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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 391-420 of 468 questions

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391 Medium

391 of 468

Application +4 | -1

The number of terms which are free from fractional powers in the expansion of

\left(a^{1 / 5}+b^{2 / 3}\right)^{45}, a \neq b

View answer and explanation

Correct option: C

Option C (4) is the correct answer because there are four terms in the expansion that are free from fractional powers.

392 Medium

392 of 468

Application +4 | -1

The coefficient of

x^{n}

in expansion of

(1+x)(1-x)^{n}

View answer and explanation

Correct option: B

(-1)^{n}(1-n)

Option B ((-1)^n(1-n)) is correct because it is the result of calculating the coefficient of x^n in the given expansion.

393 Medium

393 of 468

Application +4 | -1

x

is positive, the first negative term in the expansion of

(1+x)^{27 / 5}

View answer and explanation

Correct option: C

8 th term

Option C (8 th term) is correct because it is the first term where the factor (27/5 - r + 1) becomes negative, leading to a negative term in the expansion.

394 Medium

394 of 468

Application +4 | -1

(1+x)^n=\sum_{r=0}^n{ }^n C_r \cdot(x)^r

, then

\left(1+\frac{{ }^n C_1}{{ }^n C_0}\right) \cdot\left(1+\frac{{ }^n C_2}{{ }^n C_1}\right) \cdots\left(1+\frac{{ }^n C_n}{{ }^n C_{n-1}}\right)=

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the expression simplifies to a product of terms that results in this form.

395 Medium

395 of 468

Application +4 | -1

Sum of coefficients of all terms containing integral power of

x

in expansions of

(1-2 \sqrt{x})^{20}

View answer and explanation

Correct option: D

\frac{3^{20}+1}{2}

Option D is correct. The sum of coefficients of terms with integral powers of x is found by substituting x=1 into the expression derived from adding (1-2√x)²⁰ and (1+2√x)²⁰ and dividing by 2.

396 Medium

396 of 468

Analyzing +4 | -1

a_{r}

is the coefficient of

x^{10-r}

in expansion of

(1+x)^{10}

then

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

is:

View answer and explanation

Correct option: B

1210

Option B (1210) is correct because after simplifying the summation, it can be expressed in terms of sums of squares and cubes of integers, which have known formulas.

397 Medium

397 of 468

Application +4 | -1

The coefficient of

{x^3}{y^4}{z^5}

in the expansion of

{(xy + yz + xz)^6}

View answer and explanation

Correct option: B

Option B (60) is correct because after expanding the expression and solving the system of equations for the exponents, the coefficient simplifies to 60.

398 Medium

398 of 468

Application +4 | -1

If the polynomial

f(x)=1-x+x^2-x^3+\cdots+x^{19}+x^{20}

is expressed as

g(y)=a_0+a_1 y+a_2 y^2+\cdots+a_{20} y^{20}

, where

y=x-4

, then the value of

a_0+a_1+a_2+\cdots+a_{20}

View answer and explanation

Correct option: D

\frac{1+5^{21}}{6}

Option D (

\frac{{1+5^{21}}}{{6}}

) is correct. The value of the polynomial is found by substituting y=1 into the expression g(y) after converting f(x) to a geometric series.

399 Medium

399 of 468

Analyzing +4 | -1

Coefficient of

x^m

in the expansion of

S=(1+x)^{2 m}+x(1+x)^{2 m-1}+x^2(1 +x)^{2 m-2} +\ldots .+x^{2 m}

View answer and explanation

Correct option: B

{ }^{2 m+1} C_m

Option B,

{ }^{{2 m+1}} C_m

, is correct because the given series is a geometric progression that simplifies to

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

, and the coefficient of

x^m

{ }^{{2m+1}} C_m

400 Medium

400 of 468

Application +4 | -1

The greatest integer which divides the number

101^{100}-1

View answer and explanation

Correct option: C

10000

Option C (10000) is correct because the greatest integer that divides

101^{{100}}-1

is 10000, which is found using the binomial theorem.

