MCQ Questions for Class 11 Engineering Maths Binomial Theorem Quiz 13

Coefficient of

$x^{11}$ in the extension of

$(1+x^{2})^{4}(1+x^{3})^{7}(1+x^{4})^{12}$ is

0%
1051
0%
1106
0%
1113
0%
1120

If the variable takes the values 0,1,2,....., n with frequericies proportional to the binomial coefficients

$C\left( n,0 \right) ,C\left( n,1 \right) ,C\left( n,2 \right) ...,C\left( n,n \right)$ respectively, then the variance of the distribution is :-

0%
n
0%
$\frac { \sqrt { n } }{ 2 }$
0%
$\frac { n }{ 2 }$
0%
$\frac { n }{ 4 }$

In the expansion of

$(\dfrac{1+x}{1-x})^2$ , the coefficient of

$x^n$ will be

0%
$4n$
0%
$4n - 3$
0%
$4n - 1$
0%
none of these

Coefficient of

${ t }^{ 12 }$ in

$({ 1+t }^{ 2 }{ ) }^{ 6 }{ (1+t }^{ 6 }){ (1+t }^{ 12 })$ is:

0%
24
0%
21
0%
22
0%
23

The coefficient of

$x^n$ in the binomial expansion of

$(1-x)^{-2}$ , is

0%
$\dfrac{2^n}{2!}$
0%
$n+1$
0%
n
0%
$2n$

The coefficient of

${x}^{24}$ in the expansion of

${ \left( 1+{ x }^{ 2 } \right) }^{ 12 }\left( 1+{ x }^{ 12 } \right) \left( 1+{ x }^{ 24 } \right)$ is

0%
$^{12}{C}_{6}$
0%
$^{12}{C}_{6}+2$
0%
$^{12}{C}_{6}+4$
0%
$^{12}{C}_{6}+6$

The term independent of

$x$ in the expansion of

$\left( \sqrt { \left( \frac { x } { 3 } \right) } + \sqrt { \left( \frac { 3 } { 2 x ^ { 2 } } \right) } \right) ^ { 10 }$ is:-

0%
$5$ $/ 12$
0%
$^ { 10 } C _ { 1 }$
0%
$1$
0%
none of these

${ x }^{ m }$ occurs in the expansion of

$(x+\frac { 1 }{ { x }^{ 2 } } )^{ 2n }$ , the coefficient of

${ x }^{ m }$ , is

0%
$\frac { (2n)! }{ m!(2n-m)! }$
0%
$\frac { (2n)!3!3! }{ (2n-m)! }$
0%
$\frac { (2n)! }{ (\frac { 2n-m }{ 3 } )!(\frac { 4n+m }{ 3 } )! }$
0%
none of these

The coefficient of

$a^{3}b^{4}c$ in the expansion of

$(1+a+b-c)^{9}$ is

0%
$2\;^{9}C_{7}\cdot ^{7}C_{4}$
0%
$-2\;^{9}C_{7}\cdot ^{7}C_{3}$
0%
$^{9}C_{7}\cdot ^{7}C_{3}$
0%
$-^{9}C_{7}\cdot ^{7}C_{3}$

The 4th term in the expansion of

${ \left( \sqrt { x } +\frac { 1 }{ x } \right) }^{ 2 }$ is

0%
$110{ x }^{ \frac { 3 }{ 2 } }$
0%
$220{ x }^{ \frac { 3 }{ 2 } }$
0%
$220{ x }^{ 2 }$
0%
$110{ x }^{ 2 }$

Explanation

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

$T_4={}^{12} C_{3}(\sqrt{x})^{12-3}\left(\frac{1}{x}\right)^{3}$

$={12 C_{3}(\sqrt{x})^{9}\left(x^{-3}\right)}$

$= \frac{12 !}{9! \times 3 !}(x)^{\frac{3}{2}}$

$220 x^{\frac{3}{2}}=$ Ans

Option (b)

In how many terms in the expansion of

$(x^{1/5}+y^{1/10})^{55}$ do not have fractional power of the variable :

0%
$6$
0%
$7$
0%
$8$
0%
$10$

The coefficient of

$x^{4}$ in

$\dfrac{3x^{2}+2x}{(x^{2}+2)(x-3)}$ is

0%
$\dfrac{11}{27}$
0%
$\dfrac{77}{324}$
0%
$-\dfrac{77}{324}$
0%
$\dfrac{11}{27}$

