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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Medium

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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.

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Showing 181-210 of 468 questions

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181 Medium

181 of 468

Application +4 | -1

Number of ways of selection of 8 letters from 24 letters of which 8 are

a, 8

are

b

and the rest unlike, is given by

View answer and explanation

Correct option: C

10.2^7

Option C (

10.2^7

) is correct because it represents the number of ways to select 8 letters from the given set, calculated using generating functions and differentiation.

182 Medium

182 of 468

Application +4 | -1

If sum of coefficients in expansion of

(1+2 \mathrm{x})^{\mathrm{m}} \&(2+\mathrm{x})^{\mathrm{n}}

are respectively

6561 \& 243

, then position of point

(\mathrm{m}, \mathrm{n})

w.r.t. circle

\mathrm{x}^2+\mathrm{y}^2-4 \mathrm{x}-6 \mathrm{y}-32=0

View answer and explanation

Correct option: B

inside the circle

Option B (inside the circle) is correct because substituting m=8 and n=5 into the circle's equation results in a negative value, indicating the point lies inside the circle.

183 Medium

183 of 468

Application +4 | -1

The sum of the coefficient of

x^{2 / 3}

and

x^{-2 / 5}

in the binomial expansion of

\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9

is:

View answer and explanation

Correct option: A

21-Apr

Option A (21/4) is correct because it is the sum of the coefficients of x^(2/3) and x^(-2/5) in the binomial expansion.

184 Medium

184 of 468

Application +4 | -1

The coefficient of

x^2 y^3

in the expansion of

(1-x+y)^{20}

View answer and explanation

Correct option: D

None of these

Option D is correct because the coefficient of x²y³ is found using the multinomial theorem, and the correct coefficient is not among the given options.

185 Medium

185 of 468

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 by considering all possible combinations of powers of x that sum to 11, the coefficient is calculated to be 1113.

186 Medium

186 of 468

Analyzing +4 | -1

\frac{e^x}{1-x}=B_0+B_1 x+B_2 x^2+\ldots+B_n x^n+\ldots

, then

B_n-B_{n-1}

equals

View answer and explanation

Correct option: A

\frac{1}{n !}

Option A (

\frac{{1}}{{n !}}

) is correct because the coefficient difference can be found by expanding the given expression and comparing coefficients.

187 Medium

187 of 468

Application +4 | -1

If in the expansion of

(1+\mathrm{y})^{\mathrm{n}}

, the coefficient of

5^{\text {th }}, 6^{\text {th }}

and

7^{\text {th }}

terms are in A.P., then

\mathrm{n}

is equal to-

View answer and explanation

Correct option: B

7,14

Option B (7,14) is correct because setting up the arithmetic progression equation for the coefficients and solving the quadratic equation yields these values for n.

188 Medium

188 of 468

Application +4 | -1

\sin n x=\sum\limits_{{r=0}}^n a_r \sin ^r x

, where

n

is an odd natural number. Then

View answer and explanation

Correct option: C

a_0=0, a_1=n

Option C (

a_0=0, a_1=n

) is correct because by expanding

\sin nx

using the imaginary part of Euler's formula, the constant term is zero and the coefficient of

\sin x

n

189 Medium

189 of 468

Application +4 | -1

The unit digit of

17^{2009}+11^{2009}-7^{2009}

View answer and explanation

Correct option: A

Option A (97) is correct because it calculates the total number of terms and subtracts the number of rational terms to find the number of irrational terms.

190 Medium

190 of 468

Application +4 | -1

One value of

\alpha

for which the coefficients of the middle terms in the expansion of

(1+\alpha x)^4

and

(1-\alpha x)^6

are equal,

View answer and explanation

Correct option: A

Option A (0) is correct because by equating the coefficients of the middle terms in both expansions, we find that one possible value for alpha is 0.

191 Medium

191 of 468

Analyzing +4 | -1

n

is an integer between 0 and 21 , then the minimum value of

n !(21-n)

! is attained for

n

, is equal to

View answer and explanation

Correct option: B

Option B (10) is correct because

n!(21-n)!

is minimized when

{ }^{21} C_n

is maximized, which occurs when

n

is closest to

\frac{21}{2}

, i.e.,

n=10

n=11

192 Medium

192 of 468

Application +4 | -1

The coefficient of

x^m

(1+x)^r+(1+x)^{r+1}+(1+x)^{r+2}+\cdots+(1+x)^n, r \leq m \leq n

View answer and explanation

Correct option: A

{ }^{n+1} C_{m+1}

Option A (

{ }^{n+1} C_{m+1}

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

193 Medium

193 of 468

Application +4 | -1

{ }^n C_0,{ }^n C_1,{ }^n C_2, \ldots,{ }^n C_n

denote the binomial coefficients in the expansion of

(1+x)^n

and

p+q=1

, then

\sum\limits_{{r=0}}^n r^2{ }^n C_r p^r q^{n-r}

View answer and explanation

Correct option: C

n^2 p^2+n p q

Option C (n² p²⁺n p q) is correct because it is derived by manipulating the summation and using the binomial theorem with the given condition p+q=1.

194 Medium

194 of 468

Analyzing +4 | -1

The value of

\sum\limits_{{k=8}}^{23}(-1)^{k+1} \, \ { }^{31} C_k

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given summation simplifies to 0 using properties of binomial coefficients and their alternating sums.

195 Medium

195 of 468

Application +4 | -1

\log x=\log _{10} x

, the values of

x

for which 4 th term in the expansion of

\left[\sqrt{x^{2 /(1+\log x)}}+x^{1 / 12}\right]^6

is 200 are

View answer and explanation

Correct option: B

0.0001,10

Option B (0.0001, 10) is correct because solving the equation resulting from the binomial expansion leads to these values of x.

