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CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 4 - MCQExams.com

Determine the value of x in the expression (x+xt)5, if the third term in the expression is 10,00,000 where t=log10x.
  • 10
  • 105/2
  • Both (A)and (B)
  • None of these
The number of terms whose values depends on x in the expansion of (x22+1x2)n is
  • 2n+1
  • 2n
  • n
  • none of these
In the expansion of the expression (x+a)15, if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest?
  • T6
  • T7
  • T8
  • T9
The value of 1(n1)!+1(n3)!3!+1(n5)!5!+....
  • 2n+1n!
  • 2n1n!
  • 2n1n+1!
  • 2n+1n1!
If (1+x)n=C0+C1x+C2x2+..........+CnxR, then the sum
C0+(C0+C1)+(C0+C1+C2)+.....+(C0+C1+C2+.....+Cn1 is

  • n.2n+1
  • n.2n1
  • (n1).2n1
  • (n+1).2n+1
If n is a positive integer and Ck=nCk, find the value of nk=1k3(CkCk1)2
  • n(n1)2(n+2)12
  • n(n+1)2(n2)12
  • n(n+1)2(n+2)12
  • n(n1)2(n2)12
The sum of coefficients of x2r,r=1,2,3,..., in the expansion of (1+x)n is
  • 2n
  • 2n11
  • 2n1
  • 2n1+1
The number of terms with integral coefficients in the expansion of(71/3+51/2.x)600 is
  • 100
  • 50
  • 101
  • none of these
The sum of coefficients in the binomial expansion of (1x+2x)nis equal to 6561.The constant term in the expansion is
  • 8C4
  • 168C4
  • 6C424
  • none of these
If the 4th term in the expansion is of (px+x1)m is 2.5 for all xϵR then
  • p=52,m=3
  • p=12,m=6
  • p=12,m=6
  • none of these
The sum 10C3+11C3+12C3+..........+20C3 is equal to
  • 21C4
  • 21C4+ 10C4
  • 21C17 10C6
  • none of these
The sum of coefficients of all the integral powers of  x in the expansion of (1+2x)40 is
  • 340+1
  • 3401
  • 12 (3401)
  • 12 (340+1)
The absolute value of middle term in the expansion of (11x)n.(1x)n is
  • 2nCn
  •  2nCn
  •  2nCn1
  • none of these
The sum of last ten coeffficients in the expansion of (1+x)19 when expanded in ascending powers of x is
  • 218
  • 219
  • 21819C10
  • none of these
The middle term in the expansion of (2x332x2)2n is
  • 2nCn
  • (1)n[(2n!)/(n!)2].xn
  • 2nCn.1xn
  • none of these
The sum 20C0+20C1+20C2+20C10 is equal to
  • 220+20!(10!)2
  • 2191(2)20!(10!)2
  • 219+ 20C10
  • none of these
The number of non-zero terms in the expansion of (1+32x)9+(132x)9 is
  • 9
  • 0
  • 5
  • 10
The value of \displaystyle \sum_{j=1}^{n}(^{n+1}C_{j}-^{n}C_{j})is equal to
  • \displaystyle 2^{n}
  • \displaystyle 2^{n}+1
  • \displaystyle 3\cdot 2^{n}
  • \displaystyle 2^{n} -1
The sum \displaystyle\frac{1}{2} ^{10}\textrm{C}_{0}- ^{10}\textrm{C}_{1}+ 2\cdot  ^{10}\textrm{C}_{2}- 2^{2}\cdot  ^{10}\textrm{C}_{3}+...+2^{9}\cdot  ^{10}\textrm{C}_{10} is equal to
  • \displaystyle\frac{1}{2}
  • 0
  • \displaystyle\frac{1}{2}\cdot3^{10}
  • none of these
The number of integral terms in the expansion of \displaystyle \left ( \sqrt{3}+\sqrt[5]{5} \right )^{256} is
  • 25
  • 26
  • 24
  • None of these
If (1+x)^{2n}=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{2n}x^{2n} then
  • a_{n+1}> a_{n}
  • a_{n+1}< a_{n}
  • a_{n-3}=a_{n+3}
  • none of these
  • Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
  • Statement-1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for Statement-1
  • Statement-1 is True, Statement-2 is False
  • Statement-1 is False, Statement-2 is True
Let n\epsilon N.If (1+x)^{n}=a_{0}x+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}, and a_{n-3},a_{n-2},a_{n-1}, are in AP then
  • a_{1},a_{2},a_{3} are in AP
  • a_{1},a_{2},a_{3} are in HP
  • n=7
  • n=14
In the expansion of \left (\sqrt[3]{4}+\dfrac{1}{\sqrt[4]{6}}\right )^{20},
  • the number of rational terms =4
  • the number of irrational terms =19
  • the middle term is irrational
  • the number of irrational terms =17
The sum of the series \sum _{ r=0 }^{ 10 }{ _{  }^{ 20 }{ { C }_{ r }^{  } } } is
  • \displaystyle { 2 }^{ 19 }-\frac { 1 }{ 2 } ._{  }^{ 20 }{ { C }_{ 10 }^{  } }
  • \displaystyle { 2 }^{ 19 }+\frac { 1 }{ 2 } .{  }^{ 20 }{ { C }_{ 10 }^{  } }
  • { 2 }^{ 19 }
  • { 2 }^{ 20 }
^{ n+1 }{ { C }_{ 2 }^{  } }+2\left[ _{  }^{ 2 }{ { C }_{ 2 }^{  } }+^{ 3 }{ { C }_{ 2 }^{  } }+^{ 4 }{ { C }_{ 2 }^{  } }+...+^{ n }{ { C }_{ 2 }^{  } } \right] =
  • \displaystyle \frac { n\left( n+1 \right) \left( 2n+1 \right)  }{ 6 }
  • \displaystyle \frac { n\left( n+1 \right)  }{ 2 }
  • \displaystyle \frac { n\left( n-1 \right) \left( 2n-1 \right)  }{ 6 }
  • None of these
If \displaystyle C_{0},C_{1},C_{2},....C_{n} are binomial coefficient in the expansion of \displaystyle (1+x)^{n}, then value of  \displaystyle C_{1}+C_{4}+C_{7}+... equals
  • \displaystyle \frac{1}{3}(2^{n}+\sqrt 3 \sin \frac{n \pi}{3})
  • \displaystyle \frac{1}{3}(2^{n}-\cos \frac{n \pi}{3}+\sqrt 3 \sin \frac{n \pi}{3} )
  • \displaystyle \frac{1}{3}(2^{n}-\sqrt 3 \sin \frac{n \pi}{3})
  • \displaystyle \frac{1}{3}(2^{n}-\cos \frac{n \pi}{3}-\sqrt 3 \sin \frac{n \pi}{3} )
If the third term in the expansion of \displaystyle (\frac{1}{x}+x^{\log_{10}x})^{5} is 1,000, then x-equals
  • 10\displaystyle ^{2}
  • 10\displaystyle ^{3}
  • 10
  • None of these
If C_{0},C_{1},C_{2},.....C_{n}, are binomial coefficients,then \displaystyle\sum_{k= 0}^{n} C_{k}\:\sin kx \cos \left ( n-k \right )x equals