In the expansion of

$\left( \dfrac { x+1 }{ { x }^{ 2/3 }-{ x }^{ 1/3 }+1 } -\dfrac { x-1 }{ x-x^{ -1/2 } } \right) ^{ 10 },$ the term which does not contain x is

0%
$^{ 10 }{ { C }_{ 0 } }$
0%
$^{ 10 }{ { C }_{ 7 } }$
0%
$^{ 10 }{ { C }_{ 4 } }$
0%
none

The coefficient of

$x^{-2}$ in

$(1+3x^2+x^4)\left(1+\dfrac{1}{x}\right)^8$ is

0%
$266$
0%
$161$
0%
$162$
0%
none of these

${1 \choose 0} ^2 + {2 \choose 1}^2 + {3 \choose 2}^2 + ... + { n+1 \choose n}^2 =$

0%
$(n+1){{2n}\choose{n}}$
0%
$(\frac{n+2}{2}){{2n}\choose{n}}$
0%
$(n+2).2^{n-1}$
0%
$(n){{2n}\choose{n}}$

Number of irrational terms in the expansion of

$(\sqrt {2}+\sqrt {3})^{15}$ are

0%
$9$
0%
$7$
0%
$16$
0%
$10$

Coefficient of

$x^8$ in

$(x-1)(x-2)(x-3)..(x-10)$ is?

0%
$980$
0%
$1395$
0%
$1320$
0%
None of these

The sum of the coefficient in the expansion of

$(x+y)^{n}$ is

$4096$ . The greatest coefficient in the expansion is-

0%
$1024$
0%
$924$
0%
$824$
0%
$724$

Explanation

${ \left( { x+y } \right) ^{ n } },\, \, Sum\, \, of\, \, coefficient\, \, =4096$

$\\ When\, \, x=y=1,\, \, if\, \, n=12 \\ \Rightarrow { \left( { 1+1 } \right) ^{ 12 } }={ 2^{ 12 } }=4096 \\$

$\Rightarrow Hence,\, \, greatest\, \, coefficient\, \, \\ \, { \, ^{ n } }{ C_{ n/2 } }{ =^{ 12 } }{ C_{ 6 } }=\dfrac { { 12! } }{ { 6!6! } } =924$

Hence, this is the answer.

The coefficient of

$x^{n}$ in

$\dfrac {x+1}{(x-1)^{2}(x-2)}$ is

0%
$1-2n-\dfrac {3}{2^{n+1}}$
0%
$1-2n-\dfrac {3}{2^{n-1}}$
0%
$1+2n+\dfrac {3}{2^{n+1}}$
0%
$1+2n-\dfrac {3}{2^{n-1}}$

The coefficient of the middle term in the binomial expansion in power of

$x$ of

$(1+\alpha x)^{4}$ and of

$(1-\alpha x)^{6}$ is the same if

$\alpha$ equals-

0%
$-\dfrac{5}{3}$
0%
$\dfrac{10}{3}$
0%
$\dfrac{-3}{10}$
0%
$\dfrac{3}{5}$

Explanation

$Middle\, \, term\, \, n\, \, is\, \, =mn\, \, \, \, \, \, \, \, \, \, \, \, \, \, { \left( { 1+x } \right) ^{ n } }\, \, \, { T_{ r+1 } }{ =^{ n } }{ C_{ r } }{ \left( 1 \right) ^{ n-r } }{ x^{ r } }$

$\begin{matrix} \\ \frac { { { T_{ n } } } }{ 2 } +1\, \, \, \, \, { \left( { 1+\alpha x } \right) ^{ 4 } } \\ { T_{ 3 } }={ T_{ 2+1 } }{ =^{ 4 } }{ C_{ 2 } }{ \left( 1 \right) ^{ 2 } }{ \left( { \alpha x } \right) ^{ 2 } } \\ { \left( { i-\alpha x } \right) ^{ 6 } } \\ { T_{ 4 } }={ T_{ 3+1 } }{ =^{ 6 } }{ C_{ 3 } }{ \left( { -\alpha x } \right) ^{ 3 } } \\ { T_{ 5 } }={ T_{ 4 } } \\ ^{ 4 }{ C_{ 2 } }{ \left( { \alpha x } \right) ^{ 2 } }{ =^{ 6 } }{ C_{ 3 } }{ \left( { -\alpha x } \right) ^{ 3 } } \\ \frac { { 41 } }{ { 2121 } } { \alpha ^{ 2 } }=\frac { { 61 } }{ { 3131 } } { \left( { -\alpha } \right) ^{ 3 } } \\ \frac { { 4\times 3 } }{ 2 } { \alpha ^{ 2 } }=\frac { { 6\times 1\times 4 } }{ { 3\times 2 } } { \left( { -\alpha } \right) ^{ 3 } } \\ \frac { 6 }{ { 20 } } =-\alpha \\ \\ \end{matrix}$

$- \frac{3}{{10}} = \alpha$

Hence, Option

$C$ is the correct answer.

