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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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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 151-180 of 434 questions

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151 Easy

151 of 434

Analyzing +4 | -1

The coefficient of

{x^{101}}

in the expression

{(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + .... + {x^{500}},\,\,\,x > 0,

View answer and explanation

Correct option: A

^{501}{C_{101}}{(5)^{399}}

Option A (

^{501}{C_{101}}{(5)^{399}}

) is correct because by simplifying the expression and using binomial theorem, the coefficient of x¹⁰¹ can be found.

152 Easy

152 of 434

Analyzing +4 | -1

The coefficient of

x^{50}

in the series

\sum\limits_{r=1}^{101} r x^{r-1}(1+x)^{101-r}

View answer and explanation

Correct option: C

{ }^{102} C_{50}

Option C (

{}^{102} C_{50}

) is correct because the given series simplifies to

(1+x)^{102} - x^{101}(1+x) - 101x^{101}

, and the coefficient of

x^{50}

{}^{102} C_{50}

153 Easy

153 of 434

Understanding +4 | -1

C_0, C_1, C_2, \ldots \ldots, C_n

are binomial coefficients, (where

C_r={ }^n C_r

), then the value of

C_0-C_1+C_2-C_3+\ldots+(-1)^n C_n

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because the given expression is the binomial expansion of (1 + x)^n with x = -1, which results in 0.

154 Easy

154 of 434

Application +4 | -1

The coefficient of

{x^3}

{\left( {\sqrt {{x^5}} + \frac{3}{{\sqrt {{x^3}} }}} \right)^6},

View answer and explanation

Correct option: D

540

Option D (540) is correct because by finding the general term in the expansion and setting the power of x to 3, we can find the coefficient of

x^3

to be 540.

155 Easy

155 of 434

Application +4 | -1

The two consecutive terms in the expansion of

{\left( {3 + 2x} \right)^{74}}

whose coefficient are equal, are

View answer and explanation

Correct option: C

30,31

Option C (30,31) is correct because by equating the coefficients of two consecutive terms in the binomial expansion of

{\left( {3 + 2x} \right)^{74}}

, we find that the terms are the 30th and 31st terms.

156 Easy

156 of 434

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

{\left( {x + 3} \right)^6}

View answer and explanation

Correct option: A

Option A (18) is correct because it is the coefficient of

{x^5}

obtained by expanding the binomial expression

{\left( {x + 3} \right)^6}

using the binomial theorem.

157 Easy

157 of 434

Application +4 | -1

(1+i)^6+(1-i)^6=

View answer and explanation

Correct option: A

Option A, 0, is correct because when expanding

(1+i)^6

and

(1-i)^6

using the binomial theorem, the terms cancel each other out, resulting in 0.

158 Easy

158 of 434

Application +4 | -1

If the coefficients of

{x^7}

and

{x^8}

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

are equal, then

n

is equal to

View answer and explanation

Correct option: B

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

{x^7}

and

{x^8}

equal leads to the equation

{\frac{{\;^n{C_7} \times {2^{n - 7}}}}{{{3^7}}} = {\frac{{\;^n{C_8} \times {2^{n - 8}}}}{{{3^8}}}}

which simplifies to

n = 55

159 Easy

159 of 434

Application +4 | -1

The coefficient of

x^5

in the expansion of

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

View answer and explanation

Correct option: D

Option D (60) is correct because it is the sum of the products of the coefficients that result in x⁵ when the two expansions are multiplied.

160 Easy

160 of 434

Application +4 | -1

The middel term in the expansion of

{\left( {x - \frac{1}{x}} \right)^{18}}

View answer and explanation

Correct option: B

- _\;^{18}{C_9}

Option B (

- _\;^{18}{C_9}

) is correct because the middle term in the expansion of

{\left( {x - \frac{1}{x}} \right)^{18}}

is found by setting r = n/2 = 9 in the general term formula, and considering the negative sign.

161 Easy

161 of 434

Application +4 | -1

The digit at the unit place in the number

{19^{2005}} + {11^{2005}} - {9^{2005}}\;\mathrm{is}

View answer and explanation

Correct option: B

Option B (1) is correct because the digit at the unit place is 1, as derived from binomial expansion.

