The coefficient of ${x^{100}}$ in the expansion of $\mathop \sum \limits_{j = 0}^{200} {(1 + x)^j}$ is

$\left( {_{100}^{200}} \right)$

$\left( {_{102}^{201}} \right)$

$\left( {_{101}^{200}} \right)$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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Showing 151-180 of 951 questions

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151 Easy

151 of 951

Application +4 | -1

The remainder when

{3^{2022}}

is divided by 5 is:

View answer and explanation

Correct option: D

Option D (4) is correct because when

{3^{2022}}

is divided by 5, the remainder is 4, as determined by analyzing the pattern of powers of 3 modulo 5.

152 Easy

152 of 951

Application +4 | -1

If the absolute term in the expansion of

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

is 405 , then

k

is equal to

View answer and explanation

Correct option: B

\pm 3

Option B (

\pm

3) is correct because these values of k make the absolute term in the expansion equal to 405.

153 Medium

153 of 951

Application +4 | -1

Let

{(x + 10)^{50}} + {(x - 10)^{50}} = {a_0} + {a_1}x + {a_2}{x^2} + ... + {a_{50}}{x^{50}}

, for all

x \in R;

then

\frac{{{a_2}}}{{{a_0}}}

is equal to

View answer and explanation

Correct option: A

12.25

Option A (12.25) is correct because the ratio of coefficients a2/a0 is calculated by expanding the binomial expressions and comparing coefficients.

154 Easy

154 of 951

Analyzing +4 | -1

\sum\limits_{r=0}^n(-1)^r{ }^n C_r\left[\frac{1}{2^r}+\frac{3^r}{2^{2 r}}+\frac{7^r}{2^{3 r}}+\ldots\right.

upto

m

terms

]

=f(n)\left(1-\frac{1}{2^{m n}}\right)

\int_{-3}^3 f\left(x^3 \ln x\right) d\left(x^3 \ln x\right)

is equal to

View answer and explanation

Correct option: D

Cannot be determined

Option D (Cannot be determined) is correct because the integral cannot be calculated due to the presence of

\ln x

with a lower limit of integration at -3, which is outside the domain of the natural logarithm.

155 Easy

155 of 951

Analyzing +4 | -1

The term independent of

x

in the expansion of

{\left( {x - \frac{1}{x}} \right)^4}{\left( {x + \frac{1}{x}} \right)^3}\mathrm{\;}

View answer and explanation

Correct option: B

Option B (0) is correct because when the expression is expanded, there is no term independent of x.

156 Easy

156 of 951

Application +4 | -1

The lowest integer which is greater than

{\left( {1 + \frac{1}{{{{10}^{100}}}}} \right)^{{{10}^{100}}}}

View answer and explanation

Correct option: A

Option A (3) is correct because the expression approaches 'e' as the exponent increases, and since 'e' is between 2 and 3, the lowest integer greater than the expression is 3.

157 Medium

157 of 951

Application +4 | -1

If the

r

th term is the middle term in the expansion of

\left(x^2-\frac{1}{2 x}\right)^{20}

, then the

(r+3)

th term is

View answer and explanation

Correct option: C

-\frac{1}{2^{13}},{ }^{20} C_{13} \cdot x

Option C is correct because the (r+3)th term, which is the 14th term, is calculated using the binomial theorem and simplified to the given expression.

158 Easy

158 of 951

Analyzing +4 | -1

If the second term in the expansion of

\left(\sqrt[13]{a}+\frac{a}{\sqrt{a^{-1}}}\right)^n

14 a^{5 / 2}

, the value of

\frac{{ }^n C_3}{{ }^n C_2}

View answer and explanation

Correct option: A

Option A (4) is correct because solving for n in the given condition and then calculating the ratio of binomial coefficients results in 4.

159 Hard

159 of 951

Analyzing +4 | -1

\sum\limits_{{r=0}}^{2 n} a_r(x-2)^r=\sum\limits_{{r=0}}^{2 n} b_r(x-3)^r

and

a_k=1

for all

k \geq n

, then

b_n

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it represents the coefficient of

(x-3)^n

after transforming the given polynomial expression.

