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

The coefficient of $$x ^ { 4 }$$ in the expansion of $$\left( 1 + x + x ^ { 2 } + x ^ { 3 } \right) ^ { 11 }$$ is
  • $$900$$
  • $$909$$
  • $$990$$
  • $$999$$
In the expansion of $${ \left( { x }^{ 2 }+1+\frac { 1 }{ { x }^{ 2 } }  \right)  }^{ n }$$, $$n\in N$$, then
  • number of terms= 2n+1
  • term independent of x=$${ 2 }^{ n-1 }$$
  • coefficient of $${ x }^{ 2n-2 }$$= n
  • coefficient of $${ x }^{ 2 }$$= n
The coefficient of $${ t }^{ 50 }$$ in $$\left( 1+t \right) ^{ 41 }\left( 1-t+{ t }^{ 2 } \right) ^{ 40 }$$ is equal to
  • 1
  • 50
  • 81
  • 0
If the coefficients of second, third and fourth terms in the expansion of $$ ( 1 + x ) ^ { n } $$ are in A.P.then the value of $$ n $$ is:
  • $$5$$
  • $$6$$
  • $$7$$
  • $$9$$
The cooeficient of $${ x }^{ n }$$ in the expension of $$\left( 1-2x \right) ^{ -1/2 }$$
  • $$\frac { \left( 2n! \right) }{ { 2 }^{ n }\left( n! \right)  }$$
  • $$\frac { \left( 2n! \right) }{ { 2 }^{ n }\left( n! \right) ^{ 2 } }$$
  • $$\frac { \left( 2n! \right) }{ { 2 }^{ n+1 }\left( n! \right) ^{ 2 } }$$
  • $$\frac { \left( 2n! \right) }{ { 2 }^{n-1}\left( n! \right) ^{ 2 } }$$
In the expansion of $$\displaystyle \left ( 3 -\sqrt{\dfrac{17}{4} + 3\sqrt{2}} \right )^{15}$$  the $$11^{th}$$ term is a :
  • positive integer
  • positive irrational number
  • negative integer
  • negative irrational number
The value of m, for which the coefficients of the $$\left( {2m + 1} \right)$$ and $${\left( {4m + 5} \right)^{th}}$$ terms in the expansion $${\left( {1 + x} \right)^{10}}$$ are equal, is:
  • 3
  • 1
  • 5
  • 8
The number of terms in the expansion of $$ ( 2 x + 3 y - 4 z ) ^ { n } $$ ,is
  • $$ n + 1 $$
  • $$ n + 3 $$
  • $$ \frac { ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) } { 2 } \quad $$
  • none of these
If the second, third and fourth terms in the expansion of $${\left( {a + b} \right)^n}$$ are 135, 30 and 10/3 respectively, then the value of a is 
  • 3
  • 4
  • 5
  • 7
The sum of coefficients of integral powers of $$x$$ in the binomial expansion of $${\left(1-2\sqrt{x}\right)}^{-n}$$ is
  • $$\frac{1}{2}\left({3}^{50}-1\right)$$
  • $$\frac{1}{2}\left({2}^{50}+1\right)$$
  • $$\frac{1}{2}\left({3}^{50}+1\right)$$
  • $$\frac{1}{2}\left({3}^{50}\right)$$
The coefficient of $${ a }^{ 3 }{ b }^{ 4 }{ c }^{ 5 }$$ in the expansion of $${ (bc+ca+ab) }^{ 6 }$$ is 
  • 40
  • 60
  • 80
  • 100
In the expansion of $${\left( {3 - \sqrt {\frac{{17}}{4} + 3\sqrt 2 } } \right)^{15}}$$ the 11th term is a
  • Positive integer
  • positive irrational number
  • negative integer
  • negative irrational number
The coefficient of $${x^9}$$ in $$\left( {x - 1} \right)\left( {x - 4} \right)\left( {x - 9} \right)......\left( {x - 100} \right)$$ is 
  • -235
  • 235
  • 385
  • -385
If the third term in the binomial expansion of $$(1 + x)^m$$ is $$-\frac{1}{8}x^2$$, the the rational value of m is-
  • $$2$$
  • $$\frac{1}{2}$$
  • $$3$$
  • $$4$$
If $${C_0},{C_1},{C_2},.....,{C_n}$$ are the binomial coefficients, then  $$2{C_1}+{2^3}{C_3}+{2^5}{C_5} + ...$$ equals
  • $$\dfrac{{{3^n} + {{\left( { - 1} \right)}^n}}}{2}$$
  • $$\dfrac{{{3^n} - {{\left( { - 1} \right)}^n}}}{2}$$
  • $$\dfrac{{{3^n} + 1}}{2}$$
  • $$\dfrac{{{3^n} - 1}}{2}$$
Number of different terms in the sum $$ ( 1 + x ) ^ { 2009 } \cdot \left( 1 + x ^ { 2 } \right) ^ { 2008 } + \left( 1 + x ^ { 3 } \right) ^ { 2007 } , $$ is
  • 3683
  • 4007
  • 4017
  • 4352
The middle term of $$\left( { x-\cfrac { 1 }{ x }  } \right) ^{ 2n+1 }$$
