terms are in arithmetic progression.

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226 Easy

226 of 434

Application +4 | -1

If the middle term in the expansion of

\left(\frac{1}{x}+x \sin x\right)^{10}

equals to

7 \frac{7}{8}

then

x

is equal to

(n \in I)

View answer and explanation

Correct option: C

n \pi+(-1)^n \frac{\pi}{6}

Option C (

n \pi+(-1)^n \frac{\pi}{6}

) is correct because it is the general solution for x when the middle term of the binomial expansion equals

7 \frac{7}{8}

227 Easy

227 of 434

Analyzing +4 | -1

n

is the degree of the polynomial,

\left[\frac{2}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}\right]^8+\left[\frac{2}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}\right]^8

and

\mathrm{m}

is the coefficient of

x^n

in it, then the ordered pair

(n, m)

is equal to

View answer and explanation

Correct option: B

\left(12,(20)^4\right)

Option B (

(12, (20)^4)

) is correct because after rationalizing and simplifying, the polynomial's degree is 12 and the coefficient of

x^{12}

(20)^4

228 Easy

228 of 434

Application +4 | -1

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

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

are in A.P., then

View answer and explanation

Correct option: B

2\;{n^2} - 9\;n + 7 = 0

Option B (

2\;{n^2} - 9\;n + 7 = 0

) is correct because it is derived from the given condition that the coefficients are in A.P.

229 Easy

229 of 434

Application +4 | -1

{\left( {1 + x - 2\;{x^2}} \right)^6} = 1 + {C_1}x + {C_2}{x^2} + {C_3}{x^3} + \ldots + {C_{12}}{x^{12}},

then the value of

{C_2} + {C_4} + {C_6} + \ldots + {C_{12}},

View answer and explanation

Correct option: C

Option C (31) is correct because it is the sum of the even-indexed coefficients

{C_2} + {C_4} + {C_6} + \ldots + {C_{12}}

obtained by substituting

x = 1

and

x = -1

into the given equation and solving.

230 Easy

230 of 434

Analyzing +4 | -1

The number of rational terms in the expansion of

{\left( {\sqrt[3]{4} + \frac{1}{{\sqrt[4]{6}}}} \right)^{20}},

View answer and explanation

Correct option: A

Option A (3) is correct because there are only 3 terms where both the powers of 2 and 3 are rational numbers, which occurs when r = 8, 16, and 20.

231 Easy

231 of 434

Application +4 | -1

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

, then the value of

\sum\limits_{{r=0}}^n \sum\limits_{{s=0}}^n C_r C_s

is equal to

View answer and explanation

Correct option: B

2^{2 n}

Option B (

2^{2 n}

) is correct because the sum can be simplified to

(2^n)^2

using the binomial theorem.

232 Easy

232 of 434

Analyzing +4 | -1

For

x \in {R}, x \neq-1

, if

(1+x)^{2016}+x(1+x)^{2015}+x^2(1+x)^{2014}+\ldots+x^{2016}=\sum\limits_{{i=0}}^{2016} a_i x^i,

then

a_{17}

is equal to:

View answer and explanation

Correct option: A

\frac{2017 !}{17 ! 2000 !}

Option A (

\frac{2017 !}{17 ! 2000 !}

) is correct because it represents the coefficient of

x^{17}

in the simplified expression after summing the geometric series.

233 Easy

233 of 434

Application +4 | -1

The sum of the rational terms in the expansion of

{(\sqrt 2 + {3^{1/5}})^{10}}

View answer and explanation

Correct option: A

The correct answer is A (41) because it represents the sum of the rational terms obtained from the binomial expansion of the given expression.

234 Easy

234 of 434

Analyzing +4 | -1

The coefficient of

x^{60}

(1+x)^{51}\left(1-x+x^2\right)^{50}

View answer and explanation

Correct option: A

{ }^{50} C_{20}

Option A (

{ }^{50} C_{20}

) is correct because

(1+x)^{51}(1-x+x^2)^{50}

simplifies to

(1+x)(1+x^3)^{50}

, and the coefficient of

x^{60}

{ }^{50} C_{20}

235 Easy

235 of 434

Application +4 | -1

In the expansion of

{(1 + x + {x^2} + {x^3})^6},

then coefficient of

{x^{14}}

View answer and explanation

Correct option: B

120

Option B (120) is correct because the coefficient of

{x^{14}}

in the expansion of

{(1 + x + {x^2} + {x^3})^6}

is found by expanding and summing the appropriate terms, resulting in 120.

236 Easy

236 of 434

Application +4 | -1

The term independent of

x

in the expansion of

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

will be

View answer and explanation

Correct option: B

\frac{5}{4}

Option B (

\frac{5}{4}

) is correct because it is the constant term obtained after expanding the given binomial and setting the power of x to zero.

237 Easy

237 of 434

Application +4 | -1

If sum of the coefficients of the first, second and third terms of the expansion of

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

is 46, then the coefficient of the term that does not contain

x

View answer and explanation

Correct option: A

Option A (84) is correct because after finding m=9, the coefficient of the term independent of x in the expansion is calculated as

^{9}C_6 = 84

238 Easy

238 of 434

Application +4 | -1

The sum of the real values of x for which the middle term in the binomial expansion of

{\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}

equals 5670 is

View answer and explanation

Correct option: B

Option B (0) is correct because the real values of x are

\pm \sqrt{3}

, and their sum is 0.

239 Easy

239 of 434

Application +4 | -1

Coefficient of the term independent of

x

in the expansion

{\left( {x + \frac{1}{{{x^2}}}} \right)^6}

is equal to

View answer and explanation

Correct option: B

Option B (15) is correct because it is the coefficient of the term where the power of x is zero in the given binomial expansion.

240 Easy

240 of 434

Application +4 | -1

The number of real negative terms in the binomial expansion of

(1+i x)^{4 n-2}, n \in N, n>0, i=\sqrt{-1}

, is

View answer and explanation

Correct option: A

n

Option A (

n

) is correct because the number of real negative terms in the binomial expansion of

(1+ix)^{4n-2}

can be determined by analyzing the powers of

i

and identifying when they result in real negative values.

Revision notes and key takeaways

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