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CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 3 - MCQExams.com

Find the sum nr=1rnCrCr1
  • n(n+1)3!
  • n(n1)2
  • n(n+1)2
  • None of these
If the middle term in the expansion of (x2+1x)n is 924x6, then find the value of n
  • n=11
  • n=13
  • n=12
  • n=14
The middle term in the expansion of (xy+yx)8 is.
  • 8C5
  • 8C6
  • 8C4
  • 8C2
If the coefficient of (2r + 4)th term is equal to the coefficient of (r - 2)th term in the expansion of (1+x)18 then r=
  • 2
  • 4
  • 6
  • 8
If the coeffecient of the middle term in the expansion of (1+x)2n+2 is α and coeffecient of middle terms in the expansion of (1+x)2n+1 are β and γ, then relate α,β and γ
  • βγ=α
  • γβ=α
  • β+γ=α
  • none of these
If the coefficients of x7 and x8 in (2+x3)n are equal then n =
  • 45
  • 55
  • 35
  • 27
The middle term in the expansion of (1+x)2n is
  • 2nxn(135(2n1))n!
  • 2nxn(135(2n1))n!
  • 2nxn(135(2n1))n1
  • 2nxn(135(2n1))n1
In the second term in the expansion (13a+aa1)n is 14a52 , then the value of nC3nC2 is
  • 8
  • 12
  • 4
  • None of these
The coeffecients of the middle term in the binomial expansion in powers of x of { (1+\alpha x) }^{ 4 } and { (1+\alpha x) }^{ 6 } is the same if \alpha equals
  • -\cfrac { 5 }{ 3 }
  • \cfrac { 10 }{ 3 }
  • \cfrac { 3 }{ 10 }
  • \cfrac { 3 }{ 5 }
The middle term in the expansion of ({ { x }/{ 2+2) } }^{ 8 } is 1120, then x\varepsilon R is equal to
  • -2
  • 3
  • -3
  • 2
If { C }_{ 0 },{ C }_{ 1 },{ C }_{ 2 },...,{ C }_{ n } are the coefficients of the expansion of { \left( 1+x \right)  }^{ n }, then the value of \displaystyle \sum _{ 0 }^{ n }{ \frac { { C }_{ k } }{ k+1 }  } is
  • 0
  • \displaystyle \frac { { 2 }^{ n }-1 }{ n }
  • \displaystyle \frac { { 2 }^{ n+1 }-1 }{ n+1 }
  • None of these
The middle term in the expansion of \displaystyle \left ( 1-\frac{1}{x} \right )^{n}\left ( 1-x \right )^{n}, is
  • \displaystyle ^{2n}C_{n}
  • \displaystyle ^{-2n}C_{n}
  • \displaystyle ^{-2n}C_{n-1}
  • none of these
If A is the coefficient of the middle term in the expansion of \displaystyle (1+x)^{2n} and B and C are the coefficients of two middle terms in the expansion of \displaystyle(1+x)^{2n-1}, then
  • \displaystyle A+B=C
  • \displaystyle B+C=A
  • \displaystyle C+A=B
  • \displaystyle A+B+C=0
If n is an integer between 0 and 21, then the  minimum value of n!(21 - n)! is attained for n=
  • 1
  • 10
  • 12
  • 20
The middle term in the expansion of\displaystyle (1+x)^{2n}is
  • \displaystyle \frac{1\cdot 3\cdot5...(2n-1)}{n!}2^{n}\cdot x^{n}
  • \displaystyle \frac{1\cdot 3\cdot5...(2n-1)}{n!}2^{n}\cdot x^{n}x^{n}
  • \displaystyle ^{2n}C_{n}
  • \displaystyle ^{2n}C_{n}-1x^{n-1}
If the coefficient of x^7 in \left[ax + \left(\displaystyle\frac{1}{bx}\right)\right]^{11} is\ 55 a^{11}, then a and b satisfy the relation   
  • a + b = 1
  • a - b = 1
  • ab = 1
  • \displaystyle\frac{a}{b} = 1
The middle term in the expansion of \displaystyle \left ( x+\frac{1}{x} \right )^{10},is
  • \displaystyle ^{10}C_{1}\frac{1}{x}
  • \displaystyle ^{10}C_{5}
  • \displaystyle ^{10}C_{6}
  • \displaystyle ^{10}C_{7}x
The value of \displaystyle \frac { { _{  }^{ n }{ C } }_{ 1 }^{  } }{ 2 } +\frac { { _{  }^{ n }{ C } }_{ 3 }^{  } }{ 4 } +\frac { { _{  }^{ n }{ C } }_{ 5 }^{  } }{ 6 } +... is
  • \displaystyle \frac { { 2 }^{ n }-1 }{ n }
  • \displaystyle \frac { { 2 }^{ n }+1 }{ n }
  • \displaystyle \frac { { 2 }^{ n }-1 }{ n+1 }
  • \displaystyle \frac { { 2 }^{ n }+1 }{ n+1 }
Find the sum 1 \times 2 \times C_1 + 2 \times 3 C_2 + n (n+1)C_{n'} where C_r = ^nC_r
  • n(n+1)2^{n-1}
  • n(n+3)2^{n-2}
  • 2^n({^{2n}}C_n)
  • None of these
If(1+2x+x^2)^n = \displaystyle \sum_{r=0}^{2n} a_r x^r, then a_r=
  • (^nC_r)^2
  • ^nC_r\cdot^nC_{r+1}
  • ^{2n}C_r
  • ^{2n}C_{r+1}
Find the ratio of the coefficient of { x }^{ 10 } in { \left( 1-{ x }^{ 2 } \right)  }^{ 10 } and the term independent of x in the expansion of { \left( x-\cfrac { 2 }{ x }  \right)  }^{ 10 }
  • 1:8
  • 1:16
  • 1:24
  • 1:32
Find the coefficient of \cfrac { 1 }{ { y }^{ 2 } } in { \left( y+\cfrac { { c }}{ { y }^{ 2 } }  \right)  }^{ 10 }.
