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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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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 434 questions

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391 Easy

391 of 434

Analyzing +4 | -1

x^{10}+10 x^8 a+40 x^6 a^2+80 x^4 a^3+80 x^2 a^4+32 a^5

is equal to

View answer and explanation

Correct option: C

\left(x^2+2 a\right)^5

Option C (

(x^2+2 a)^5

) is correct because it matches the binomial expansion of the given expression.

392 Easy

392 of 434

Analyzing +4 | -1

f(x)=\sum\limits_{r=1}^n\left\{r^2\left({ }^n C_r-{ }^n C_{r-1}\right)+(2 r+1)^n C_r\right\}

and

f(30)=30(2)^\lambda

, then the value of

\lambda

View answer and explanation

Correct option: C

Option C (5) is correct, but the provided solution steps are incomplete. A full solution is not possible without further information or context.

393 Easy

393 of 434

Understanding +4 | -1

1 + \frac{1}{3}x + \frac{{1.4}}{{3.6}}{x^2} + \frac{{1.4.7}}{{3.6.9}}{x^3} + ...

is equal to

View answer and explanation

Correct option: D

{\left( {1 - x} \right)^{ - 1/3}}

Option D (

{\left( {1 - x} \right)^{ - 1/3}}

) is correct because the given series matches the binomial expansion of

{\left( {1 - x} \right)^{ - 1/3}}

394 Easy

394 of 434

Application +4 | -1

For

\left| x \right| < 1

, the constant term in the expansion of

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

View answer and explanation

Correct option: D

- \frac{1}{2}

Option D (

- \frac{1}{2}

) is correct because it represents the constant term obtained after expanding the given expression using partial fractions and binomial expansions.

395 Easy

395 of 434

Application +4 | -1

({2^{3n}} - 1)

will be divisible by

\left( {\forall n \in N} \right)

View answer and explanation

Correct option: C

Option C (7) is correct because

{2^{3n}} - 1

can be rewritten as

{8^n} - 1

, which is always divisible by 7.

396 Easy

396 of 434

Application +4 | -1

In the expansion of

{\left( {\frac{a}{x} + bx} \right)^{12}}

,the coefficient of

{x^{ - 10}}

will be

View answer and explanation

Correct option: C

12{a^{11}}b

Option C (

12{a^{11}}b

) is correct because it is the coefficient of

{x^{ - 10}}

obtained from the binomial expansion of the given expression.

397 Easy

397 of 434

Application +4 | -1

The 4th term from the end in the expansion of

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

View answer and explanation

Correct option: C

\frac{672}{x^3}

Option C (672/x³) is correct because it is the 7th term from the beginning, calculated using the binomial theorem.

398 Easy

398 of 434

Application +4 | -1

The coefficients of

{x^2}{y^2},\;yz{t^2}

and

\mathrm{\;}xyzt

in the expansion of

{\left( {x + y + z + t} \right)^4}

are in the ratio

View answer and explanation

Correct option: B

1:2:4

Option B (

1:2:4

) is correct because the coefficients can be calculated using the multinomial theorem, and their ratio is

6:12:24

which simplifies to

1:2:4

399 Easy

399 of 434

Application +4 | -1

^{n-1} C_r=\left(k^2-3\right) \cdot{ }^n C_{r+1}

then

k

belongs to

View answer and explanation

Correct option: D

(\sqrt{3}, 2]

Option D ((

\sqrt{3}, 2]

) is correct because the given condition leads to

3 < k^2 \leq 4

, which means

k

belongs to the interval

(\sqrt{3}, 2]

400 Easy

400 of 434

Application +4 | -1

The sum of the coefficients of all even degree terms is

x

in the expansion of

{(x + \sqrt {{x^3} - 1} )^6}\, + \,{(x - \sqrt {{x^3} - 1} )^6}\,,

(x > 1)

is equal to

View answer and explanation

Correct option: D

Option D (24) is correct because it accurately calculates the sum of the coefficients of all even degree terms in the given expansion.

