$\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + . \ldots {\rm{is\;equal\;to}}$

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

$\frac{{{2^{n + 1}} - 1}}{{n + 1}}$

Integral part of $(2+\sqrt{5})^{2 n+1}$ is $(n \in N)$

an even or an odd number depending upon the value of $n$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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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 241-270 of 951 questions

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241 Easy

241 of 951

Application +4 | -1

The 8 th term of

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

, when expanded in ascending power of

x

, is

View answer and explanation

Correct option: A

\frac{228096}{x^3}

Option A (

\frac{{228096}}{{x^3}}

) is correct because it is the 8th term in the expansion.

242 Easy

242 of 951

Application +4 | -1

Let

{T_n}

denotes the number of triangles which can be formed using the vertices of a regular polygon of

n

sides. If

{T_{n + 1}} - {T_n} = 21

then

n

equals

View answer and explanation

Correct option: B

The correct answer is B (7) because it satisfies the given condition

{T_{n + 1}} - {T_n} = 21

where

{T_n}

represents the number of triangles formed using vertices of a regular polygon.

243 Medium

243 of 951

Application +4 | -1

If the 4 th term in the expansion of

\left(p x+x^{-1}\right)^m

is 2.5 for all

x \in R

, then

View answer and explanation

Correct option: B

p=\frac{1}{2}, m=6

Option B (

p=\frac{1}{2}, m=6

) is correct because by equating the power of x to zero and comparing the constant term, we can find the values of p and m.

244 Easy

244 of 951

Application +4 | -1

t_n=n \cdot n

! then

\sum_{n=1}^{30} t_n=

View answer and explanation

Correct option: B

31 !-1

Option B (

31 !-1

) is correct because the summation telescopes, resulting in

31! - 1! = 31! - 1

245 Hard

245 of 951

Analyzing +4 | -1

The coefficient of

{x^{50}}

in the expression

{\left( {1 + x} \right)^{1000}} + 2\;x{\left( {1 + x} \right)^{999}} + 3\;{x^2}{\left( {1 + x} \right)^{998}} + \ldots + 1001\;{x^{1000}}

View answer and explanation

Correct option: C

{\;^{1002}}{C_{50}}

The correct answer is C (

{\;^{1002}}{C_{50}}

) because the given series can be simplified to a single binomial term using summation techniques.

246 Easy

246 of 951

Application +4 | -1

a > 0, b > 0

and

a^2+b=2

, then the maximum value of the term independent of

x

in the expansion of

\left(a x^{\frac{1}{6}}+b x^{-\frac{1}{3}}\right)^9

View answer and explanation

Correct option: B

Option B (84) is correct because the maximum value of the term independent of x is found by using the binomial theorem and AM >= GM inequality.

247 Easy

247 of 951

Application +4 | -1

The coefficient of

x

in the expansion of

\left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right) \ldots \ldots \left( {1 + 100x} \right)

View answer and explanation

Correct option: A

5050

Option A (5050) is correct because the coefficient of x is the sum of the integers from 1 to 100, which is calculated as

\frac{{100(100 + 1)}}{2} = 5050

248 Medium

248 of 951

Application +4 | -1

The coefficient of

x^5

in the expansion of

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

View answer and explanation

Correct option: B

Option B,

\left(\begin{array}{c}34 \\ 3\end{array}\right)

, is correct because it represents the number of non-negative integer solutions to the given equation.

249 Easy

249 of 951

Application +4 | -1

The remainder when

7^{2022}+3^{2022}

is divided by 5 is:

View answer and explanation

Correct option: C

The correct answer is C (3) because when

7^{2022}+3^{2022}

is divided by 5, the remainder is 3.

250 Medium

250 of 951

Application +4 | -1

Let

\left[ x \right]

denote the greatest integer less than or equal to

x

. If

x = {\left( {\sqrt 3 + 1} \right)^5}

, then

\left[ x \right]

is equal to

View answer and explanation

Correct option: D

152

Option D (152) is correct because it is the greatest integer less than or equal to the given expression after expansion and approximation.

251 Medium

251 of 951

Application +4 | -1

(1+\mathrm{x})^n=\mathrm{C}_0+\mathrm{C}_1 \mathrm{x}+\mathrm{C}_2 \mathrm{x}^2+\ldots+\mathrm{C}_n \mathrm{x}^{\mathrm{n}}

, then

\mathrm{C}_0 \mathrm{C}_1+\mathrm{C}_1 \mathrm{C}_2+\mathrm{C}_2 \mathrm{C}_3+\ldots .+\mathrm{C}_{n-1} \mathrm{C}_n

is equal to-

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the given expression can be interpreted as the coefficient of

x^{n-1}

in the expansion of

(1+x)^n(1+x)^n

, which simplifies to

{}^{2n}C_{n-1}

252 Medium

252 of 951

Application +4 | -1

If it is known that the third term of the binomial expansion

\left(x+x^{\log _{10} x}\right)^5

10^6

then

x

is equal to

View answer and explanation

Correct option: A

Option A (10) is correct because solving the equation formed by the third term of the binomial expansion leads to x=10.

253 Easy

253 of 951

Application +4 | -1

The coefficient of

{x^{ - 7}}

in the expansion of

{\left[ {ax - \frac{1}{{b{x^2}}}} \right]^{11}}

will be

View answer and explanation

Correct option: B

\frac{{462{a^5}}}{{{b^6}}}

Option B is correct because the coefficient of

{x^{-7}}

is found by determining the appropriate term in the binomial expansion and calculating its value.

