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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 421-434 of 434 questions

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421 Easy

421 of 434

Application +4 | -1

In the expansion of

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

, the constant term is

View answer and explanation

Correct option: B

-{ }^{15} C_6

Option B (

-{ }^{15} C_6

) is correct because the constant term is found by determining the value of r that makes the exponent of x equal to zero, and then calculating the corresponding binomial coefficient.

422 Easy

422 of 434

Application +4 | -1

In the expansion of

{\left( {2{x^2} - \frac{1}{x}} \right)^{12}}

, the term independent of

x

View answer and explanation

Correct option: C

9th

Option C (9th) is correct because the term independent of

x

occurs when the power of

x

is 0, which happens at the 9th term in the expansion.

423 Easy

423 of 434

Application +4 | -1

If the term independent of

x

in the expansion of

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

is 405 , then

k

equals

View answer and explanation

Correct option: B

3,-3

Option B (

3, -3

) is correct because the term independent of x is found by setting the power of x to zero and solving for k.

424 Easy

424 of 434

Application +4 | -1

Sum of the series

2 \mathrm{C}_0+\frac{\mathrm{C}_1}{2} 2^2+\frac{\mathrm{C}_2}{3} 2^3+\frac{\mathrm{C}_3}{4} 2^4+\ldots \ldots \cdot \frac{\mathrm{C}_{\mathrm{n}}}{\mathrm{n}+1} 2^{\mathrm{n}+1}

equals-

View answer and explanation

Correct option: B

\frac{3^{\mathrm{n}+1}-1}{\mathrm{n}+1}

The correct answer is B (

\frac{{3^{n+1}-1}}{{n+1}}

) because the given series can be expressed as a summation involving binomial coefficients, which simplifies to the given expression.

425 Easy

425 of 434

Analyzing +4 | -1

{(2{x^2} - x - 1)^5} = {a_0} + {a_1}x + {a_2}{x^2} + ... + {a_{10}}{x^{10}},\;\mathrm{then\;}{a_2} + {a_4} + {a_6} + {a_8} + {a_{10}}

is equal to

View answer and explanation

Correct option: A

Option A (15) is correct because by substituting x = 1 and x = -1 into the given equation and adding the resulting equations, we can isolate and find the value of

{a_2} + {a_4} + {a_6} + {a_8} + {a_{10}}

426 Easy

426 of 434

Application +4 | -1

The sum of the magnitudes of the coefficients in the expansion of

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

View answer and explanation

Correct option: D

{4^n}

Option D is correct. The sum of the magnitudes of the coefficients is found by substituting x = -1 into the expression.

427 Easy

427 of 434

Analyzing +4 | -1

Sum of the series

\sum\limits_{{k=0}}^n \frac{(2 n) !}{(k !)^2(n-k !)^2}

equals

View answer and explanation

Correct option: B

\left({ }^{2 n} C_n\right)^2

Option B (

\left({ }^{2 n} C_n\right)^2

) is correct because the given sum can be simplified to

\left({ }^{2 n} C_n\right) \sum_{k=0}^n\left({ }^n C_k\right)^2

, which equals

\left({ }^{2 n} C_n\right)^2

428 Easy

428 of 434

Application +4 | -1

The coefficient of

{x^4}

in the expansion of

{\left( {1 + x + {x^2} + {x^3}} \right)^{11}},

View answer and explanation

Correct option: C

990

Option C (990) is correct because it is the sum of the products of binomial coefficients that contribute to the x⁴ term in the expansion.

429 Easy

429 of 434

Application +4 | -1

If the fourth term in the expansion of

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

is equal to 200 and

x > 1,

, then

x

is equal to

View answer and explanation

Correct option: B

Option B (10) is correct because the value of x is found by setting the fourth term in the given expansion equal to 200 and solving the resulting equation for x, given that x > 1.

430 Easy

430 of 434

Application +4 | -1

{\;^n}{C_{12}} = {\;^n}{C_6}

, then

{\;^n}{C_2}

is equal to

View answer and explanation

Correct option: B

153

Option B (153) is correct. Using the property that if

^nC_x = ^nC_y

, then either x=y or x+y=n, we can find n and then calculate

^nC_2

431 Easy

431 of 434

Analyzing +4 | -1

The total number of terms in the expansion of

(x+a)^{200}+(x-a)^{200}

after simplification is

View answer and explanation

Correct option: A

101

Option A (101) is correct because after simplification, only the even-powered terms remain, resulting in 101 terms.

432 Easy

432 of 434

Application +4 | -1

If the fourth term in the expansion of

{(x + {x^{{{\log }_2}x}})^7}

is 4480, then the value of x, Where

x \in \mathrm{N}

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because substituting x = 2 into the binomial expansion and solving the resulting equation yields a consistent result.

433 Easy

433 of 434

Application +4 | -1

The sum of the coefficients of all the integral powers of

x

in the expansion of

(1+2 \sqrt{x})^{40}

View answer and explanation

Correct option: D

\frac{1}{2}\left(3^{40}+1\right)

Option D (

\frac{1}{2}\left(3^{40}+1\right)

) is correct because it represents the sum of the coefficients of all integral powers of x in the expansion of

(1+2 \sqrt{x})^{40}

434 Easy

434 of 434

Application +4 | -1

\left| x \right| < \frac{1}{2}

, then the coefficient of

{x^r}

in the expansion of

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

, is

View answer and explanation

Correct option: D

\left( {2r + 1} \right){2^r}

Option D (

(2r + 1){2^r}

) is correct because it represents the coefficient of

{x^r}

after expanding and simplifying the given expression.

Revision notes and key takeaways

Review BINOMIAL THEOREM in your textbook or class notes, then use this set to check your understanding.

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