401 Medium

401 of 468

Application +4 | -1

(1+x)^n=c_0+c_1 x+c_2 x^2+\ldots+c_n x^n

, then

c_0-\frac{1}{2} c_1+\frac{1}{3} c_2-\ldots+(-1)^n \frac{1}{n+1} c_n=

View answer and explanation

Correct option: D

\frac{1}{n+1}

Option D is correct because the given expression can be simplified using integration and the binomial theorem.

402 Medium

402 of 468

Application +4 | -1

The digit at the unit place in the number

{19^{2005}} + {11^{2005}} - {9^{2005}}

View answer and explanation

Correct option: B

Option B (1) is correct because analyzing the unit digits of each term reveals that the unit digit of the entire expression is 1.

403 Medium

403 of 468

Application +4 | -1

If coefficients of

2 \mathrm{nd}, 3 \mathrm{rd}

and 4 th terms in the expansion of

(1+x)^{2 n}

are in AP, 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 the coefficients being in AP leads to the given quadratic equation after simplification.

404 Medium

404 of 468

Application +4 | -1

The coefficient of

x^{10}

in the expansion of

\left(3 x^2-\frac{1}{x^2}\right)^{15}

View answer and explanation

Correct option: B

-\frac{15 !}{10 ! 5 !} 3^{10}

Option B (

-\frac{15 !}{10 ! 5 !} 3^{10}

) is correct because it represents the coefficient of

x^{10}

in the binomial expansion of

\left(3 x^2-\frac{1}{x^2}\right)^{15}

, calculated using the binomial theorem.

405 Medium

405 of 468

Application +4 | -1

There are at least 10 books of each of the subjects Mathematics, Physics and Chemistry. In how many ways 5 books can be chosen from these books?

View answer and explanation

Correct option: A

The correct answer is 21 because it represents the number of ways to choose 5 books from the three subjects.

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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

406 Medium

406 of 468

Application +4 | -1

1+x^4+x^5=\sum\limits_{{i=0}}^5 a_i(1+x)^i

, for all

x

{R}

, then

a_2

is:

View answer and explanation

Correct option: A

-4

Option A (-4) is correct because by substituting

x

with

x-1

and comparing coefficients, we can find the value of

a_2

407 Medium

407 of 468

Analyzing +4 | -1

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

, then

\sum\limits_{0 \le i} {\sum\limits_{ < j \le n} {{{({C_i}{ + C_j})}^2}} }

is equal to

View answer and explanation

Correct option: A

(n-1) \cdot{ }^{2 n} C_n+2^{2 n}

Option A (

(n-1) \cdot{ }^{2 n} C_n+2^{2 n}

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

408 Medium

408 of 468

Application +4 | -1

The coefficient of

x^{7}

in the expansion of

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

View answer and explanation

Correct option: B

-144

Option B (-144) is correct because after factoring and expanding, the coefficient of x⁷ is -144.

409 Medium

409 of 468

Application +4 | -1

The term independent of

x

{\left\{ {\sqrt {\frac{x}{3}} + \frac{3}{{2\;{x^2}}}} \right\}^{10}},

View answer and explanation

Correct option: C

\frac{5}{4}

Option C (

\frac{5}{4}

) is the correct answer because it represents the term independent of x after expanding the binomial and setting the power of x to zero.

410 Medium

410 of 468

Analyzing +4 | -1

The coefficient of

\lambda^n \mu^n

in the expansion of

(1+\lambda)^n(1+\mu)^n(\lambda+\mu)^n

View answer and explanation

Correct option: D

\sum\limits_{{r=0}}^n\left({ }^n C_r\right)^3

Option D (

\sum\limits_{{r=0}}^n\left({ }^n C_r\right)^3

) is correct because the coefficient of

\lambda^n \mu^n

in the expansion is the sum of the cubes of the binomial coefficients.

