Back to BINOMIAL THEOREM

Your Progress

0 / 434 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 301-330 of 434 questions

Preparing practice and saved progress…

301 Easy

301 of 434

Analyzing +4 | -1

Let

{(1 + x + 2{x^2})^{20}} = {a_0} + {a_1}x + {a_2}{x^2} + ..... + {a_{40}}{x^{40}}

, then

{a_1} + {a_3} + {a_5} + .... + {a_{37}}

is equal to

View answer and explanation

Correct option: B

{2^{19}}({2^{20}} - 21)

Option B (

{2^{19}}({2^{20}} - 21)

) is correct because it is derived by substituting x = 1 and x = -1 into the polynomial and then subtracting the resulting equations.

302 Easy

302 of 434

Application +4 | -1

Let

{\left( {1 + x} \right)^n} = 1 + {a_1}x + {a_2}{x^2} + .... + {a_n}{x^n}

. If

{a_1},\;{a_2}\,\mathrm{and\;}{a_3}

are in AP, then the value of

n

View answer and explanation

Correct option: D

Option D (7) is correct because by equating the given condition that a1, a2, and a3 are in AP, we can solve for n, which results in n=7.

303 Easy

303 of 434

Application +4 | -1

Let

{C_1},\;{C_2},\;{C_3}

are the usual binomial coefficients. Let

S = {C_1} + 2{C_2} + 3{C_3} + ... + n{C_n}

, then

S

equals

View answer and explanation

Correct option: C

n{2^{n - 1}}

Option C (

n{2^{n - 1}}

) is correct because it is the sum of the given series of binomial coefficients.

304 Easy

304 of 434

Application +4 | -1

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

then the value of

{a_0} + {a_2} + {a_4} + .... + {a_{2n}}

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because substituting x = 1 and x = -1 into the given expression and adding them allows us to isolate the desired sum.

305 Easy

305 of 434

Analyzing +4 | -1

If the coefficient of the middle term in the expansion of

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

p

and the coefficients of middle term in the expansion of

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

are

q

and

r

, then

View answer and explanation

Correct option: C

p = q + r

Option C (

p = q + r

) is correct because the middle term's coefficient in the expansion of

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

is the sum of the two middle terms' coefficients in the expansion of

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

306 Easy

306 of 434

Application +4 | -1

\left| x \right| < 1

, then the coefficient of

{x^6}

in the expansion of

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

View answer and explanation

Correct option: A

Option A (3) is correct because the coefficient of x⁶ in the expansion of (1 + x + x²)^(-3) is calculated to be 3.

307 Easy

307 of 434

Application +4 | -1

The value of

{\left( {0.99} \right)^{15}}

View answer and explanation

Correct option: B

0.8601

Option B (0.8601) is correct because using the binomial approximation, we can calculate the value of (0.99)¹⁵ to be approximately 0.8601.

308 Easy

308 of 434

Application +4 | -1

The Binomial coefficient of the 4 th term in the expansion of

(x-q)^5

View answer and explanation

Correct option: C

Option C (10) is correct as it represents the binomial coefficient of the 4th term in the expansion of (x-q)⁵.

309 Easy

309 of 434

Application +4 | -1

The total number of terms in the expansion of

(a+b+c+d)^n, n \in N

View answer and explanation

Correct option: C

\frac{(n+1)(n+2)(n+3)}{6}

Option C (

\frac{{(n+1)(n+2)(n+3)}}{{6}}

) is correct because it represents the number of terms in the expansion of (a+b+c+d)^n, which is a combination with repetition problem.

310 Easy

310 of 434

Application +4 | -1

The smallest natural number

n

, such that the coefficient of x in the expansion of

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

^n{C_{23}}

, is

View answer and explanation

Correct option: D

Option D (38) is correct because it is the smallest natural number n that satisfies the condition for the coefficient of x in the expansion to be

^n{C_{23}}

311 Easy

311 of 434

Analyzing +4 | -1

\sum\limits_{\mathop {i,j = 0}\limits_{i \ne j} }^n {{\,^n}{C_i}^n{C_j}}

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it represents the total number of ways to choose i and j from 0 to n, excluding the cases where i equals j.