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

196 Medium

196 of 468

Analyzing +4 | -1

{\left( {1 + x - 2{x^2}} \right)^6} = 1 + {a_1}x + {a_2}{x^2} + ... + {a_{12}}{x^{12}}

, then the expression

{a_2} + {a_4} + {a_6} + ... + {a_{12}}

has the value

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the value of the expression is 31, which is not among the provided options.

197 Medium

197 of 468

Application +4 | -1

If the coefficient of

x^7

\left(a x-\frac{1}{b x^2}\right)^{13}

and the coefficient of

x^{-5}

\left(a x+\frac{1}{b x^2}\right)^{13}

are equal, then

a^4 b^4

is equal to:

View answer and explanation

Correct option: C

Option C (22) is correct because after equating the coefficients of x⁷ and x^-5 and simplifying, we find that a⁴b⁴ equals 22.

198 Medium

198 of 468

Application +4 | -1

The coefficient of

x^{11}

in the expansion of

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

View answer and explanation

Correct option: A

-25344

Option A (-25344) is correct because it is the coefficient of x¹¹ obtained after applying the binomial theorem and solving for the appropriate term.

199 Medium

199 of 468

Application +4 | -1

x > 0

, then the number of positive terms in the expansion of

(1+i x)^{4 n}

View answer and explanation

Correct option: A

n+1

Option A (n+1) is correct because the positive terms occur when the power of i is a multiple of 4, resulting in n+1 positive terms.

200 Medium

200 of 468

Analyzing +4 | -1

If the constant term in the expansion of

\left(3 x^3-2 x^2+\frac{5}{x^5}\right)^{10}

2^k \cdot I

, where

I

is an odd integer, then the value of

k

is equal to

View answer and explanation

Correct option: D

The correct answer is D (9) because the value of k is determined by finding the constant term in the multinomial expansion and extracting the power of 2.

201 Medium

201 of 468

Application +4 | -1

If the digits at ten's and hundred's places in (11)

{ }^{2016}

are

\mathrm{x}

and

\mathrm{y}

respectively, then the ordered pair

(x, y)

is equal to :

View answer and explanation

Correct option: C

(6,1)

Option C ((6,1)) is correct because it represents the digits at the ten's and hundred's places in the expansion of (11)

^{2016}

, which are 6 and 1 respectively.

202 Medium

202 of 468

Application +4 | -1

If the second term in the expansion

{\left[ {\sqrt[{13}]{a} + \frac{a}{{\sqrt {{a^{ - 1}}} }}} \right]^n}

is,

14{a^{5/2}}

then the value of

\frac{{{n_{{C_3}}}}}{{{n_{{C_2}}}}}

View answer and explanation

Correct option: A

Option A (4) is correct because by comparing the second term in the expansion with the given value, we find

n = 14

, and then

\frac{{{n_{{C_3}}}}}{{{n_{{C_2}}}}} = 4

203 Medium

203 of 468

Understanding +4 | -1

The number of distinct terms in the expansion of

\left(x_1+x_2+\cdots+x_n\right)^3

View answer and explanation

Correct option: B

{ }^{n+2} C_3

Option B (

{ }^{n+2} C_3

) is correct because the number of distinct terms corresponds to the number of non-negative integer solutions to the equation

\lambda_1 + \lambda_2 + ... + \lambda_n = 3

204 Medium

204 of 468

Analyzing +4 | -1

The value of

\sum\limits_{1 \le i < } {\sum\limits_{j \le n - 1} {{{(i.j)}^n}{C_i}{.^n}{C_j}} }

is equal to

View answer and explanation

Correct option: A

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

Option A is correct because the summation can be simplified using properties of binomial coefficients and results in

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

205 Medium

205 of 468

Application +4 | -1

If the coefficients of three consecutive terms of

(1+x)^{n+5}

are in the ratio 5:10:14, then

n

is equal to

View answer and explanation

Correct option: B

Option B (6) is correct because the given ratio of consecutive coefficients leads to a system of equations that can be solved for n.

206 Medium

206 of 468

Application +4 | -1

7^{103}

is divided by 25 , then the remainder is

View answer and explanation

Correct option: C

Option C (18) is correct because when

7^{{103}}

is divided by 25, the remainder is 18, which is found using modular arithmetic.

207 Medium

207 of 468

Analyzing +4 | -1

The value of

\sum\limits_{{r=0}}^n r(n-r) \cdot\left({ }^n C_r\right)^2

is equal to

View answer and explanation

Correct option: D

n^2.{ }^{2 n-2} C_n

Option D (

n^2.{ }^{2 n-2} C_n

) is correct because the given summation can be simplified using combinatorial identities to match this expression.

208 Medium

208 of 468

Analyzing +4 | -1

Coefficient of

x^{50}

in the expansion of

(1+x)^{41}\left(1-x+x^2\right)^{40}

View answer and explanation

Correct option: A

Option A (0) is correct because the expression can be simplified to

(1+x)(1-x^3)^{40}

, and the coefficient of

x^{50}

in this expression is 0.

209 Medium

209 of 468

Application +4 | -1

\frac{1}{1 ! 10 !}+\frac{1}{3 ! 8 !}+\frac{1}{5 ! 6 !}+\ldots+\frac{1}{11 ! 1 !}=\frac{2^k}{11 !}

then

\frac{k}{2}

is equal to

View answer and explanation

Correct option: A

Option A (5) is correct because the value of k is 10, and k/2 equals 5, after simplifying the given expression using binomial coefficients.

210 Medium

210 of 468

Application +4 | -1

n

is an odd natural number, then

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

equals

View answer and explanation

Correct option: A

Option A (0) is the correct answer because the sum evaluates to zero when n is an odd natural number.

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