401 Easy

401 of 434

Application +4 | -1

The number of terms in the expansion of

(\sqrt{5}+\sqrt[4]{11})^{124}

which are integers, is equal to

View answer and explanation

Correct option: D

Option D (32) is correct because there are 32 integer terms in the expansion, determined by the conditions on r.

402 Easy

402 of 434

Application +4 | -1

The coefficient of

{x^m}

{\left( {1 + x} \right)^p} + {\left( {1 + x} \right)^{p + 1}} + \ldots + {\left( {1 + x} \right)^n},p \le m \le n

View answer and explanation

Correct option: A

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

The correct answer is

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

because the coefficient of

{x^m}

can be found by manipulating the sum into a single term and then finding the coefficient of

{x^{m+1}}

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

403 Easy

403 of 434

Application +4 | -1

If the ratio of the 7th term from the beginning to the seventh term from the end in the expansion of

{\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^x}

\frac{1}{6}

then

x

, is

View answer and explanation

Correct option: A

Option A (9) is correct because it satisfies the given ratio condition of the 7th terms from the beginning and end of the expansion.

404 Easy

404 of 434

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

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

View answer and explanation

Correct option: B

{\left( { - 1} \right)^n}

Option B (

{\left( { - 1} \right)^n}

) is correct because the given expression simplifies to

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

, and the coefficient of

{x^n}

is then

{\left( { - 1} \right)^n}

405 Easy

405 of 434

Application +4 | -1

Find the coefficient of

x^{13}

in the expansion of

(1-x)^5\left(1+x+x^2+x^3\right)^4

View answer and explanation

Correct option: A

Option A (4) is correct because it is the coefficient of

x^{13}

after simplifying the expression

(1-x)^5(1+x+x^2+x^3)^4

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406 Easy

406 of 434

Analyzing +4 | -1

(1+a x)^n=1+10 x+40 x^2+\ldots

, then value of

\frac{a+n}{a-n}

View answer and explanation

Correct option: D

-7 / 3

Option D (-7 / 3) is correct because by comparing coefficients and solving the resulting equations, we find a=2 and n=5, leading to the correct value of (a+n)/(a-n).

407 Easy

407 of 434

Application +4 | -1

\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2 \ldots \ldots+a_{2 n} x^{2 n}

. Then value of

a_0+a_3+a_6 \ldots \ldots

. is equal to

View answer and explanation

Correct option: C

3^{n-1}

Option C (

3^{n-1}

) is correct because by substituting

x=1, \omega, \omega^2

and adding them, we can isolate the desired sum.

408 Easy

408 of 434

Application +4 | -1

If the

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

term in the expansion of

{\left( {\frac{{{a^{1/3}}}}{{{b^{1/6}}}} + \frac{{{b^{1/2}}}}{{{a^{1/6}}}}} \right)^{21}}

has equal exponents of both

a

and

b

, then value of

r

View answer and explanation

Correct option: B

Option 9 is correct. By equating the exponents of a and b in the (r+1)th term, we find r=9.

409 Easy

409 of 434

Analyzing +4 | -1

X = \left\{ {{4^n} - 3n - 1:n \in N} \right\}

and

Y = \left\{ {9\left( {n - 1} \right):n \in N} \right\}

then

X \cup Y

is equal to:

View answer and explanation

Correct option: B

Option B (Y) is correct because

X \subseteq Y

, therefore

X \cup Y = Y

410 Easy

410 of 434

Analyzing +4 | -1

The sum to infinity of the series

1+\frac{2}{3}+\frac{6}{3^2}+\frac{10^3}{3^3}+\frac{14}{3^4}+\ldots .

. , is

View answer and explanation

Correct option: B

Option B (3) is correct because the sum to infinity of the series is 3.

