The coefficient of $x^{10}$ in the expansion of $(1+2 x)^{-7}$ is

$2^{10}\left(\begin{array}{c}16 \\ 9\end{array}\right)$

$\left(\begin{array}{c}16 \\ 6\end{array}\right)$

If the expansion in powers of $x$ of the function $\frac{1}{(1-a x)(1-b x)}$ is $a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots$, then $a_n$ is

$\frac{b^{\mathrm{n}}-a^n}{b-a}$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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691 Medium

691 of 951

Application +4 | -1

The value of

\sum\limits_{k = 1}^\infty {\sum\limits_{r = 0}^k {\frac{1}{{{3^k}}}{(^k}{C_r})} }

View answer and explanation

Correct option: C

Option C (2) is correct because the double summation simplifies to a geometric series with a sum of 2.

692 Medium

692 of 951

Analyzing +4 | -1

The value of

\sum\limits_{{r=0}}^n(-1)^r \cdot r^2 \cdot{ }^n C_r

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the value of the summation simplifies to 0.

693 Easy

693 of 951

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.

694 Medium

694 of 951

Application +4 | -1

The coefficient of

x^{10}

in the expansion of

(1+2 x)^{-7}

View answer and explanation

Correct option: B

2^{10}\left(\begin{array}{l}16 \\ 10\end{array}\right)

Option A (

2^{10}\left(\begin{array}{l}17 \\ 10\end{array}\right)

) is incorrect. Option B (

2^{10}\left(\begin{array}{l}16 \\ 10\end{array}\right)

) is the correct answer because it is derived from the general term in the binomial expansion of

(1+2x)^{-7}

695 Medium

695 of 951

Analyzing +4 | -1

(1+x)^n=C_0+C_1 x+\ldots+C_n x^n

, then the value of

\sum\limits_{0 \le r} {\sum\limits_{ < s \le n} {{C_r}{C_r}} }

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\left[2^{2 n}-{ }^{2 n} C_n\right]

Option A (

\frac{1}{2}\left[2^{2 n}-{ }^{2 n} C_n\right]

) is the correct answer because it is derived from the binomial identity and combinatorial arguments.

696 Easy

696 of 951

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.

697 Easy

697 of 951

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.

698 Medium

698 of 951

Understanding +4 | -1

The sum of the series

\sum \limits_{r = 0}^{10} {\;^{20}}{C_r}

, is

View answer and explanation

Correct option: C

{2^{19}} + \frac{1}{2}{\;^{20}}{C_{10}}

Option C (

{2^{19}} + \frac{1}{2}{\;^{20}}{C_{10}}

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

699 Medium

699 of 951

Understanding +4 | -1

\left(1+2 x+x^2\right)^n=\sum\limits_{{r=0}}^{2 n} a_r x^r

, then

a_r

is equal to

View answer and explanation

Correct option: C

{ }^{2n} C_r

Option C (

{ }^{2n} C_r

) is correct because the given expression simplifies to

(1+x)^{2n}

, and

a_r

is the coefficient of

x^r

in its binomial expansion.

700 Easy

700 of 951

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.

701 Medium

701 of 951

Application +4 | -1

{\left( {1 + 2x + 3{x^2}} \right)^{10}} = {a_0} + {a_1}\;x + {a_2}\;{x^2} + \ldots + {a_{20}}\;{x^{20}},

then

{a_1}

equals

View answer and explanation

Correct option: B

Option B (20) is correct because the coefficient of x in the expansion of

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

is 20.

702 Medium

702 of 951

Analyzing +4 | -1

n

is positive integer and

(3 \sqrt{3}+5)^{2 n+1}=\alpha+\beta

, where

\alpha

is an integer and

0 < \beta < 1

, then

View answer and explanation

Correct option: A

\alpha

is an even integer

Option A is correct because when we consider the conjugate expression and add it to the original, the irrational parts cancel out, resulting in an even integer.

703 Easy

703 of 951

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.

704 Medium

704 of 951

Application +4 | -1

In the expansion of

(1+x)^{18}

, if the coefficients of

(2 r+4)^{\mathrm{th}}

and

(r-2)^{\mathrm{th}}

terms are equal, then the value of

r

is :

View answer and explanation

Correct option: D

Option D (6) is correct because by equating the coefficients using the property of binomial coefficients, we find that r = 6.

705 Easy

705 of 951

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.

