In the expansion of $${ \left( { x }^{ 2 }+1+\frac { 1 }{ { x }^{ 2 } } \right) }^{ n }$$, $$n\in N$$, then

term independent of x=$${ 2 }^{ n-1 }$$

coefficient of $${ x }^{ 2n-2 }$$= n

coefficient of $${ x }^{ 2 }$$= n

The cooeficient of $${ x }^{ n }$$ in the expension of $$\left( 1-2x \right) ^{ -1/2 }$$

$$\frac { \left( 2n! \right) }{ { 2 }^{ n }\left( n! \right) ^{ 2 } }$$

$$\frac { \left( 2n! \right) }{ { 2 }^{ n }\left( n! \right) }$$

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

$$\frac { \left( 2n! \right) }{ { 2 }^{n-1}\left( n! \right) ^{ 2 } }$$

The number of terms in the expansion of $$ ( 2 x + 3 y - 4 z ) ^ { n } $$ , is

$$ \frac { ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) } { 2 } \quad $$

The sum of coefficients of integral powers of $$x$$ in the binomial expansion of $${\left(1-2\sqrt{x}\right)}^{-n}$$ is

$$\frac{1}{2}\left({3}^{50}+1\right)$$

$$\frac{1}{2}\left({3}^{50}-1\right)$$

$$\frac{1}{2}\left({2}^{50}+1\right)$$

$$\frac{1}{2}\left({3}^{50}\right)$$

If $${x^{15}} - {x^{13}} + {x^{11}} - {x^9} + {x^7} - {x^5} + {x^3} - x = 7$$ then

$${x^{16}}$$ is less then $$15$$

$${x^{16}}$$ greater then $$15$$

nothing can be said about $${x^{16}}$$

MCQ Questions for Class 11 Engineering Maths Binomial Theorem Quiz 14

The coefficient of

x^{4}

$x ^ { 4 }$ in the expansion of

{(1 + x + x^{2} + x^{3})}^{11}

$\left( 1 + x + x ^ { 2 } + x ^ { 3 } \right) ^ { 11 }$ is

0%
$900$
0%
$909$
0%
$990$
0%
$999$

In the expansion of

{(x^{2} + 1 + \frac{1}{x^{2}})}^{n}

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

n \in N

$n\in N$ , then

0%
number of terms= 2n+1
0%
term independent of x= ${ 2 }^{ n-1 }$
0%
coefficient of ${ x }^{ 2n-2 }$ = n
0%
coefficient of ${ x }^{ 2 }$ = n

The coefficient of

t^{50}

${ t }^{ 50 }$ in

{(1 + t)}^{41} {(1 - t + t^{2})}^{40}

$\left( 1+t \right) ^{ 41 }\left( 1-t+{ t }^{ 2 } \right) ^{ 40 }$ is equal to

0%
1
0%
50
0%
81
0%
0

If the coefficients of second, third and fourth terms in the expansion of

(1 + x)^{n}

$( 1 + x ) ^ { n }$ are in A.P.then the value of

n

$n$ is:

0%
$5$
0%
$6$
0%
$7$
0%
$9$

The cooeficient of

x^{n}

${ x }^{ n }$ in the expension of

{(1 - 2 x)}^{- 1 / 2}

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

0%
$\frac { \left( 2n! \right) }{ { 2 }^{ n }\left( n! \right) }$
0%
$\frac { \left( 2n! \right) }{ { 2 }^{ n }\left( n! \right) ^{ 2 } }$
0%
$\frac { \left( 2n! \right) }{ { 2 }^{ n+1 }\left( n! \right) ^{ 2 } }$
0%
$\frac { \left( 2n! \right) }{ { 2 }^{n-1}\left( n! \right) ^{ 2 } }$

In the expansion of

{(3 - \sqrt{\frac{17}{4} + 3 \sqrt{2}})}^{15}

$\displaystyle \left ( 3 -\sqrt{\dfrac{17}{4} + 3\sqrt{2}} \right )^{15}$ the

11^{t h}

$11^{th}$ term is a :

0%
positive integer
0%
positive irrational number
0%
negative integer
0%
negative irrational number

The value of m, for which the coefficients of the

(2 m + 1)

$\left( {2m + 1} \right)$ and

{(4 m + 5)}^{t h}

${\left( {4m + 5} \right)^{th}}$ terms in the expansion

{(1 + x)}^{10}

${\left( {1 + x} \right)^{10}}$ are equal, is:

0%
3
0%
1
0%
5
0%
8

The number of terms in the expansion of

(2 x + 3 y - 4 z)^{n}

$( 2 x + 3 y - 4 z ) ^ { n }$ ,is

0%
$n + 1$
0%
$n + 3$
0%
$\frac { ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) } { 2 } \quad$
0%
none of these