The number of terms which are free from radical signs in the expansion of

$(y^{1/5} + x^{1/10})^{55}$ is

0%
$5$
0%
$6$
0%
$7$
0%
None of these

The number of irrational terms in the expansion of

$\left( 5 ^ { 1 / 6 } + 2 ^ { 1 / 8 } \right) ^ { 100 }$ is :

0%
$96$
0%
$97$
0%
$98$
0%
$99$

The value of the sum

$\sum _{ j=0 }^{ 8 }{8 \choose j}{ \frac { 1 }{ (j+1)(j+2) } }$ is

0%
$\frac { 1003 }{ 90 }$
0%
$\frac { 1013 }{ 90 }$
0%
$\frac { 1023 }{ 90 }$
0%
$\frac { 1033 }{ 90 }$

Find the number of terms in the expression

$5x^ {2}-x^ {2}y+2xy+7$

0%
$3$
0%
$4$
0%
$1$
0%
$2$

The coefficient of

$x^2$ in expansion of the product
(2

$x^2$ ).(

$(1 + 2x + 3x^2)^6$ +

$(1 4x^2)^6$ ) is :

0%
107
0%
106
0%
108
0%
155

The sum of the co-efficients of all odd degree terms in the expansion of

${ \left( x+\sqrt { { x }^{ 3 }-1 } \right) }^{ 5 }+{ \left( x-\sqrt { { x }^{ 3 }-1 } \right) }^{ 5 }$ ,

$\left(x>1\right)$

0%
$2$
0%
$-1$
0%
$0$
0%
$1$

Explanation

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

${ }^{5} c_{r} x^{5-r}\left(\sqrt{x^{3}-1}\right)^{r}=$ General term

${ }^{5} c_{r} x^{5-r}\left(-\sqrt{x^{3}-1}\right)^{r}=$ General term

For degree terms

$2\left[{ }^{5} C_{0} \cdot x^{5}+{ }^{5} c_{2} x^{3}\left(x^{3}-1\right)+{ }^{5} c_{4} x\left(x^{3}-1\right)^{2}\right.]$

$2\left(x^{5}+10 x^{6}-10 x^{3}+5 x^{7}-10 x^{4}+5 x\right]$

Considering only odd degree term,

$=2(1-10+5+5)$

=2 Ans

option (a)

$^{26}C_{0}+^{26}C_{1}+^{26}C_{2}+...+^{26}C_{13}$ is equal to

0%
$2^{25}$
0%
$2^{25}+\dfrac {1}{2}^{26}C_{13}$
0%
$2^{13}$
0%
$2^{13}+\dfrac {1}{2}^{26}C_{13}$

Number of terms which are rational in the expansion of

$(\sqrt[4]{5}+\sqrt[3]{4})^{100}$ is

0%
$10$
0%
$8$
0%
$9$
0%
$11$

Explanation

General term

$T_{r+1}={}^{100} C_{2}(5)^{\frac{100-r}{4}} \cdot(4)^{\frac{r}{3}}$

In this case, for rational tems., power of

$5 and 4$ should be integer

$\therefore \frac{100-r}{4} and \frac{r}{3}$ should be integers

For

$r=6,0,12 , 24,30,36,42,4.8,54,60$

powers will be integer.

