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

Find the sum $$\displaystyle\sum _{ r=1 }^{ n }{ \cfrac { { r }^{ n }{ C }_{ r } }{ { C }_{ r-1 } }  } $$
  • $$\displaystyle\cfrac { n(n+1) }{ 3! } $$
  • $$\displaystyle\cfrac { n(n-1) }{ 2 } $$
  • $$\displaystyle\cfrac { n(n+1) }{ 2 } $$
  • None of these
If the middle term in the expansion of $$({ { x }^{ 2 } }+\dfrac{1}{x} )^{ n }$$ is $$924\,{ x }^{ 6 }$$, then find the value of $$n$$
  • $$n=11$$
  • $$n=13$$
  • $$n=12$$
  • $$n=14$$
The middle term in the expansion of $$\left (\dfrac{x}{y}+\dfrac{y}{x}  \right )^{8}$$ is.
  • $$^{8}\textrm{C}_{5}$$
  • $$^{8}\textrm{C}_{6}$$
  • $$^{8}\textrm{C}_{4}$$
  • $$^{8}\textrm{C}_{2}$$
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 $$\alpha $$ and coeffecient of middle terms in the expansion of $${ (1+x) }^{ 2n+1 }$$ are $$\beta $$ and $$\gamma $$, then relate $$\alpha, \beta$$ and $$ \gamma $$
  • $$\beta -\gamma =\alpha $$
  • $$\gamma -\beta =\alpha $$
  • $$\beta +\gamma =\alpha $$
  • none of these
If the coefficients of $$x^7$$ and $$x^8$$ in $$\displaystyle \left ( 2 + \frac{x}{3} \right )^n$$ are equal then n $$=$$
  • 45
  • 55
  • 35
  • 27
The middle term in the expansion of $${ (1+x) }^{ 2n }$$ is
  • $$\displaystyle\frac { { 2 }^{ n }{ x }^{ n }(1\cdot 3\cdot 5\cdot \cdot \cdot (2n-1)) }{ n! } $$
  • $$\displaystyle\frac { { 2n }{ x }^{ n }(1\cdot 3\cdot 5\cdot \cdot \cdot (2n-1)) }{ n! } $$
  • $$\displaystyle\frac { { 2n }{ x }^{ n }(1\cdot 3\cdot 5\cdot \cdot \cdot (2n-1)) }{ n-1 }$$
  • $$\displaystyle\frac { { 2 }^{ n }{ x }^{ n }(1\cdot 3\cdot 5\cdot \cdot \cdot (2n-1)) }{ n-1 } $$
In the second term in the expansion $$\displaystyle { \left( \sqrt [ 13 ]{ a } +\frac { a }{ \sqrt { { a }^{ -1 } }  }  \right)  }^{ n }$$ is $$\displaystyle 14{ a }^{ \frac { 5 }{ 2 }  }$$ , then the value of $$\displaystyle \frac { _{  }^{ n }{ { C }_{ 3 }^{  } } }{ ^{ n }{ { C }_{ 2 }^{  } } } $$ 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$$_r$$ = $$^n$$C$$_r$$. 
  • $$n(n+1)2^{n-1}$$
  • $$n(n+3)2^{n-2}$$
  • $$2^n({^{2n}}C_n)$$
  • 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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