If the second, third and fourth terms in the expansion of

{(a + b)}^{n}

${\left( {a + b} \right)^n}$ are 135, 30 and 10/3 respectively, then the value of a is

0%
3
0%
4
0%
5
0%
7

The sum of coefficients of integral powers of

x

$x$ in the binomial expansion of

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

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

0%
$\frac{1}{2}\left({3}^{50}-1\right)$
0%
$\frac{1}{2}\left({2}^{50}+1\right)$
0%
$\frac{1}{2}\left({3}^{50}+1\right)$
0%
$\frac{1}{2}\left({3}^{50}\right)$

The coefficient of

a^{3} b^{4} c^{5}

${ a }^{ 3 }{ b }^{ 4 }{ c }^{ 5 }$ in the expansion of

{(b c + c a + a b)}^{6}

${ (bc+ca+ab) }^{ 6 }$ is

0%
40
0%
60
0%
80
0%
100

In the expansion of

{(3 - \sqrt{\frac{17}{4} + 3 \sqrt{2}})}^{15}

${\left( {3 - \sqrt {\frac{{17}}{4} + 3\sqrt 2 } } \right)^{15}}$ the 11th term is a

0%
Positive integer
0%
positive irrational number
0%
negative integer
0%
negative irrational number

The coefficient of

x^{9}

${x^9}$ in

(x - 1) (x - 4) (x - 9) . . . . . . (x - 100)

$\left( {x - 1} \right)\left( {x - 4} \right)\left( {x - 9} \right)......\left( {x - 100} \right)$ is

0%
-235
0%
235
0%
385
0%
-385

If the third term in the binomial expansion of

(1 + x)^{m}

$(1 + x)^m$ is

- \frac{1}{8} x^{2}

$-\frac{1}{8}x^2$ , the the rational value of m is-

0%
$2$
0%
$\frac{1}{2}$
0%
$3$
0%
$4$

C_{0}, C_{1}, C_{2}, . . . . ., C_{n}

${C_0},{C_1},{C_2},.....,{C_n}$ are the binomial coefficients, then

2 C_{1} + 2^{3} C_{3} + 2^{5} C_{5} + . . .

$2{C_1}+{2^3}{C_3}+{2^5}{C_5} + ...$ equals

0%
$\dfrac{{{3^n} + {{\left( { - 1} \right)}^n}}}{2}$
0%
$\dfrac{{{3^n} - {{\left( { - 1} \right)}^n}}}{2}$
0%
$\dfrac{{{3^n} + 1}}{2}$
0%
$\dfrac{{{3^n} - 1}}{2}$

Explanation

We have,

$(1+x)^n =C_0+C_1x +C_2x^2 +C_3x^3+....+C_n x^n\quad...(1)\ \ \ \text{ and }\\(1-x)^n =C_0-C_1x+C_2x^2-C_3x^3+.....+(-1)^n \cdot C_n x^n\quad...(2)$

Subtracting equation

$(2)$ from

$(1)$

$\Rightarrow (1+x)^n -(1-x)^n =2[C_1x+C_3x^3 +C_5x^5+....]$

$\Rightarrow \dfrac{1}{2}[(1+x)^n -(1-x)^n ] =C_1x+C_3x^3+C_5x^5+.....$

Putting

$x=2$ , we get,

$2C_1 +2^3 C_3+ 2^5 C_5 +...=\dfrac{3^n -(-1)^n}{2}$

Hence, the correct answer is option (B).

Number of different terms in the sum

$( 1 + x ) ^ { 2009 } \cdot \left( 1 + x ^ { 2 } \right) ^ { 2008 } + \left( 1 + x ^ { 3 } \right) ^ { 2007 } ,$ is

0%
3683
0%
4007
0%
4017
0%
4352

The middle term of

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

0%
$^{ 2n+1 }{ { c }_{ n }.x }$
0%
$^{ 2n+1 }{ { c }_{ n } }$
0%
${ (-1) }^{ n\quad 2n+1 }{ c }_{ n }$
0%
${ (-1) }^{ n\quad 2n+1 }{ c }_{ n }.x$

The coefficient of

${x^{49}}$ in the expansion of

$\left( {x - 1} \right)\left( {x - \frac{1}{2}} \right)\left( {x - \frac{1}{{{2^2}}}} \right).....\left( {x - \frac{1}{{{2^{49}}}}} \right)$ is equal to