312 Easy

312 of 434

Analyzing +4 | -1

If the coefficients of the three successive terms in the binomial expansion of

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

are in the ratio

1: 7: 42

, then the first of these terms in the expansion is:

View answer and explanation

Correct option: C

7^{\text {th }}

The correct answer is C (7th) because by setting up ratios of consecutive binomial coefficients and solving for r, we find that the first term is the 7th term.

313 Easy

313 of 434

Application +4 | -1

If sum of the coefficients in the expansion of

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

is 128, then coefficient of

x

in the expansion of

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

View answer and explanation

Correct option: C

Option D (35) is correct because the sum of coefficients being 128 implies n=7, and the coefficient of x in the expansion is 35.

314 Easy

314 of 434

Application +4 | -1

The value of

2\left(n c_1\right)+2^3\left(n c_3\right)+2^5\left(n c_5\right) \ldots . . .2 n

' being an integers is

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it represents the closed-form expression for the given series, derived from the binomial theorem.

315 Easy

315 of 434

Application +4 | -1

The sum of coefficients in the expansion of

(7 x-8 y)^{149}

View answer and explanation

Correct option: B

-1

Option B (-1) is correct because the sum of the coefficients in the expansion is found by substituting x=1 and y=1 into the expression, which results in (7-8)¹⁴⁹ = (-1)¹⁴⁹ = -1.

Explore Other Chapters in MATHEMATICS

View All →

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

316 Easy

316 of 434

Application +4 | -1

If in the expansion of

{\left( {a - 2b} \right)^n}.

. The sum of the 5th and 6th term is zero, then the value of

\frac{a}{b}

View answer and explanation

Correct option: B

\frac{{2\left( {n - 4} \right)}}{5}

Option B (

\frac{{2\left( {n - 4} \right)}}{5}

) is correct because it is the result obtained after simplifying the equation derived from the given condition that the sum of the 5th and 6th terms is zero.

317 Easy

317 of 434

Application +4 | -1

The expression

\frac{1}{{\sqrt[3]{{6 - 3x}}}}

is equal to

View answer and explanation

Correct option: B

{6^{ - 1/3}}\left[ {1 + \frac{x}{6} + \frac{{2{x^2}}}{{{6^2}}} + ...} \right]

Option B (

{6^{ - 1/3}}\left[ {1 + \frac{x}{6} + \frac{{2{x^2}}}{{{6^2}}} + ...} \right]

) is correct because by rewriting the expression and applying the binomial theorem, the expansion matches option B.

318 Easy

318 of 434

Application +4 | -1

r

and

n

are positive integers such that

r > 1,n > 2

and coefficients of

{\left( {r + 2} \right)^{\mathrm{th}}}

term and

{\left( {3r} \right)^{\mathrm{th}}}

term in the expansion of

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

are equal, then

n

equals

View answer and explanation

Correct option: C

2r

Option C (2r) is correct because equating the coefficients and solving for n yields n = 2r.

319 Easy

319 of 434

Analyzing +4 | -1

The total number of irrational terms in the binomial expansion of

{\left( {{7^{1/5}} - {3^{1/10}}} \right)^{60}}

View answer and explanation

Correct option: C

Option C (54) is correct because by finding the number of rational terms in the binomial expansion, we can subtract that from the total number of terms to find the number of irrational terms.

320 Easy

320 of 434

Application +4 | -1

If the 4th term in the expansion of

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

is independent of

x

, then

n

is equal to

View answer and explanation

Correct option: B

Option B (6) is correct because setting the power of x to zero in the 4th term of the expansion

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

allows us to solve for n, which equals 6.

321 Easy

321 of 434

Application +4 | -1

\lambda=\frac{1 \times 2^2+2 \times 3^2+3 \times 4^2+\ldots \text { upto } 99 \text { terms }}{1+2+3+\ldots \text { upto } 100 \text { terms }}

then

\lambda

equals

View answer and explanation

Correct option: A

4983

Option A (4983) is correct because after simplifying the summation, we get

\frac{{n(n+1)}}{{2}} - \frac{{2n+1}}{{3}}

which equals 4983 when n=100.