$\therefore \quad Ans =10$

option (a)

$| x | < 2 / 3$ then the fourth term in the expansion of

$\left( 1 + \frac { 3 } { 2 } x \right) ^ { 1 / 2 }$ is

0%
$\frac { 27 } { 128 } x ^ { 3 }$
0%
$-\frac { 27 } { 128 } x ^ { 3 }$
0%
$\frac { 81 } { 256 } x ^ { 3 }$
0%
$-\frac { 81 } { 256 } x ^ { 3 }$

${ (1+x-{ 2x }^{ 2 }) }^{ 6 }=1+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+{ C }_{ 3 }{ x }^{ 3 }+...+{ C }_{ 12 }{ x }^{ 12 }$ , then the value of

${ C }_{ 2 }+{ C }_{ 4 }+{ C }_{ 6 }+...+{ C }_{ 12 }$ is

0%
30
0%
32
0%
31
0%
None of these

The sum of the coefficients in the expansion (6a-5b) when n is a positive integer, is

0%
${ 10 }^{ \circ }$
0%
1
0%
${ 11 }^{ n }$
0%
0

Coefficient of

${ x }^{ 2009 }in(1+x+{ x }^{ 2 }+{ x }^{ 3 }+{ x }^{ 4 }{ ) }^{ 1001 }{ \left( 1-x \right) }^{ 1002 }$ IS...

0%
${ 4. }^{ 1001 }{ c }_{ 501 }$
0%
- 2009
0%
0
0%
None of these

The number of terms which are free from radical signs in the expansion of

$\left( \frac { 1 }{ { y }^{ 4 } } +\frac { 1 }{ { y }^{ 8 } } \right)$ is:

0%
5
0%
6
0%
7
0%
non of these

If sum of the coefficient in the expression of

${ (-3{ x }^{ 2 }+\frac { 2 }{ x } ) }^{ 2n+1 }$ is 'a' then the values of 'b' for which roots of the equation

${ x }^{ 2 }+bx+6a=0$ are integral

0%
{-7,-5,5,7}
0%
{-7,-1,1,7}
0%
{-5,-1,1,5}
0%
none of these

$\left | x \right |$ < 1, then the coefficient of

$x^n$ in the expansion of

$(1 + x + x^2 + x^3 + .....)^2$ is

0%
n
0%
n 1
0%
n + 2
0%
n + 1

If the number of terms in the expansion of

${ \left( 1-\dfrac { 2 }{ X } +\dfrac { 4 }{ { X }^{ 2 } } \right) }^{ n },x\neq 0,$ is 28, then sum coefficients of all the terms in this expansion,is:

0%
2187
0%
243
0%
729
0%
64

Coefficient of

$x^ {15}$ in

$(1+x+x^ {3}+x^ {4})$ ^ {n} is

0%
$\displaystyle \sum _{ r=0 }^{ 5 }{ ^{ n } } { C }_{ 5-r }.^{ n }{ C }_{ 3r }$
0%
$\displaystyle \sum _{ r=0 }^{ 5 }{ ^{ n } } { C }_{ 5r }$
0%
$\displaystyle \sum _{ r=0 }^{ 5 }{ ^{ n } } { C }_{ 2r }$
0%
$\displaystyle \sum _{ r=0 }^{ 3 }{ ^{ n } } { C }_{ 3-r }.^{ n }{ C }_{ 5r }$

The 11

$^{th}$ term from last, in expansion of (2x+

$\frac{1}{^{x2}}$ is

0%
$^{25}C_{15}$ $\frac{2^{10}}{x^{20}}$
0%
$^{-25}C_{15}\frac{2^{10}}{x^{20}}$
0%
$^{25}C_{14}\frac{2^{11}}{x^{11}}$
0%
$^{25}C_{15}\frac{2^{10}}{x^{20}}$

The sum of rational terms in

$(\sqrt{2}+\sqrt[3] {3} +\sqrt[6] {5})^{10}$

0%
$12632$
0%
$1260$
0%
$126$
0%
none of these

The number of distinct terms in the expansion of

${ \left( { x+y }^{ 2 } \right) }^{ 13 }+{ \left( { x }^{ 2 }+y \right) }^{ 14 }$ is...

0%
29
0%
28
0%
25
0%
27

The number of terms whose values depend on

$x$ in the expansion of

$\left( x ^ { 2 } - 2 + \frac { 1 } { x ^ { 2 } } \right) ^ { n }$ is

0%
$2 n + 1$
0%
2 $n$
0%
$n$
0%
None of these

Find the middle terms of the equation of

$\left(x^4 - \dfrac{1}{x^3}\right)^{11}$ .

0%
$-462\ x^9, 462\ x^2$
0%
$-462\ x^8, 462\ x^4$
0%
$462\ x^7, -462\ x^3$
0%
None of these

Explanation

Given expression is

$\left(x^4-\dfrac{1}{x^3}\right)^{11}$

Here index,

$n=11$ (odd)

so, there are two middle terms, which are

$\left(\dfrac{11+1}{2}\right)^{th}$ i.e,

$6^{th}$ term and

$\left(\dfrac{11+3}{2}\right )^{th}$ i.e

$7^{th}$ term.