411 Easy

411 of 434

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

{\left( {1 + {x^2}} \right)^5}{\left( {1 + x} \right)^4},

View answer and explanation

Correct option: B

Option B (60) is correct because by expanding the product of binomials and finding the terms that contribute to x⁵, the coefficient is 60.

412 Easy

412 of 434

Application +4 | -1

a,\;b,\;c

are in AP, then the sum of the coefficients of

{\left\{ {1 + {{\left( {a{x^2} - 2bx + c} \right)}^2}} \right\}^{1973}}

View answer and explanation

Correct option: D

Option D (1) is correct because when

a, b, c

are in AP,

a - 2b + c = 0

, and substituting

x = 1

into the expression results in 1.

413 Easy

413 of 434

Application +4 | -1

{C_0}{C_r} + {C_1}{C_{r + 1}} + {C_2}{C_{r + 2}} + ... + {C_{n - r}}{C_n}

is equal to

View answer and explanation

Correct option: A

\frac{{\left( {2n} \right)!}}{{\left( {n - r} \right)!\left( {n + r} \right)!}}

Option A (

\frac{{\left( {2n} \right)!}}{{\left( {n - r} \right)!\left( {n + r} \right)!}}

) is correct because it represents the coefficient of x^(n+r) in the expansion of (1+x)⁽²ⁿ⁾.

414 Easy

414 of 434

Application +4 | -1

The number of non zero terms in the expansion of

(1+3 \sqrt{2} x)^9+(1-3 \sqrt{2} x)^9

View answer and explanation

Correct option: C

Option C (5) is correct because only the even-powered terms survive after adding the two expansions.

415 Easy

415 of 434

Application +4 | -1

The first 3 terms in the expansion of

{\left( {1 + ax} \right)^n}\;\left( {n \ne 0} \right)

are

1,\;6x,

and

16{x^2}

. Then, the value of

a

and

n

are respectively

View answer and explanation

Correct option: C

\frac{2}{3}

and 9

Option C (

\frac{2}{3}

and 9) is correct because these values satisfy the equations derived from the given expansion.

416 Easy

416 of 434

Application +4 | -1

If the expansion of

{\left( {1 + x} \right)^{20}},

the coefficients of

{r^{th}}

and

{\left( {r + 4} \right)^{th}}

terms are equal, then the value of

r

, is

View answer and explanation

Correct option: C

Option C (9) is correct because the equality of binomial coefficients implies either the indices are equal or their sum equals the upper index.

417 Easy

417 of 434

Application +4 | -1

The value of

x

in the expansion

{\left[ {x + {x^{{{\log }_{10}}x}}} \right]^5}

, if the third term in the expansion is 1000000, is

View answer and explanation

Correct option: A

Option A (10) is correct because substituting x=10 into the third term of the expansion satisfies the given condition that the third term is 1000000.

418 Easy

418 of 434

Analyzing +4 | -1

Statement-1

\sum\limits_{r=0}^n(r+1){ }^n C_r=(n+2) \cdot 2^{n-1}

Statement - 2

\sum\limits_{{r=0}}^n(r+1)^n C_r x^r

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

View answer and explanation

Correct option: A

Statement-1 is true, Statement-2 is true; Statement-2 is a
correct explanation for Statement-1

Option A is correct because both Statement-1 and Statement-2 are true, and Statement-2 provides a general formula that, when evaluated at x=1, proves Statement-1.

419 Easy

419 of 434

Application +4 | -1

The number of terms with integral coefficients in the expansion of

{\left( {{7^{1/3}} + {5^{1/2}}x} \right)^{600}},

View answer and explanation

Correct option: C

101

Option C (101) is correct because the number of terms with integral coefficients corresponds to the number of multiples of 6 between 0 and 600, inclusive.

420 Easy

420 of 434

Application +4 | -1

In the expansion of

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

the number of terms, is

View answer and explanation

Correct option: D

Option D (4) is correct because after simplification and combining like terms, the expansion results in a polynomial with only 4 terms.

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