0%
$- 2\left( {1 - \frac{1}{{{2^{50}}}}} \right)$
0%
$+ \,\,ve\,\,coefficient\,\,of\,\,x$
0%
$- \,\,ve\,\,coefficient\,\,of\,\,x$
0%
$- 2\left( {1 - \frac{1}{{{2^{49}}}}} \right)$

The coefficient of

$x^{49}$ in the product

$(x-1)(x-3) \dots(x-99)$ is

0%
$-99^{2}$
0%
$1$
0%
$-2500$
0%
$99^2$

The number of terms in the expansion of

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

0%
$2 n$
0%
$3 n$
0%
$2 n + 1$
0%
None

If the number of terms in

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

$401$ , then

$n$ is greater then

0%
$201$
0%
$200$
0%
$199$
0%
$None\ of\ these$

The coefficient x

$^2 in (1 + 2x + 3x^2 +..)^{-3/2}$ is

0%
21
0%
25
0%
26
0%
None of these

In the expansion of

$\left( { x }^{ 3 }-\dfrac { 1 }{ { x }^{ 2 } } \right) ^{ n },\quad n\quad \epsilon \quad N,$ if the sum of the coefficient of

${ x }^{ 5 }\quad and\quad { x }^{ 10 }\quad$ is 0 then n is

0%
25
0%
20
0%
15
0%
none of these

The sum of the coefficients of even powers of

$x$ in the expansion of

$\left( 1 + x + x ^ { 2 } + x ^ { 3 } \right) ^ { 5 }$ is

0%
$512$
0%
$-512$
0%
$1024$
0%
$-1024$

After simplification, the total number of terms in the expansion of

$( x + \sqrt { 2 } ) ^ { 4 } + ( x - \sqrt { 2 })^4$ is-

0%
10
0%
5
0%
4
0%
3

The number of integral terms in expansion

$(\sqrt [2]{3} + \sqrt [ 8 ]{ 5 } )^{256}$ is

0%
32
0%
33
0%
34
0%
35

The number of distinct terms in the expansion of

$(x+y^ {2})^ {13}+(x^ {2}+y)^ {14}$ is

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

$r$ and

$n$ are positive integers;

$r > 1, n > 2$ , and the coefficient of

$(r + 2)^{th}$ term and

$3r^{th}$ term in the expansion of

$(1 + x)^{2n}$ are equal, then

$n$ equals

0%
$3r$
0%
$3r + 1$
0%
$2r$
0%
$2r + 1$

The coefficient of the term independent of

$x$ in the expansion of

$(1 - x)^2 \cdot \left(x + \dfrac{1}{x}\right)^{10}$ is

0%
$^{11}C_3$
0%
$^{10}C_3$
0%
$^{10}C_4$
0%
None of these

${x^{15}} - {x^{13}} + {x^{11}} - {x^9} + {x^7} - {x^5} + {x^3} - x = 7$ then

0%
${x^{16}}$ is $15$
0%
${x^{16}}$ is less then $15$
0%
${x^{16}}$ greater then $15$
0%
nothing can be said about ${x^{16}}$

Number of irrational terms in the expansion of

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

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

Total number of terms in the expansion of [(1+x)

$^{100}$ +(1+x

$^{2}$ )

$^{100}$ +(1+x

$^{3}$ )

$^{100}$ ] is

0%
303
0%
201
0%
196
0%
301

Coefficient of

$x^{25}$ in

$(1+x+x^2+x^3+....x^{10})^7$ is

0%
$^{31}C_{15}-7.^{20}C_{14}$
0%
$^{31}C_{14}-7.^{20}C_{14}$
0%
$31$
0%
None of these

In the expansion of

${ x }^{ 2 }{ \left( \sqrt { x } +\frac { \lambda }{ { x }^{ 2 } } \right) }^{ 10 }$ . The coefficient of

${ x }^{ 2 }$ is

$720$ then

$\lambda$ is

0%
4
0%
9
0%
2
0%
3

The coefficient of

$\dfrac{1}{x}$ in the expansion of

$(1+x)^n \Bigg (1 + \dfrac{1}{x} \Bigg )^n$ is

0%
$\dfrac{n!}{(n-1)!(n+1)!}$
0%
$\dfrac{2n!}{(n-1)!(n+1)!}$
0%
$\dfrac{2n!}{(2n-1)!(2n+1)!}$
0%
None of these

The coefficient of t

$t ^ { 4 }$ in

$\left( \frac { 1 - t ^ { 6 } } { 1 - t } \right) ^ { 3 }$ is

0%
18
0%
12
0%
9
0%
15

The number of distinct terms in the expansion of

$\left(x + \dfrac { 1 }{ x } + x^{ 2 } + \dfrac { 1 }{ x^{ 2 } }\right)^{ 15 }$ is/are