254 Medium

254 of 951

Application +4 | -1

If the third term in the expansion of

\left(\frac{1}{x}+x^{\log _{10} x}\right)^5

x > 1

, is 1000, then value of

x

View answer and explanation

Correct option: C

100

Option C (100) is correct because solving the equation derived from the third term of the binomial expansion yields x=100.

255 Easy

255 of 951

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

\frac{{\left( {a - bx} \right)}}{{{e^x}}}

View answer and explanation

Correct option: A

\frac{{{{\left( { - 1} \right)}^n}}}{{n!}}\left( {a + bn} \right)

Option A is correct because it represents the coefficient of x^n in the expansion of the given function using the Taylor series expansion of e^(-x).

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256 Medium

256 of 951

Application +4 | -1

If the coefficient of the

x^7

and

x^8

in the expansion of

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

are equal then the value of

{n}

is equal to

View answer and explanation

Correct option: C

Option C (55) is correct because setting the coefficients of

x^7

and

x^8

equal in the binomial expansion of

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

and solving for n yields 55.

257 Easy

257 of 951

Application +4 | -1

The expansion of

\frac{1}{{{{\left( {4 - 3x} \right)}^{1/2}}}}

by Only One Correct Option will be valid, if

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the expansion is valid only when

|\frac{3}{4}x| < 1

, which means

-\frac{4}{3} < x < \frac{4}{3}

, and none of the provided options match this range.

258 Medium

258 of 951

Application +4 | -1

Let

n \in N

. If

(1+x)^n=a_0+a_1 x+a_2 x^2+\ldots+a_n x^n

and

a_{n-3}, a_{n-2}, a_{n-1}

are in

\mathrm{AP}

, then

View answer and explanation

Correct option: A

a_1, a_2, a_3

are in

\mathrm{AP}

Option A (

a_1, a_2, a_3

are in AP) is correct because binomial coefficients equidistant from the beginning and the end are equal, implying

a_1, a_2, a_3

are in AP.

259 Medium

259 of 951

Analyzing +4 | -1

\sum\limits_{{r=0}}^n(-1)^r \frac{{ }^n C_r}{{ }^{r+3} C_r}

is equal to

View answer and explanation

Correct option: B

\frac{3 !}{2(n+3)}

Option C (

\frac{{3 !}}{{(n+3)}}

) is the correct answer. After simplifying the summation, the expression reduces to this form.

260 Easy

260 of 951

Application +4 | -1

x

is so small that

\left(\frac{1+3 x}{1-2 x}\right)^2=1+k x

, then

k=

View answer and explanation

Correct option: C

Option C (10) is correct because it is the coefficient of x in the expansion of the given expression, considering x is very small.

261 Medium

261 of 951

Application +4 | -1

(1+a x)^n=1+8 x+24 x^2+\ldots \ldots

; then

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

is equal to

View answer and explanation

Correct option: C

-1 / 3

Option C (-1/3) is correct because by comparing coefficients in the binomial expansion, we can solve for a and n and then calculate (a-n)/(a+n).

262 Medium

262 of 951

Application +4 | -1

The value of

\sqrt{e}

will be

View answer and explanation

Correct option: A

1.648

Option A (1.648) is correct because it is the approximate value of

\sqrt{e}

calculated using the Taylor series expansion of

e^x

263 Medium

263 of 951

Application +4 | -1

Let

{S}_{{n}}={ }^{{n}} {C}_0+{ }^{{n}} {C}_3+{ }^{{n}} {C}_6+{ }^{{n}} {C}_9+\ldots \ldots

, then the value of

{S}_{{n}}

, if

{n}

is not a multiple of 3, is

View answer and explanation

Correct option: B

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

Option B is correct because the value of S_n, when n is not a multiple of 3, is (2^n - (-1)^n) / 3, which is derived using complex roots of unity.

264 Medium

264 of 951

Application +4 | -1

The value of

\sum\limits_{{r=1}}^n \frac{r \cdot{ }^n C_r}{{ }^n C_{r-1}}

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the summation simplifies to an arithmetic series after simplifying the binomial coefficients.

265 Easy

265 of 951

Application +4 | -1

\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + . \ldots \mathrm{is\;equal\;to}

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the given series can be expressed as an integral, and the result simplifies to

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

266 Medium

266 of 951

Analyzing +4 | -1

Integral part of

(2+\sqrt{5})^{2 n+1}

(n \in N)

View answer and explanation

Correct option: A

an even number

Option A (an even number) is correct because the integral part of the expression is always an even number.

267 Medium

267 of 951

Analyzing +4 | -1

y = \frac{1}{3} + \frac{{1.3}}{{3.6}} + \frac{{1.3.5}}{{3.6.9}} + \ldots ,

then the value of

{y^2} + 2\;y

View answer and explanation

Correct option: A

Option A (2) is correct because by comparing the given series with the binomial expansion and simplifying, we find that

{y^2} + 2y = 2

268 Easy

268 of 951

Application +4 | -1

The greatest term in the expansion of

{\left( {1 + 3x} \right)^{54}},

where

x = \frac{1}{3}

View answer and explanation

Correct option: A

{T_{28}}

Option A (

{T_{28}}

) is correct because it is the greatest term in the expansion, determined by the given condition and the binomial theorem.

269 Medium

269 of 951

Application +4 | -1

2^{60}

when divided by 7 leave the remainder

View answer and explanation

Correct option: A

Option A (1) is correct because the remainder is found by rewriting the expression and applying the binomial theorem.

270 Easy

270 of 951

Analyzing +4 | -1

If the fourth term of

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

is equal to 200 and

x>1, x

is equal to

View answer and explanation

Correct option: B

10

Option B (10) is correct because substituting the given condition into the binomial expansion and solving the resulting logarithmic equation yields x = 10.

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