160 Medium

160 of 951

Application +4 | -1

The 14th term from the end in the expansion of

{\left( {\sqrt x - \sqrt y } \right)^{17}}

View answer and explanation

Correct option: C

{\;^{17}}{C_4}{x^{13/2}}{y^2}

Option C (

{\;^{17}}{C_4}{x^{13/2}}{y^2}

) is correct because the 14th term from the end is the 5th term from the beginning in the binomial expansion.

161 Medium

161 of 951

Analyzing +4 | -1

\left[\lim \limits_{{n \rightarrow \infty}} \sum\limits_{{r=0}}^n\left({ }^n C_r \cdot \frac{1}{(r+4) \cdot n^r}\right)\right]

is equal to
(where [.] denotes greatest integer function):

View answer and explanation

Correct option: A

Option A (0) is correct because the determinant becomes zero as two rows are identical after summation.

162 Medium

162 of 951

Analyzing +4 | -1

\alpha=\frac{5}{2 ! 3}+\frac{5 \cdot 7}{3 ! 3^2}+\frac{5 \cdot 7 \cdot 9}{4 ! 3^3}+\ldots

, then

\alpha^2+4 \alpha

is equal to

View answer and explanation

Correct option: B

Option B (23) is correct because after simplifying the series and using the binomial theorem, the expression

\alpha^2 + 4\alpha

evaluates to 23.

163 Medium

163 of 951

Application +4 | -1

The coefficient of

{x^{100}}

in the expansion of

\sum \limits_{j = 0}^{200} {(1 + x)^j}

View answer and explanation

Correct option: D

\left( {_{100}^{201}} \right)

Option D (

\left( {_{100}^{201}} \right)

) is correct because the sum of binomial coefficients simplifies to this after applying the hockey-stick identity.

164 Medium

164 of 951

Analyzing +4 | -1

Let

P(x)

be a polynomial with real coefficients such that

\int\limits_{0}^1 x^m P(1-x) d x=0 \, \forall \, m \in {N} \cup\{0\}

, then

View answer and explanation

Correct option: D

P(x)=0

Option D (

P(x)=0

) is correct because the given integral condition implies that the polynomial must be identically zero.

165 Medium

165 of 951

Understanding +4 | -1

The total number of different terms in the expansion of

\left(x_1+x_2+x_3+\ldots+x_n\right)^m

is equal to

View answer and explanation

Correct option: B

{ }^{n+m-1} C^{n-1}

Option B (

{ }^{n+m-1} C^{n-1}

) is correct because it represents the number of non-negative integer solutions to the equation

y_1 + y_2 + ... + y_n = m

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166 Easy

166 of 951

Analyzing +4 | -1

The sum

\sum \limits_{i = 0}^m \left( {\begin{array}{cccccccccccccccccccc} {10}\\ i \end{array}} \right)\left( {\begin{array}{cccccccccccccccccccc} {20}\\ {m - i} \end{array}} \right),\;\;\left( {\mathrm{where},\;\left( {\begin{array}{cccccccccccccccccccc} \mathrm{p}\\ q \end{array}} \right) = 0\;\mathrm{if}\;p < q} \right)

is maximum, when

m

View answer and explanation

Correct option: C

Option C (15) is correct because the sum is the coefficient of

x^m

(1+x)^{30}

, which is maximum when

m = 30/2 = 15

167 Easy

167 of 951

Application +4 | -1

The value of

{\left( {1.002} \right)^{12}}

upto fourth place of decimal is

View answer and explanation

Correct option: A

1.0242

Option A (1.0242) is correct because it is the value of (1.002)¹² calculated up to the fourth decimal place using the binomial theorem.

168 Medium

168 of 951

Application +4 | -1

The constant term in the expansion of

{\left( {1 + x} \right)^{10}}{\left( {1 + \frac{1}{x}} \right)^{12}}

View answer and explanation

Correct option: A

{\;^{22}}{C_{10}}

Option A (

{{\;^{22}}}{C_{10}}

) is correct because the constant term in the expansion of

{\left( {1 + x} \right)^{10}}{\left( {1 + \frac{1}{x}} \right)^{12}}

is the coefficient of

x^{12}

{\left( {1 + x} \right)^{22}}

, which is

{{\;^{22}}}{C_{12}} = {{\;^{22}}}{C_{10}}

169 Easy

169 of 951

Application +4 | -1

The number of rational terms in the expansion of

\left(5^{1 / 6}+7^{1 / 9}\right)^{1824}

View answer and explanation

Correct option: B

102

Option B (102) is correct because it counts the number of terms in the binomial expansion that result in rational numbers, which occurs when the powers of 5^(1/6) and 7^(1/9) are integers.