162 Easy

162 of 434

Analyzing +4 | -1

If the magnitude of the coefficient of

{x^7}

in the expansion of

{\left( {{x^2} + \frac{1}{{bx}}} \right)^8}

,where

a,b

,are positive numbers,is equal to the magnitude of the
coefficient of

{x^{ - 7}}

in the of

{\left( {ax - \frac{1}{{b{x^2}}}} \right)^8}

,then

a

,and

b

are connected by the relation

View answer and explanation

Correct option: A

ab = 1

The correct answer is A (

ab = 1

) because equating the magnitudes of the coefficients of

x^7

and

x^{-7}

leads to the relation

ab = 1

163 Easy

163 of 434

Application +4 | -1

If the number of terms in

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

is 401 then

n

is greater than

View answer and explanation

Correct option: B

200

Option B (200) is correct because the number of terms in the expansion is related to n by the formula 2n+1, which equals 401 in this case.

164 Easy

164 of 434

Application +4 | -1

The number of dissimilar terms in the expansion of

{(a + b)^n}

n + 1

therefore no of dissimilar terms of the expansion

{\left( {a + b + c} \right)^{12}}

View answer and explanation

Correct option: D

The correct answer is D (91) because the number of dissimilar terms in the expansion of

{(a + b + c)^{12}}

is calculated using the formula

{\;^{n + r - 1}}{C_{r - 1}}

, where n is the power and r is the number of variables.

165 Easy

165 of 434

Application +4 | -1

The sum of rational terms in the expansion of

\left(\sqrt{2}+3^{1 / 15}\right)^{10}

View answer and explanation

Correct option: B

Option B (41) is correct because the sum of rational terms in the expansion is obtained by identifying terms where both the power of 2 and 3 are integers, which occurs only for r=0 and r=10.

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166 Easy

166 of 434

Understanding +4 | -1

The coefficient of

x^k(0 \leq k \leq n)

in the expansion of

E=1+(1+x)+(1+x)^2+\cdots+(1+x)^n

View answer and explanation

Correct option: A

{ }^{n+1} C_{k+1}

Option A (

{ }^{n+1} C_{k+1}

) is correct because it represents the coefficient of

x^k

after summing the geometric series and simplifying.

167 Easy

167 of 434

Application +4 | -1

{\left( {1 + x} \right)^n} = {\;^n}{C_0} + {\;^n}{C_1}x + {\;^n}{C_2}{x^2} + ... + {\;^n}{C_n}{x^n},

then

\frac{{{\;^n}{C_1}}}{{{\;^n}{C_0}}} + \frac{{2{\;^n}{C_2}}}{{{\;^n}{C_1}}} + \frac{{\;{3^n}{C_3}}}{{{\;^n}{C_2}}} + ... + \frac{{n{\;^n}{C_n}}}{{{\;^n}{C_{n - 1}}}}

is equal to

View answer and explanation

Correct option: C

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

The correct answer is

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

because substituting values for n and checking the options reveals that only option C satisfies the condition.

168 Easy

168 of 434

Application +4 | -1

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

, then

\mathrm{C}_1+\mathrm{C}_3+\mathrm{C}_5+\ldots .

. is equal to-

View answer and explanation

Correct option: D

2^{n-1}

Option D (

2^{n-1}

) is correct because adding and subtracting the binomial expansion with x=1 and x=-1 isolates the sum of odd-indexed coefficients.

169 Easy

169 of 434

Analyzing +4 | -1

{(1 + x - 2{x^2})^6} = 1 + {a_1}x + {a_2}{x^2} + ... + {a_{12}}{x^{12}},

then the value of

{a_2} + {a_4} + ... + {a_{12}}\;

View answer and explanation

Correct option: A

The correct answer is A (31) because it is obtained by substituting x = 1 and x = -1 into the given equation and solving for the required sum.

170 Easy

170 of 434

Analyzing +4 | -1

The value of

\frac{19^3+6^3+(3)(19)(6)(25)}{3^6+6(243)(2)+(15)(81)(4)+(20)(27)(8)+(15)(9)(16)+(6)(3)(32)+2^6}

View answer and explanation

Correct option: D

Option D (1) is correct because the numerator can be written as

(19+6)^3 = 25^3

and the denominator as

(3+2)^6 = 5^6

, so the fraction simplifies to 1.

171 Easy

171 of 434

Application +4 | -1

The values of

a

and

b

so that the coefficient of

x^n

in the expansion of

\frac{a+b x}{(1-x)^2}

may be

2 n+1

, are

View answer and explanation

Correct option: A

1,1

Option A (1,1) is correct because by expanding the given expression and equating the coefficient of x^n to 2n+1, we get a=1 and b=1.