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706 Medium

706 of 951

Analyzing +4 | -1

Coefficient of

x^{2009}

\left(1+x+x^2+x^3+x^4\right)^{1001}(1-x)^{1002}

View answer and explanation

Correct option: A

Option A (0) is correct because the coefficient of x²⁰⁰⁹ in the simplified expression is zero.

707 Easy

707 of 951

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.

708 Easy

708 of 951

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.

709 Medium

709 of 951

Application +4 | -1

If the expansion in powers of

x

of the function

\frac{1}{(1-a x)(1-b x)}

a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots

, then

a_n

View answer and explanation

Correct option: C

\frac{b^{n+1}-a^{n+1}}{b-a}

Option C (

\frac{{b^{n+1}-a^{n+1}}}{{b-a}}

) is correct because it represents the coefficient of

x^n

in the expansion of the given function using partial fractions or series expansion.

710 Medium

710 of 951

Application +4 | -1

\mathrm{C}_{\mathrm{r}}=\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right)

, then

\sum_{\mathrm{r}=1}^{\mathrm{n}}(-1)^{\mathrm{r}-1} \frac{\mathrm{C}_{\mathrm{r}}}{\mathrm{r}}=

View answer and explanation

Correct option: C

\sum_{r=1}^n \frac{1}{r}

Option C (

\sum_{{r=1}}^n \frac{{1}}{{r}}

) is correct because it is the result of integrating the given binomial series.

711 Easy

711 of 951

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.

712 Easy

712 of 951

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.

713 Easy

713 of 951

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.

714 Medium

714 of 951

Application +4 | -1

The sum of the rational terms in the binomial expansion of

\left(2^{\frac{1}{2}}+3^{\frac{1}{5}}\right)^{10}

is:

View answer and explanation

Correct option: D

Option D (41) is correct because only the first and last terms of the expansion are rational, and their sum is 41.

715 Medium

715 of 951

Analyzing +4 | -1

A={ }^{2 n} C_0{ }^{2 n} C_1+{ }^{2 n} C_1{ }^{2 n-1} C_1+{ }^{2 n} C_2{ }^{2 n-2} C_1+\ldots

, then

A

View answer and explanation

Correct option: C

n \cdot 2^{2 n}

Option C is correct because the expression can be interpreted as the coefficient of x in a binomial expansion, which simplifies to

n \cdot 2^{2 n}

716 Easy

716 of 951

Application +4 | -1

The coefficient of

\left(a^3 \cdot b^6 \cdot c^8 \cdot d^9 \cdot e \cdot f\right)

in the expansion of

(a+b+c-d-e-f)^{31}

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the coefficient is calculated using the multinomial theorem, and the provided options do not match the correct calculation.

717 Medium

717 of 951

Application +4 | -1

{\left( {1 + x - 3{x^2}} \right)^{10}} = 1 + {a_1}x + {a_2}{x^2} + ... + {a_{20}}{x^{20}},

then

{a_2} + {a_4} + {a_6} + ... + {a_{20}}

is equal to

View answer and explanation

Correct option: C

\frac{{{3^{10}} - 1}}{2}

Option C (

\frac{{{3^{10}} - 1}}{2}

) is correct because it is the value obtained by substituting x = 1 and x = -1 into the given polynomial and solving for the sum of even-powered coefficients.

718 Easy

718 of 951

Application +4 | -1

If the expansion of

{\left( {\frac{{3\sqrt x }}{7} - \frac{5}{{2x\sqrt x }}} \right)^{13n}}

contains a term independent of

x

in 14th term,then

n

should be

View answer and explanation

Correct option: D

Option D (4) is correct because the value of n that makes the 14th term independent of x is found by setting the exponent of x in the 14th term to zero and solving for n.

719 Medium

719 of 951

Application +4 | -1

3^{51}

when divided by 8 leaves the remainder,

View answer and explanation

Correct option: D

Option D (3) is correct because 3⁽⁵¹⁾ can be written as 3 * (3²)²⁵ = 3 * (9)²⁵ = 3 * (8+1)²⁵, and when divided by 8, the remainder is 3.

720 Easy

720 of 951

Application +4 | -1

The term independent of

x

in the expansion of

\left( {1 - {x^2} + 3{x^3}} \right){\left( {\frac{5}{2}{x^3} - \frac{1}{{5{x^2}}}} \right)^{11}},\,\,x \ne 0

is:

View answer and explanation

Correct option: B

\frac{{33}}{{200}}

Option B (33/200) is correct because it is the constant term obtained after expanding the binomial and multiplying by the polynomial.

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