$T_6=T_{5+1}=^{11}C_5(x^4)^{11-5}\left(\dfrac{-1}{x^3}\right)^5$

$=\dfrac{11!}{6!5!}(x^4)^6 \left(\dfrac{-1}{x^{15}}\right)$

$=\dfrac{-11\times 10\times 9\times 8\times 7\times 6!}{6!\times 5\times 4\times 3\times 2} \dfrac{x^{24}}{x^{15}}$

$=-462 x^9$

$T_7=T_{6+1}=^{11}C_6(x^4)^{11-6}\left(\dfrac{-1}{x^3}\right)^6$

$=\dfrac{11!}{5!\times 6!}(x^4)^5 \left(\dfrac{1}{x^{18}}\right)$

$=462 x^2$

The coefficient of

$x^{5}$ in the expansion of

$\left(x^{2}-x-2\right)^{5}$ is

0%
$-83$
0%
$-82$
0%
$-81$
0%
$0$

Explanation

${\textbf{Step -1: Identify binomial coefficients and number of terms in a binomial expansion.}}$

$(x^2-x-2)^5=(x^2-2x+x-2)^5$

$=(x(x-2)+1(x-2))^5$

$=((x-2)(x+1))^5$

$=(x-2)^5(x+1)^5$

$=[^{5}C_{0}x^5+^{5}C_{1}x^4.(-2)+^{5}C_{2}x^3.(-2)^2+^{5}C_{3}x^2.(-2)^3+^{5}C_{4}x.(-2)^4+(-2)^5]$

$\times[^{5}C_{0}x^5+^{5}C_{1}x^4+^{5}C_{2}x^3+^{5}C_{3}x^2+^{5}C_{4}x+1]$

$\therefore\text{coefficient of }x^5\text{ in the expansion of the product }(x-2)^5(x+1)^5$

$=-2^5+1+^5C_2 .^5C_3(-2)^3+^5C_3 .^5C_2(-2)^2+^5C_4 .^5C_1(-2)^1+^5C_1 .^5C_4(-2)^4$

$=-32+1-800+400-50+400$

$=-81$

${\textbf{Hence, option C is correct.}}$

In the expansion of

$\left(1+3 x+2 x^{2}\right)^{6}$ the coefficient of

$x^{11}$ is

0%
$144$
0%
$288$
0%
$216$
0%
$576$

The coefficient of

${ x }^{ 1 }$ in the expansion of

$\left( 1-2x \right) ^{ -1/2 }$ is

0%
$\dfrac { \left( 2r \right) ! }{ \left( r! \right) ^{ 2 } }$
0%
$\dfrac { \left( 2r \right) ! }{ { 2 }^{ r }\left( r! \right) ^{ 2 } }$
0%
None of these
0%
$\dfrac { \left( 2r \right) ! }{ { 2 }^{ r }\left( r+1 \right) !\left( c-1 \right) ! }$

The coefficient of

$x^m$ in the expansion of

$(1+)^{m+n}$ is -

0%
$\frac { m!n! }{ (m+n)! }$
0%
$(m+n)!$
0%
$\frac { (m+n)! }{ m!n! }$
0%
none of these

The fifth term of

$\left(1 - \dfrac{2x}{3}\right)^{3/4}$ is

0%
$\dfrac{-5x^4}{1152}$
0%
$\dfrac{5x^4}{1152}$
0%
$-\dfrac{5x^4}{1052}$
0%
$\dfrac{5x^4}{1052}$

If the middle term in the expansion of

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

$924{ x }^{ 6 }$ then n=

0%
10
0%
12
0%
14
0%
None of these

the number of terms in the expansion of

${\left( {x + y - x} \right)^3}$ is

0%
4
0%
6
0%
9
0%
10

The value of

$x,$ for which the

$6 th$ term in the expansion of

$\int 2^{\log 2 \sqrt{\left(9^{x-1}+7\right)}}+\frac{1}{_{0}^{(1 / 5) \log _{2}\left(3^{x-1}+1\right)}} )^{7}$ is 84 is equal to

0%
$4$ , $3$
0%
$0$ , $3$
0%
$0$ , $2$
0%
$1$ , $2$

CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 13 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 13 - MCQExams.com

0:0:1

Practice Class 11 Engineering Maths Quiz Questions and Answers

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