0%
$255$
0%
$61$
0%
$127$
0%
$None of these$

If x=1/3, then the greatest term in the expansion of

$(1+4x)^{8}$ is

0%
$56(\frac{3}{4})^{4}$
0%
$56(\frac{4}{3})^{4}$
0%
$56(\frac{3}{4})^{5}$
0%
$56(\frac{2}{5})^{4}$

$\left (5^{\dfrac {1}{2}} + 7^{\dfrac {1}{6}}\right )^{642}$ contains

$n$ integral terms then

$n$ is

0%
$108$
0%
$106$
0%
$107$
0%
$109$

The coefficient of

${ x }^{ 4y }\quad in\quad the\quad expansion\quad of\quad (x-1)\left( x-\frac { 1 }{ 2 } \right) \left( x-\frac { 1 }{ { 2 }^{ 2 } } \right) ......\left( x-\frac { 1 }{ { 2 }^{ 49 } } \right)$ is equal to

0%
$-2\left( 1-\frac { 1 }{ { 2 }^{ 50 } } \right)$
0%
+ve cocfficient of x.
0%
-ve coefficient of x
0%
$-2\left( 1-\frac { 1 }{ { 2 }^{ 49 } } \right)$

The coefficient of

$x ^ { 15 }$ in the product of

$( 1 - x )( 1 - 2 x ) \left( 1 - 2 ^ { 2 } x \right) \dots \ldots \ldots \ldots \left( 1 - 2 ^ { 15 } x \right)$ is equal to

0%
$2 ^ { 105 } - 2 ^ { 121 }$
0%
$2 ^ { 121 } - 2 ^ { 105 }$
0%
$2 ^ { 120 } - 2 ^ { 104 }$
0%
none of these

The 13th term of

$\left( 9x-\frac { 1 }{ 3\sqrt { x } } \right) ^{ 18 }\quad is\quad$

0%
17682
0%
18564
0%
18564 $x^{ 6 }$
0%
none of these

The coefficient of

${ x }^{ 98 }\quad$ in the expression of

$\left( \times -1 \right) ({ \times }-2).......{ (\times }-100)$ must be

0%
${ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+.....+{ 100 }^{ 2 }$
0%
$({ 1 }+{ 2 }+{ 3 }+.....+{ 100 }^{ 2 })-({ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+.....+{ 100 }^{ 2 })$
0%
$\frac { 1 }{ 2 } ({ 1 }+{ 2 }+{ 3 }+.....+{ 100) }^{ 2 }-({ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+.....+{ 100 }^{ 2 })$
0%
none of these

The coefficient of

${ x }^{ 4 }$ in

${ \left( \dfrac { x }{ 2 } -\dfrac { 3 }{ { x }^{ 2 } } \right) }^{ 10 }$ is

0%
$\dfrac { 450 }{ 263 }$
0%
$\dfrac { 405 }{ 256 }$
0%
$\dfrac { 504 }{ 259 }$
0%
none of these

the coefficient of

$x ^ { 5 }$ in the expansion of

$( 1 + x ) ^ { 10 }.( 1 + \cfrac { 1 } { x }) ^ { 20 }$ is

0%
$^ { 30 } C _ { 5 }$
0%
$^ { 10 }{ C } _ { 5 }$
0%
$^ { 20 } { C } _ { 5 }$
0%
$^ { 30 } { C } _ { 20 }$

The coefficient of

$x^{m}$ in

$(1 + x)^{p} + (1 + x)^{p + 1} + .... + (1 + x)^{n}, p\leq m\leq n$ is

0%
$^{n + 1}C_{m + 1}$
0%
$^{n - 1}C_{m - 1}$
0%
$^{n}C_{m}$
0%
$^{n}C_{m + 1}$

The sum of the remaining terms is the group after

$2000th$ term in which

$2000th$ term in

0%
$1088$
0%
$1008$
0%
$1040$
0%
none of there

The coefficient of

$x^{9}$ in the expansion of (1+x)

$(1+x^{2})(1+x^{3})....(1+x^{100})$ is____.

0%
2
0%
4
0%
6
0%
8
0%
None of these.

The first integral term in the expansion of

$(\sqrt{3}+\sqrt[3]{2})^{9}$ , is its

0%
2nd term
0%
3 rd term
0%
4th term
0%
5th term

The sum of the coefficients in the expansion of

${ (1+x-{ 3x }^{ 2 }) }^{ 171 }$ is

0%
0
0%
1
0%
-1
0%
None of these

CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 14 - 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 14 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

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