170 Medium

170 of 951

Application +4 | -1

Value of '

a

', if

17^{\text {th}}

and

18^{\text {th}}

terms in the expansion of

(2+a)^{50}

are equal, is

View answer and explanation

Correct option: A

Option A (1) is correct because equating the 17th and 18th terms in the expansion of

(2+a)^{50}

leads to the equation

\frac{{ }^{50} {C}_{16}}{{ }^{50} {C}_{17}} \times 2={a}

, which gives

a=1

171 Easy

171 of 951

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: C

199

Option C (199) is correct because the number of terms in the expansion is related to n, and solving for n shows it's greater than 199.

172 Medium

172 of 951

Analyzing +4 | -1

The coefficient of

x^{301}

(1+x)^{500}+x(1+x)^{499}

+x^2(1+x)^{498}+\ldots \ldots+x^{500}

View answer and explanation

Correct option: B

{ }^{501} C_{200}

The coefficient of x³⁰¹ is

{ }^{501} C_{200}

. By summing the geometric series and using the identity

{ }^n C_r + { }^n C_{r+1} = { }^{n+1} C_{r+1}

, the coefficient can be simplified to

{ }^{501} C_{200}

173 Medium

173 of 951

Analyzing +4 | -1

The coefficients of

{x^6}

in the expansion of

\log \left\{ {{{\left( {1 + x} \right)}^{1 + x}}{{\left( {1 - x} \right)}^{1 - x}}} \right\},

View answer and explanation

Correct option: A

\frac{1}{{15}}

Option A (

\frac{1}{{15}}

) is correct because the coefficient of x⁶ in the expansion is calculated correctly using the logarithmic properties and series expansion.

174 Medium

174 of 951

Application +4 | -1

The coefficient of

a^8 b^4 c^9 d^9

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

View answer and explanation

Correct option: C

2520

Option C (2520) is correct because the coefficient is found by using multinomial theorem and simplifying the expression.

175 Medium

175 of 951

Application +4 | -1

Coefficient of

x^7

in the expansion of

\left(1-2 x+x^3\right)^6

View answer and explanation

Correct option: B

360

Option B (360) is correct because it is the sum of the coefficients of x⁷ obtained from the multinomial expansion of the given expression.

176 Easy

176 of 951

Application +4 | -1

\frac{{{\;^8}{C_0}}}{6} = {\;^8}{C_1} + {\;^8}{C_2}.6 - {\;^8}{C_3}{.6^2} + \ldots + {\;^8}{C_8}{.6^7}\;\mathrm{is\;equal\;to}

View answer and explanation

Correct option: D

\frac{{{5^8}}}{6}

Option D (5⁸/6) is correct because the given expression can be simplified using the binomial theorem.

177 Medium

177 of 951

Analyzing +4 | -1

x+y=1

, then

\sum_{r=0}^n r^{2 n} C, x^r y^{n-r}

equals:

View answer and explanation

Correct option: C

n x(n x+y)

Option C (n x(n x+y)) is correct because the given summation can be simplified using binomial theorem properties and the condition x+y=1.

178 Medium

178 of 951

Application +4 | -1

The coefficient of

x^n

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

View answer and explanation

Correct option: B

\frac{(-2)^n}{n !}

Option B (

\frac{(-2)^n}{n !}

) is the correct answer because it is the coefficient of

x^n

in the expansion of the given expression.

179 Medium

179 of 951

Application +4 | -1

Value of

\sum\limits_{{k=0}}^n{ }^n C_k \sin (k x) \cos (n-k) x

View answer and explanation

Correct option: A

2^{n-1} \sin (n x)

Option A (

2^{n-1} \sin (n x)

) is correct because the summation can be simplified using trigonometric identities and the binomial theorem.

180 Easy

180 of 951

Understanding +4 | -1

The coefficient of the middle term in the expansion of

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

, is

View answer and explanation

Correct option: A

\frac{{1.3.5 \ldots \left( {2n - 1} \right)}}{{n\;!}}{2^n}

Option A is correct because the coefficient of the middle term in the expansion of

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

is derived using binomial theorem and simplification.

Revision notes and key takeaways

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