172 Easy

172 of 434

Analyzing +4 | -1

The term independent of

x

in the expansion of

{\left( {\frac{{2\sqrt x }}{5} - \frac{1}{{2x\sqrt x }}} \right)^{11}}\mathrm{is}

View answer and explanation

Correct option: D

No term

Option D (No term) is correct because the power of x in the general term is never zero for any integer value of r, indicating no term is independent of x.

173 Easy

173 of 434

Application +4 | -1

Coefficient of

x^5

in the expansion of

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

View answer and explanation

Correct option: D

Option D (60) is correct because the coefficient of

x^5

in the expansion of

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

is obtained by multiplying the relevant terms and summing them, resulting in 60.

174 Easy

174 of 434

Analyzing +4 | -1

The

\operatorname{sum} S={ }^{20} C_2+2 \cdot{ }^{20} C_3+3 \cdot{ }^{20} C_4+\ldots+19 \cdot{ }^{20} C_{20}

is equal to

View answer and explanation

Correct option: C

1+9 \cdot 2^{20}

The correct answer is C (

1+9 \cdot 2^{20}

) because it is derived by differentiating the binomial expansion and substituting x=1.

175 Easy

175 of 434

Application +4 | -1

Coefficient of

x^n

in the expansion of

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

is-

View answer and explanation

Correct option: C

\frac{2^n}{n !}

Option C (

\frac{2^n}{n !}

) is correct because the coefficient of

x^n

in the expansion is the sum of products of coefficients, which simplifies to the given expression.

176 Easy

176 of 434

Analyzing +4 | -1

The value of

\sum\limits_{{k=0}}^n\left({ }^{4 n+1} C_k+{ }^{4 n+1} C_{2 n-k}\right)

View answer and explanation

Correct option: A

2^{4 n}+{ }^{4 n+1} C_{2 n}

Option A (

2^{4 n}+{ }^{4 n+1} C_{2 n}

) is correct because the summation can be simplified to

2^{4n} + { }^{4n+1}C_{2n}

using properties of binomial coefficients.

177 Easy

177 of 434

Analyzing +4 | -1

If coefficient of

x^n

in the expansion of

(1+x)^{101}

\left(1-x+x^2\right)^{100}

is non-zero, then

n

cannot be of the form

View answer and explanation

Correct option: C

3 \lambda+2

Option C (

3 \lambda+2

) is correct because the expansion of the given expression only contains terms of the form

x^{3 \lambda}

x^{3 \lambda+1}

178 Easy

178 of 434

Application +4 | -1

If in the expansion of

\left(2^x+\frac{1}{4^x}\right)^n, T_3 / T_2=7

and the sum of the coefficient of

2^{\text {nd }}

and

3^{\text {rd }}

terms is 36 , then the value of

x

View answer and explanation

Correct option: A

-1 / 3

Option A (-1/3) is correct because solving the given conditions

{ }^n C_1+{ }^n C_2=36

and

\frac{{ }^n C_2(2^x)^{n-2} (\frac{{1}}{{4^x}})^2}{{ }^n C_1(2^x)^{n-1} (\frac{{1}}{{4^x}})^1}=7

leads to

x = -1/3

179 Easy

179 of 434

Application +4 | -1

If in the binomial expansion of

(1-x)^m(1+x)^n

, the coefficients of

x

and

x^2

are respectively

3

and

-4,

then the ratio

m: n

is equal to:

View answer and explanation

Correct option: D

7: 10

Option D (

7: 10

) is correct because by setting up equations based on the coefficients of

x

and

x^2

, we can solve for

m

and

n

and find their ratio.

180 Easy

180 of 434

Understanding +4 | -1

\alpha

and

\beta

are the coefficients of

x^r

and

x^{n-r}

respectively in the expansion of

(1+x)^n

, then

View answer and explanation

Correct option: A

\alpha=\beta

Option A (

a_p=a_q

) is correct because

a_r

represents the coefficient of

x^r

in the expansion of

(1+x)^{p+q}

, and

a_p = { }^{p+q}C_p

and

a_q = { }^{p+q}C_q

. Since

{ }^{n}C_r = { }^{n}C_{n-r}

, and

{ }^{p+q}C_p = { }^{p+q}C_{(p+q)-p} = { }^{p+q}C_q

, therefore

a_p = a_q

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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