  • 210{ c }^{ 4 }
  • 210{ c }^{ 5 }
  • 120{ c }^{ 3 }
  • 120{ c }^{ 4 }
The Value of ^nC_1+^{n+1}C_2+^{n+2}C_3+...+^{n+m-1}C_m is equal 
  • ^{m+n}C_{n-1}
  • ^{m+n}C_{n+1}
  • ^mC_1+^{m+1}C_2+^{m+2}C_3+...+^{n+m-1}C_n
  • ^{m+n}C_{m}-{1}
In the expansion of (7^{\frac 13} + 11^{\frac 19})^{6561},
  • there are exactly 730 rational terms
  • there are exactly 5832 irrational terms
  • the term which involves greatest binomial coefficient is irrational
  • the term which involves greatest binomial coefficients is rational
If { \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+...+{ C }_{ n }{ x }^{ n }, then \displaystyle 2{ C }_{ 0 }+{ 2 }^{ 2 }.\frac { { C }_{ 1 } }{ 2 } +{ 2 }^{ 3 }.\frac { { C }_{ 2 } }{ 3 } +...+{ 2 }^{ n+1 }.\frac { { C }_{ n } }{ n+1 } =
  • \displaystyle \frac { { 3 }^{ n+1 }-1 }{ n+1 }
  • \displaystyle \frac { { 3 }^{ n }-1 }{ n }
  • \displaystyle \frac { { 3 }^{ n+2 }-1 }{ n+2 }
  • None of these
Determine the value of x in the expression of { ( 2+x) }^{ 5 }, if the second term in the expansion is 240
  • (24)^\frac{1}{4}
  • 6
  • 3
  • None of the above
Find the (n+1)^{th} term from the end in the expansion of { \left( x-\cfrac { 1 }{ x }  \right)  }^{ 2n }
  • { \left( -1 \right) }^{ n }. { _{ }^{ 2n }{ C } }_{ n }
  • { \left( -1 \right) }^{ n+1 }. { _{ }^{ 2n }{ C } }_{ n-1 }
  • { \left( -1 \right) }^{ n }. { _{ }^{ 2n }{ C } }_{ n -1}
  • None of these
Find the middle term in the expansion of { \left( \cfrac { 2x }{ 3 } +\cfrac { 3 }{ 2x }  \right)  }^{ 10 }.