411 Medium

411 of 468

Application +4 | -1

|x| < 1

, then the coefficient of

x^n

in expansion of

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

View answer and explanation

Correct option: D

n+1

Option D (

n+1

) is correct because the coefficient of

x^n

in the expansion of

(1-x)^{-2}

n+1

412 Medium

412 of 468

Application +4 | -1

If sum of middle terms is

S

in the expansion of

\left(2 a-\frac{a^2}{4}\right)^9

, then the value(s) of

S

is/are

View answer and explanation

Correct option: D

\left(\frac{63}{32}\right) a^{13}(8-a)

Option D is correct because the sum of the middle terms in the binomial expansion is calculated by finding T5 and T6, then simplifying the expression.

413 Medium

413 of 468

Application +4 | -1

The coefficient of

{x^4}

in the expansion of

{\left( {1 + x + {x^3} + {x^4}} \right)^{10}},

View answer and explanation

Correct option: D

310

Option D (310) is correct because the coefficient of x⁴ is found by multiplying the relevant terms in the expansion of (1+x)¹⁰ and (1+x³)¹⁰.

414 Medium

414 of 468

Application +4 | -1

If the coefficients of

{x^3}

and

{x^4}

in the expansion of

(1 + ax + b{x^2}){(1 - 2x)^{18}}

in powers of

x

are both zero, then

(a,\,b)

is equal to

View answer and explanation

Correct option: D

\left( {16,\,\frac{{272}}{3}} \right)

Option D

\left( {16,\frac{{272}}{3}} \right)

is correct because solving the equations formed by setting the coefficients of

x^3

and

x^4

to zero yields these values for a and b.

415 Medium

415 of 468

Application +4 | -1

Coefficient of

x^8

in the expansion of

\left(1+2 x-x^3\right)^6

View answer and explanation

Correct option: A

528

Option A (528) is correct because it is the sum of the coefficients of x⁸ obtained from the multinomial expansion of the given expression.

416 Medium

416 of 468

Application +4 | -1

The value of

\sum\limits_{{r=0}}^n{ }^n C_r \sin r x \cdot \cos (n-r) x

is equal to

View answer and explanation

Correct option: A

2^{n-1} \cdot \sin (n x)

Option A (

2^{n-1} \cdot \sin (n x)

) is correct because the summation can be simplified using trigonometric identities and binomial theorem.

417 Medium

417 of 468

Analyzing +4 | -1

Let

f(x)=\left(\sqrt{x^2+1}+\sqrt{x^2-1}\right)^6

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

. Then,

View answer and explanation

Correct option: B

f(x)

has exactly two terms

Option B (

f(x)

has exactly two terms) is correct because after expanding and simplifying the expression,

f(x)

reduces to

64x^6 - 48x^2

, which has two terms.

418 Medium

418 of 468

Application +4 | -1

The largest term in the expansion of

(2+3 x)^{25}

where

x=2

is its.

View answer and explanation

Correct option: C

20^{\text {th }}

term

Option C (

20^{\text {th }}

term) is correct because the largest term is found by determining when the ratio of consecutive terms is greater than or equal to 1.

419 Medium

419 of 468

Application +4 | -1

The approximate value of

\frac{1}{\sqrt{23}}

correct to four decimal places is

View answer and explanation

Correct option: B

0.2085

Option B (0.2085) is the correct approximate value of

\frac{1}{\sqrt{23}}

correct to four decimal places, obtained through binomial approximation.

420 Medium

420 of 468

Application +4 | -1

{\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + ...,

then the values of

a

and

n

are

View answer and explanation

Correct option: A

\left( {2,4} \right)

Option A ((2,4)) is correct because by comparing the coefficients of x and x² in the given expansion with the binomial expansion of (1 + ax)^n, we can solve for a and n.

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