  • $$^{ 2n+1 }{ { c }_{ n }.x }$$
  • $$^{ 2n+1 }{ { c }_{ n } }$$
  • $${ (-1) }^{ n\quad 2n+1 }{ c }_{ n }$$
  • $${ (-1) }^{ n\quad 2n+1 }{ c }_{ n }.x$$
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 
  • $$ - 2\left( {1 - \frac{1}{{{2^{50}}}}} \right)$$
  • $$ + \,\,ve\,\,coefficient\,\,of\,\,x$$
  • $$ - \,\,ve\,\,coefficient\,\,of\,\,x$$
  • $$ - 2\left( {1 - \frac{1}{{{2^{49}}}}} \right)$$
The coefficient of $$  x^{49}  $$ in the product $$ (x-1)(x-3) \dots(x-99)  $$ is
  • $$-99^{2}$$
  • $$1$$
  • $$-2500$$
  • $$99^2$$
The number of terms in the expansion of  $$\left( x ^ { 2 } + 1 + \dfrac { 1 } { x ^ { 2 } } \right) ^ { n } , n \in  { N } ,$$  is :
  • $$2 n $$
  • $$3 n $$
  • $$2 n + 1$$
  • 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
  • $$201$$
  • $$200$$
  • $$199$$
  • $$None\ of\ these$$
The coefficient x$$^2 in (1 + 2x + 3x^2 +..)^{-3/2}$$ is
  • 21
  • 25
  • 26
  • 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 
  • 25
  • 20
  • 15
  • 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
  • $$512$$
  • $$-512$$
  • $$1024$$
  • $$-1024$$
After simplification, the total number of terms in the expansion of $$ ( x + \sqrt { 2 } ) ^ { 4 } + ( x - \sqrt { 2 })^4 $$ is-
  • 10
  • 5
  • 4
  • 3
The number of integral terms in expansion $$(\sqrt [2]{3} + \sqrt [ 8 ]{ 5 } )^{256}$$ is 
  • 32
  • 33
  • 34
  • 35
The number of distinct terms in the expansion of $$(x+y^ {2})^ {13}+(x^ {2}+y)^ {14}$$ is
  • $$27$$
  • $$29$$
  • $$28$$
  • $$25$$
If $$r$$ 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
  • $$3r$$
  • $$3r + 1$$
  • $$2r$$
  • $$2r + 1$$
The coefficient of the term independent of $$x$$ in the expansion of $$(1 - x)^2 \cdot \left(x + \dfrac{1}{x}\right)^{10}$$ is
  • $$^{11}C_3$$
  • $$^{10}C_3$$
  • $$^{10}C_4$$
  • None of these
If $${x^{15}} - {x^{13}} + {x^{11}} - {x^9} + {x^7} - {x^5} + {x^3} - x = 7$$ then 
  • $${x^{16}}$$ is $$15$$
  • $${x^{16}}$$ is less then $$15$$
  • $${x^{16}}$$ greater then $$15$$
  • nothing can be said about $${x^{16}}$$
Number of irrational terms in the expansion of $$\left( \sqrt { 2 } +\sqrt { 3 }  \right) ^{ 15 }\quad are$$
  • 9
  • 7
  • 16
  • 10
Total number of terms in the expansion of [(1+x)$$^{100}$$+(1+x$$^{2}$$)$$^{100}$$+(1+x$$^{3}$$)$$^{100}$$] is 
  • 303
  • 201
  • 196
  • 301
Coefficient of $$x^{25}$$ in $$(1+x+x^2+x^3+....x^{10})^7$$ is 
  • $$^{31}C_{15}-7.^{20}C_{14}$$
  • $$^{31}C_{14}-7.^{20}C_{14}$$
  • $$31$$
  • 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
  • 4
  • 9
  • 2
  • 3
The coefficient of $$\dfrac{1}{x}$$ in the expansion of $$(1+x)^n \Bigg (1 + \dfrac{1}{x} \Bigg )^n$$ is 
  • $$\dfrac{n!}{(n-1)!(n+1)!}$$
  • $$\dfrac{2n!}{(n-1)!(n+1)!}$$
  • $$\dfrac{2n!}{(2n-1)!(2n+1)!}$$
  • None of these
The coefficient of t $$ t ^ { 4 } $$ in $$ \left( \frac { 1 - t ^ { 6 } } { 1 - t } \right) ^ { 3 } $$ is
  • 18
  • 12
  • 9
  • 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
  • $$255$$
  • $$61$$
  • $$127$$
  • $$None of these$$
If x=1/3, then the greatest term in the expansion of $$(1+4x)^{8}$$ is
  • $$56(\frac{3}{4})^{4}$$
  • $$56(\frac{4}{3})^{4}$$
  • $$56(\frac{3}{4})^{5}$$
  • $$56(\frac{2}{5})^{4}$$
$$\left (5^{\dfrac {1}{2}} + 7^{\dfrac {1}{6}}\right )^{642}$$ contains $$n$$ integral terms then $$n$$ is
  • $$108$$
  • $$106$$
  • $$107$$
  • $$109$$
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
  • $$-2\left( 1-\frac { 1 }{ { 2 }^{ 50 } } \right) $$
  • +ve cocfficient of x.
  • -ve coefficient of x
  • $$-2\left( 1-\frac { 1 }{ { 2 }^{ 49 } } \right) $$
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
  • $$2 ^ { 105 } - 2 ^ { 121 }$$
  • $$2 ^ { 121 } - 2 ^ { 105 }$$
  • $$2 ^ { 120 } - 2 ^ { 104 }$$
  • none of these
The 13th term of $$\left( 9x-\frac { 1 }{ 3\sqrt { x }  }  \right) ^{ 18 }\quad is\quad $$
  • 17682
  • 18564
  • 18564$$x^{ 6 }$$
  • none of these
The coefficient of $${ x }^{ 98 }\quad $$ in the expression of $$\left( \times -1 \right) ({ \times }-2).......{ (\times }-100)$$ must be
  • $${ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+.....+{ 100 }^{ 2 }$$
  • $$({ 1 }+{ 2 }+{ 3 }+.....+{ 100 }^{ 2 })-({ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+.....+{ 100 }^{ 2 })$$
  • $$\frac { 1 }{ 2 } ({ 1 }+{ 2 }+{ 3 }+.....+{ 100) }^{ 2 }-({ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+.....+{ 100 }^{ 2 })$$
  • none of these
The coefficient of $${ x }^{ 4 }$$ in $${ \left( \dfrac { x }{ 2 } -\dfrac { 3 }{ { x }^{ 2 } }  \right)  }^{ 10 }$$ is
  • $$\dfrac { 450 }{ 263 } $$
  • $$\dfrac { 405 }{ 256 } $$
  • $$\dfrac { 504 }{ 259 } $$
  • none of these
the coefficient of $$x ^ { 5 }$$ in the expansion of $$( 1 + x ) ^ { 10 }.( 1 + \cfrac { 1 } { x }) ^ { 20 }$$ is
  • $$^ { 30 } C _ { 5 }$$
  • $$^ { 10 }{ C } _ { 5 }$$
  • $$^ { 20 } { C } _ { 5 }$$
  • $$^ { 30 } { C } _ { 20 }$$
The coefficient of $$x^{m}$$ in $$(1 + x)^{p} + (1 + x)^{p + 1} + .... + (1 + x)^{n}, p\leq m\leq n$$ is
  • $$^{n + 1}C_{m + 1}$$
  • $$^{n - 1}C_{m - 1}$$
  • $$^{n}C_{m}$$
  • $$^{n}C_{m + 1}$$
The sum of the remaining terms is the group after $$2000th$$ term in which $$2000th$$ term in 
  • $$1088$$
  • $$1008$$
  • $$1040$$
  • none of there
The coefficient of $$x^{9}$$ in the expansion of (1+x)$$(1+x^{2})(1+x^{3})....(1+x^{100})$$ is____.
  • 2
  • 4
  • 6
  • 8
  • None of these.
The first integral term in the expansion of $$(\sqrt{3}+\sqrt[3]{2})^{9}$$, is its
  • 2nd term
  • 3 rd term
  • 4th term
  • 5th term
The sum of the coefficients in the expansion of $${ (1+x-{ 3x }^{ 2 }) }^{ 171 }$$ is 
  • 0
  • 1
  • -1
  • None of these
0:0:1


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