  • 210
  • 630
  • 252
  • 756
The fourth term in the expansion of { \left( px+\cfrac { 1 }{ x }  \right)  }^{ n } is \cfrac { 5 }{ 2 } . Then,
  • n=6
  • n=7
  • p=\dfrac{1}{2}
  • p =\dfrac{1}{4}
If { \left( 1+2x+x^{ 2 } \right)  }^{ n }=\sum _{ r=0 }^{ 2n }{ { a }_{ r }{ x }^{ r } } , then { a }_{ r }=
  • { \left( _{  }^{ n }{ { C }_{ r }^{  } } \right)  }^{ 2 }
  • ^{ n }{ { C }_{ r }^{  } }.^{ n }{ { C }_{ r+1 }^{  } }
  • ^{ 2n }{ { C }_{ r }^{  } }
  • ^{ 2n }{ { C }_{ r+1 }^{  } }
The middle term in the expansion of { \left( 1+x \right)  }^{ 2n } is , n being a positive integer is
  • \cfrac { \left\{ 1.3.5....(2n) \right\} { 2 }^{ n } }{ n! } { x }^{ n }\\ \quad \quad \quad \quad \quad \quad \quad \quad
  • \cfrac { \left\{1.3.5....(2n) \right\} { 2 }^{ n }n! }{ n! } { x }^{ n }\\ \quad \quad \quad \quad \quad \quad \quad \quad
  • \cfrac { \left\{ 1.3.5....(2n-1) \right\} { 2 }^{ n }n! }{ n! } { x }^{ n }\\ \quad \quad \quad \quad \quad \quad \quad \quad
  • \cfrac { \left\{ 1.3.5....(2n-1) \right\} { 2 }^{ n } }{ n! } { x^n }\\ \quad \quad \quad \quad \quad \quad \quad \quad
If n>2, then find the value of { C }_{ 1 }{ \left( a-1 \right)  }^{ 2 }-{ C }_{ 2 }{ \left( a-2 \right)  }^{ 2 }+{ C }_{ 3 }{ \left( a-3 \right)  }^{ 2 }-.....+{ \left( -1 \right)  }^{ n-1 }{ C }_{ n }{ \left( a-n \right)  }^{ 2 } where { C }_{ r } stands for \quad { _{  }^{ n }{ C } }_{ r }
  • { a }^{ 3}
  • { a }
  • \dfrac{ a }{2}
  • { a }^{ 2 }
find the 7th term in the expansion of { \left( 4x-\frac { 1 }{ 2\sqrt { x }  }  \right)  }^{ 13 }
  • 439296{ x }^{ 7 }
  • 439296{ x }^{ 4 }
  • 439396{ x }^{ 7 }
  • 43396{ x }^{ 4 }
The 4th term from the end in the expansion of { \left( \cfrac { { x }^{ 3 } }{ 2 } -\cfrac { 2 }{ { x }^{ 2 } }  \right)  }^{ 7 } is

  • 35x
  • 70x^{2}
  • 35x^{2}
  • 70x
The middle term in the expansion of { \left( \cfrac { a }{ x } +bx \right)  }^{ 12 } is
  • 924{ a }^{ 6 }{ b }^{ 6 }
  • 924{ a }^{ 6 }{ b }^{ 5 }
  • 924{ a }^{ 5 }{ b }^{ 5 }
  • 924{ a }^{ 5 }{ b }^{ 6 }
The 8^{th} term of \displaystyle { \left( 3x+\frac { 2 }{ 3{ x }^{ 2 } }  \right)  }^{ 12 }, when expanded ina scending power of x, is
  • \displaystyle \frac { 228096 }{ { x }^{ 3 } }
  • \displaystyle \frac { 228096 }{ { x }^{ 9 } }
  • \displaystyle \frac { 328179 }{ { x }^{ 3 } }
  • None of these
Find the middle term in the expansion of { \left( 3x-\cfrac { { x }^{ 3 } }{ 6 }  \right)  }^{ 9 }.

  • {_{ }^{ 9 }{ C } }_{ 6 }{ \left( 3x \right) }^{ 5 }
  • { _{ }^{ 9 }{ C } }_{ 5 }{ \left( 3x \right) }^{ 4}
  • Both A & B
  • none of the above
If { \left( 8+3\sqrt { 7 }  \right)  }^{ n }=\alpha +\beta   where n and \alpha are positive integers and \beta is a positive proper fraction,then
  • (1-\beta)(\alpha+\beta)=1
  • (1+\beta)(\alpha+\beta)=1
  • (1-\beta)(\alpha-\beta)=1
  • (1+\beta)(\alpha-\beta)=1
If { \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+...+{ C }_{ n }{ x }^{ n }, then \displaystyle \sum _{ 0\le i\le  }^{  }{ \sum _{ j\le n }^{  }{ { \left( { C }_{ i }+{ C }_{ j } \right)  }^{ 2 } } = }
  • \left( n-1 \right) ._{  }^{ 2n }{ { C }_{ n }^{  } }+{ 2 }^{ 2n }
  • n._{  }^{ 2n }{ { C }_{ n }^{  } }+{ 2 }^{ 2n }
  • \left( n+1 \right) ._{  }^{ 2n }{ { C }_{ n }^{  } }+{ 2 }^{ 2n }
  • None of these
Find the middle term in the expansion of { \left( \cfrac { x }{ a } -\cfrac { a }{ x }  \right)  }^{ 21 }