322 Easy

322 of 434

Application +4 | -1

The value of

2\;{C_0} + \frac{{{2^2}}}{2}{C_1} + \frac{{{2^3}}}{3}{C_2} + \frac{{{2^4}}}{4}{C_3} + \ldots + \frac{{{2^{11}}}}{{11}}{C_{10}},

View answer and explanation

Correct option: A

\frac{{{3^{11}} - 1}}{{11}}

Option A (

\frac{{{3^{11}} - 1}}{{11}}

) is correct because it represents the sum of the given series, which can be simplified using binomial theorem and integration.

323 Easy

323 of 434

Analyzing +4 | -1

A={ }^{2 n} C_0 \cdot{ }^{2 n} C_1+{ }^{2 n} C_1 \cdot{ }^{2 n-1} C_1+{ }^{2 n} C_2 \cdot{ }^{2 n-2} C_1 +\ldots, \text { then } A \text { is }

View answer and explanation

Correct option: C

n \cdot 2^{2 n}

Option C (

n \cdot 2^{2 n}

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

324 Easy

324 of 434

Application +4 | -1

The coefficient of

{t^{24}}

in the expansion of

{\left( {1 + {t^2}} \right)^{12}}\left( {1 + {t^{12}}} \right)\left( {1 + {t^{24}}} \right)

View answer and explanation

Correct option: A

{\;^{12}}{C_6} + 2

Option A (

{\;^{12}}{C_6} + 2

) is correct because it represents the sum of the coefficients that contribute to the

{t^{24}}

term in the expansion.

325 Easy

325 of 434

Application +4 | -1

Coefficient of

x

in the expansion of

{\left( {{x^2} + \frac{a}{x}} \right)^5}

View answer and explanation

Correct option: B

10{a^3}

Option B (

10{a^3}

) is correct because it is the coefficient of x obtained after expanding the given binomial expression and solving for the term with x.

326 Easy

326 of 434

Application +4 | -1

\left(1+x-3 x^2\right)^{2145}=a_0+a_1 x+a_2 x^2+\ldots

, then

a_0-a_1+a_2-a_3+\ldots

ends with

View answer and explanation

Correct option: B

Option B (3) is correct because substituting

x=-1

into the given expression and simplifying yields a number that ends in 3.

327 Easy

327 of 434

Application +4 | -1

If the value of

x

is so small that

{x^2}

and greater powers can be neglected, then

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

is equal to

View answer and explanation

Correct option: B

1 - \frac{5}{6}x

Option B (

1 - \frac{5}{6}x

) is correct because it is the simplified form of the given expression when higher powers of x are neglected.

328 Easy

328 of 434

Analyzing +4 | -1

n

is an integer with

0 \leq n \leq 11

, then the minimum value of

n !(11-n) !

is attained when a value of

n

equals

View answer and explanation

Correct option: B

Option B (5) is correct because

n!(11-n)!

is minimized when

{ }^{11}C_n = \frac{11!}{n!(11-n)!}

is maximized, which occurs when n is closest to the middle value, 5 or 6.

329 Easy

329 of 434

Application +4 | -1

Number of irrational terms in the expansion of

\left(4+3^{1 / 4}\right)^{15}

View answer and explanation

Correct option: A

Option A (12) is correct because the number of irrational terms is found by subtracting the number of rational terms from the total number of terms in the binomial expansion.

330 Easy

330 of 434

Application +4 | -1

{a_k}

is the coefficient of

\;{x^k}

in the expansion of

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

for

k = 0,\;1,2,\;

….,

2n

then

View answer and explanation

Correct option: D

n{.3^n}

Option D (

n{.3^n}

) is correct because differentiating the given expression and substituting x=1 yields the required sum of coefficients.

Revision notes and key takeaways

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

Showing 301-330 of 434 questions

Your Progress

Binomial theorem practice · Easy

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice