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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 241-270 of 434 questions

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241 Easy

241 of 434

Application +4 | -1

In the expansion of

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

, the coefficient of

{x^0}

View answer and explanation

Correct option: B

- 20

Option B (-20) is correct. The constant term in the expansion of (x - 1/x)⁶ is -20, found by determining the term where the power of x is 0.

242 Easy

242 of 434

Analyzing +4 | -1

\frac{1}{{1!\left( {n - 1} \right)!}} + \frac{1}{{3!\left( {n - 3} \right)!}} + \frac{1}{{5!\left( {n - 5} \right)!}} + ...

is equal to

View answer and explanation

Correct option: B

\frac{{{2^{n - 1}}}}{{n!}}

The correct answer is B (

\frac{{{2^{n - 1}}}}{{n!}}

) because it is derived from recognizing the sum as part of the binomial expansion of

(1+1)^n

and

(1-1)^n

243 Easy

243 of 434

Application +4 | -1

If the constant term in the expansion of

\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0,

\alpha \times 2^8 \times \sqrt[5]{3},

then

25 \alpha

is equal to:

View answer and explanation

Correct option: C

693

The correct answer is C (693). The constant term in the binomial expansion is found by setting the power of x to zero and solving for the term.

244 Easy

244 of 434

Application +4 | -1

The coefficient of

x^{53}

\sum\limits_{{r=0}}^{100}{ }^{100} C_r(x-3)^{100-r} 2^r

View answer and explanation

Correct option: C

{ }^{-100} C_{53}

Option C (

{}^{-100} C_{53}

) is correct because the given expression simplifies to (x-1)¹⁰⁰, and the coefficient of x⁵³ is found using the binomial theorem.

245 Easy

245 of 434

Application +4 | -1

In the expansion of

{(1 + x)^{30}}

, the sum of the coefficients of odd powers of

x

View answer and explanation

Correct option: D

{2^{29}}

Option D (

{2^{29}}

) is correct because the sum of coefficients of odd powers of x in the expansion of

{(1 + x)^{30}}

is equal to

{2^{n-1}}

, where n is the power.

246 Easy

246 of 434

Application +4 | -1

The coefficient of the middle term in the expansion of

{\left( {x + 2y} \right)^6}

View answer and explanation

Correct option: B

8\left( {{\;^6}{C_3}} \right)

Option B (8(^6C_3)) is correct because the coefficient of the middle term in the expansion of (x + 2y)⁶ is 8 times 6C3.

247 Easy

247 of 434

Analyzing +4 | -1

\sum\limits_{r=0}^{n-1} \frac{{ }^n C_r}{{ }^n C_r+{ }^n C_{r+1}}

is equal to

View answer and explanation

Correct option: A

\frac{n}{2}

Option A (n/2) is correct because simplifying the summation using binomial coefficient identities leads to a straightforward calculation.

248 Easy

248 of 434

Application +4 | -1

If the coefficient of

x

\left(x^2+k / x\right)^5

is 270, then the value of

k

View answer and explanation

Correct option: A

Option A (3) is correct because the coefficient of

x

\left(x^2+k / x\right)^5

being 270 leads to the equation

{ }^5 C_3 k^3=270

, which gives

k=3

249 Easy

249 of 434

Analyzing +4 | -1

The term independent of

x

in the expansion of

\left(1-\frac{1}{2 x}+3 x^7\right)\left(x^2+\frac{1}{3 x}\right)^{10}

View answer and explanation

Correct option: A

10 / 3^8

Option A (

10 / 3^8

) is correct because it is the term independent of x after expanding the given expression and finding the constant term.

250 Easy

250 of 434

Analyzing +4 | -1

The number of values of '

r

' satisfying the equation

{ }^{39} C_{3 r-1}-{ }^{39} C_{r^2}={ }^{39} C_{r^2-1}-{ }^{39} C_{3 r}

View answer and explanation

Correct option: B

Option B (2) is correct because solving the given equation using properties of combinations yields two valid solutions for 'r': 3 and 5.

251 Easy

251 of 434

Application +4 | -1

The coefficient of

{x^{ - 10}}\;\mathrm{in\;}{\left( {{x^2} - \frac{1}{{{x^3}}}} \right)^{10}}

View answer and explanation

Correct option: B

210

Option B (210) is correct because it is the coefficient of the term with x^-10 in the given binomial expansion.

252 Easy

252 of 434

Application +4 | -1

Coefficient of

{x^{19}}

in the polynomial

\left( {x - 1} \right)\left( {x - 2} \right) \ldots ..\left( {x - 20} \right)

is equal to

View answer and explanation

Correct option: B

- 210

Option B (-210) is correct because it is the negative sum of the roots of the polynomial, which corresponds to the coefficient of x¹⁹.

253 Easy

253 of 434

Application +4 | -1

n

is an odd natural number, then number of zeros at the end of

99^n+1

View answer and explanation

Correct option: A

Option A (2) is correct because the last two digits of 99^n + 1 are 01 when n is odd, resulting in exactly two zeros at the end.

254 Easy

254 of 434

Application +4 | -1

If the binomial coefficient of the third term in the expansion of

\left(9 x-\frac{1}{\sqrt{3 x}}\right)^m

is 105 , then the third term in the expansion is

View answer and explanation

Correct option: A

35 \cdot 9^{13} \cdot x^{12}

Option A (

35 \cdot 9^{13} \cdot x^{12}

) is correct because it is the third term in the expansion.

255 Easy

255 of 434

Application +4 | -1

If the coefficient of

{x^2}

and

{x^3}

in the expansion of

{\left( {3 + ax} \right)^9}

be same, then the value of

a

View answer and explanation

Correct option: D

\frac{9}{7}

Option D (

\frac{9}{7}

) is correct because equating the coefficients of

{x^2}

and

{x^3}

in the expansion of

{\left( {3 + ax} \right)^9}

leads to the equation

{{\;^9}{C_2}{3^7}{a^2} = {\;^9}{C_3}{3^6}{a^3}}

, which simplifies to

a = \frac{9}{7}

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256 Easy

256 of 434

Application +4 | -1

The coefficient of

y

in the expansion of

{\left( {{y^2} + c/y} \right)^5},

View answer and explanation

Correct option: C

10\;{c^3}

Option C (

10\;{c^3}

) is correct because when expanding

{\left( {{y^2} + c/y} \right)^5}

, the term containing

y

has a coefficient of

10c^3

257 Easy

257 of 434

Application +4 | -1

{\left( {1 + x} \right)^{15}} = {a_0} + {a_1}x + . \ldots + {a_{15}}{x^{15}}

, then

\sum \limits_{r = 1}^{15} r\frac{{{a_r}}}{{{a_{r - 1}}}}

is equal to

View answer and explanation

Correct option: C

120

Option C (120) is correct because by expanding the binomial and simplifying the summation, the result equals 120.

258 Easy

258 of 434

Application +4 | -1

The coefficient of

{x^{53}}

in the expansion

\sum \limits_{m = 0}^{100} {\;^{100}}{C_m}{\left( {x - 3} \right)^{100 - m}}{.2^m},\;\mathrm{is}

View answer and explanation

Correct option: C

- {\;^{100}}{C_{53}}

Option C (-^{100}C_{53}) is correct because the given summation simplifies to (x-1)¹⁰⁰, and the coefficient of x⁵³ is -^{100}C_{53}.

259 Easy

259 of 434

Application +4 | -1

The sum of the series

{\;^{20}}{C_0} - {\;^{20}}{C_1} + {\;^{20}}{C_2} - {\;^{20}}{C_3} + ... + {\;^{20}}{C_{10}}\,\mathrm{is}

View answer and explanation

Correct option: B

\frac{1}{2}{\;^{20}}{C_{10}}

Option B (

\frac{1}{2}{\;^{20}}{C_{10}}

) is correct because the given series can be simplified using properties of binomial coefficients and the binomial theorem.

260 Easy

260 of 434

Application +4 | -1

The coefficient of

{x^n}

in the expansion of

{(1 + 2x + 3{x^2} + . \ldots )^{1/2}}

View answer and explanation

Correct option: D

Option D (1) is correct because the given expression simplifies to (1-x)^(-1), and the coefficient of x^n in its expansion is 1.

261 Easy

261 of 434

Application +4 | -1

The coefficient of

a^{10} b^7 c^3

in the expansion of

(b c+c a+a b)^{10}

View answer and explanation

Correct option: C

120

Option C (120) is correct because the coefficient is found using the multinomial theorem and solving for p, q, and r.

262 Easy

262 of 434

Analyzing +4 | -1

Let

S

denote the set of real numbers

x

such that

\left(x+\sqrt{x^2-1}\right)^3+\left(x-\sqrt{x^2-1}\right)^3=2 \cos \left(3 \cos ^{-1} x\right)

Then

S

contains

View answer and explanation

Correct option: D

No element

Option D (No element) is correct because the domain of

\sqrt{x^2-1}

and

\cos^{-1}x

have no common elements that satisfy the equation.

263 Easy

263 of 434

Application +4 | -1

The number of integral terms in the expansion of

\left(5^{\frac{1}{2}}+7^{\frac{1}{6}}\right)^{642}

View answer and explanation

Correct option: B

108

Option B (108) is correct because the number of integral terms is found by determining the multiples of 6 within the range and adding 1 for the initial term.

264 Easy

264 of 434

Application +4 | -1

{a_1},{a_2},{a_3},{a_4}

are the coefficients of any four consecutive terms in the expansion of

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

, then

\frac{{{a_1}}}{{{a_1} + {a_2}}} + \frac{{{a_3}}}{{{a_3} + {a_4}}}

is equal to

View answer and explanation

Correct option: C

\frac{{2{a_2}}}{{{a_2} + {a_3}}}

Option C (

\frac{{2{a_2}}}{{{a_2} + {a_3}}}

) is correct because the given expression can be simplified using binomial coefficient identities to arrive at this result.

265 Easy

265 of 434

Application +4 | -1

\left( {1 + \frac{{{C_1}}}{{{C_0}}}} \right)\left( {1 + \frac{{{C_2}}}{{{C_1}}}} \right)\left( {1 + \frac{{{C_3}}}{{{C_2}}}} \right) \ldots \left( {1 + \frac{{{C_n}}}{{{C_{n - 1}}}}} \right)\;

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

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

266 Easy

266 of 434

Application +4 | -1

Least positive integer greater than

(1+0.0001)^{1{0^4}}

View answer and explanation

Correct option: A

Option A (3) is correct because the expression is approximately equal to e, which is between 2 and 3, so the least integer greater than it is 3.

267 Easy

267 of 434

Application +4 | -1

What is the sum of the coefficient of

{({x^2} - x - 1)^{99}}?

View answer and explanation

Correct option: C

- 1

Option C (-1) is correct because substituting x = 1 into the expression gives the sum of the coefficients, which evaluates to -1.

268 Easy

268 of 434

Analyzing +4 | -1

If the number of terms in the expansion of

\left(1+5 x+10 x^2+10 x^3+5 x^4+x^5\right)^{20}

m

, then unit's place of

2^m

View answer and explanation

Correct option: A

Option A (2) is correct because the number of terms in the expansion is 101, and the unit's digit of

2^{101}

is 2.

269 Easy

269 of 434

Application +4 | -1

The sum of the last eight coefficient in the expansion of

{\left( {1 + x} \right)^{15}}

View answer and explanation

Correct option: C

{2^{14}}

Option C (

{2^{14}}

) is correct because the sum of the last eight coefficients in the expansion of

{\left( {1 + x} \right)^{15}}

is half of the total coefficients, which equals

{2^{14}}

270 Easy

270 of 434

Application +4 | -1

6^{83}+8^{83}

is divided by 49 , the remainder is

View answer and explanation

Correct option: C

Option C (35) is correct because by applying the binomial theorem and simplifying, the remainder when

6^{83}+8^{83}

is divided by 49 is 35.

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