  • { _{ }^{ 20 }{ C } }_{ 10 }\cfrac { x }{ a } , { _{ }^{ 21 }{ C } }_{ 10 }\cfrac { a }{ x }
  • { _{ }^{ 20 }{ C } }_{ 9 }\cfrac { x }{ a } , { _{ }^{ 21 }{ C } }_{ 10 }\cfrac { a }{ x }
  • { _{ }^{ 21 }{ C } }_{ 10 }\cfrac { x }{ a } , -{ {}_{ }^{ 21 }{ C } }_{ 10 }\cfrac { a }{ x }
  • { _{ }^{ 21 }{ C } }_{ 9 }\cfrac { x }{ a } , { _{ }^{ 21 }{ C } }_{ 10 }\cfrac { a }{ x }
If the second term in the expansion { \left[ a^{\dfrac {1}{13}} +\dfrac { a }{ \sqrt { { a }^{ -1 } }  }  \right]  }^{ n } is 14\ { a }^{ 5/2 }, then the value of \dfrac {^{n}C_{3}}{^{n}C_{2}} is
  • 4
  • 3
  • 12
  • 6
If the number of terms in { \left( x+1+\cfrac { 1 }{ x }  \right)  }^{ n }\quad (n\in { I }^{ + }) is 401, then n is greater than
  • 201
  • 200
  • 199
  • none of these
If \displaystyle{ a }_{ n }=\sum _{ r=0 }^{ n }{ \cfrac { 1 }{ { _{  }^{ n }{ C } }_{ r } }  } then \displaystyle\sum _{ r=0 }^{ n }{ \cfrac { r }{ { _{  }^{ n }{ C } }_{ r } }  } equals
  • (n-1){ a }_{ n }
  • n{ a }_{ n }
  • \cfrac { 1 }{ 2 } n{ a }_{ n }
  • \cfrac { (n-1) }{ 2 } { a }_{ n }
The total number of terms in the expansion of { \left( x+a \right)  }^{ 100 }+{ \left( x-a \right)  }^{ 100 }  after simplification is

  • 202
  • 51
  • 50
  • 49
Let n and k be  positive integers such that \displaystyle n\ge \frac { k\left( k+1 \right)  }{ 2 } . The number of solution \left( { x }_{ 1 },{ x }_{ 2 },..,{ x }_{ k } \right) \ge 1;{ x }_{ 2 }\ge 2,...,{ x }_{ k }\ge k all integers satisfying { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 }+...+{ x }_{ k }=n is
  • ^{ m }{ { C }_{ k-1 } }
  • ^{ m }{ { C }_{ k } }3
  • ^{ m }{ { C }_{ k+1 } }
  • None of these
The number of irrational terms in the expansion of { \left( { 2 }^{ \dfrac 15 }+{ 3 }^{ \dfrac {1}{10} } \right)  }^{ 55 } is

  • 47
  • 56
  • 50
  • 48
If \cfrac { { _{  }^{ n }{ C } }_{ r }+4{ _{  }^{ n }{ C } }_{ r+1 }+6{ _{  }^{ n }{ C } }_{ r+2 }+4{ _{  }^{ n }{ C } }_{ r+3 }+{ _{  }^{ n }{ C } }_{ r+4 } }{ { _{  }^{ n }{ C } }_{ r }+3{ _{  }^{ n }{ C } }_{ r+1 }+3{ _{  }^{ n }{ C } }_{ r+2 }+{ _{  }^{ n }{ C } }_{ r+3 } } =\cfrac { n+k }{ r+k } . Find the value of k
  • 2
  • 4
  • 6
  • 8
In the expansion of \left (x+ \sqrt{x^{2}-1}\right )^{6}+ \left (x- \sqrt{x^{2}-1}\right )^{6},the number of terms is
  • 7
  • 14
  • 6
  • 4
The number of real negative terms in the binomial expansion of \left ( 1+ix \right )^{4n-2}, n\epsilon N, x>0, is
  • n
  • n+1
  • n-1
  • 2n
Find the sum of the series \displaystyle\sum _{ r=0 }^{ n }{ { \left( -1 \right)  }^{ n } }  { _{  }^{ n }{ C } }_{ r }\left[ \cfrac { 1 }{ { 2 }^{ r } } +\cfrac { { 3 }^{ r } }{ { 2 }^{ 2r } } +\cfrac { { 7 }^{ r } }{ { 2 }^{ 3r } } +\cfrac { { 15 }^{ r } }{ { 2 }^{ 4r } } +...upto\: m\: terms \right]
  • \displaystyle\cfrac { \left( { 2 }^{ mn }+1 \right) }{ \left( { 2 }^{ n }+1 \right) \left( { 2 }^{ mn } \right) }
  • \displaystyle\cfrac { \left( { 2 }^{ mn }-1 \right) }{ \left( { 2 }^{ n }-1 \right) \left( { 2 }^{ mn } \right) }
  • \displaystyle\cfrac { \left( { 2 }^{ mn }+1 \right) }{ \left( { 2 }^{ n }-1 \right) \left( { 2 }^{ mn } \right) }
  • \displaystyle\cfrac { \left( { 2 }^{ mn }-1 \right) }{ \left( { 2 }^{ n }+1 \right) \left( { 2 }^{ mn } \right) }
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