  • 2^{n} \sin nx
  • 2^{n+1}\sin \left ( n+1 \right )x
  • 2^{n-1}\sin nx
  • 2^{n+1}\sin nx
If coefficient of x^{100} in 1+\left ( 1+x \right )\left ( 1+x \right )^{2}+.....+\left ( 1+x \right )^{n}\left ( if\:n \geq 100\right ) is C_{101}^{201} then the value of n equals

  • 202
  • 100
  • 200
  • 201
If \displaystyle \left ( 1+x+x^{2} \right )^{n}=\sum_{r=0}^{2n} a_{r}x^{r}=a_{0}+a_{1}x+a_{2}x^{2}+...+a^{2n}x^{2n} and
\displaystyle P=a_{0}+a_{3}+a_{6}+...
\displaystyle Q=a_{1}+a_{4}+a_{7}+...
\displaystyle R=a_{2}+a_{5}+a_{8}+...
then the set of values of P, Q, R are respectively equals
  • \displaystyle (1 ,1, 1)
  • \displaystyle (3^{n},3^{n},3^{n})
  • \displaystyle (3^{n+1},3^{n+1},3^{n+1})
  • \displaystyle (3^{n-1},3^{n-1},3^{n-1})
The evaluated value of \displaystyle \sum_{i=0}^{n} \sum_{j=1}^{n} \displaystyle ^{n}C_{j} \displaystyle ^{j}C_{i}, \displaystyle i\leq j
  • \displaystyle 3^{n}+1
  • \displaystyle 3^{n}-1
  • \displaystyle 3^{n+1}+1
  • None of these
If \displaystyle C_{r}=^{n}C_{r} and \displaystyle (C_{0}+C_{1})(C_{1}+C_{2})...(C_{n-1}+C_{n})=k \displaystyle (C_{0} C_{1}C_{2}...C_{n}) then the value of \displaystyle k equals
  • \displaystyle \frac{(n+1)^{n+1}}{n!}
  • \displaystyle \frac{(n+1)^{n}}{n.n!}
  • \displaystyle \frac{(n)^{n}}{n!}
  • \displaystyle \frac{(n+1)^{n}}{n!}
If \displaystyle P be the sum of odd term and  \displaystyle Q that of even terms in the expansion of  \displaystyle (x+a)^{n} , then the value of  \displaystyle [(x+a)^{2n}-(x-a)^{2n}] equals
  • \displaystyle PQ
  • \displaystyle 2PQ
  • 4 \displaystyle PQ
  • None of these
The sum of the coefficients of all odd exponets of \displaystyle x in the product of \displaystyle (1-x +x^{2}-x^{3}+x^{4}+...-x^{49}+x^{50})\times(1+x+x^{2}+x^{3}+...+x^{50}) equals
  • 1
  • 0
  • -1
  • None of these
If C_{0},C_{1},C_{2}....,C_{n} are Binomial Coefficients, such that \displaystyle S_{n}=\sum_{r=0}^{n}\frac{1}{{C_{r}}^{n}} and \displaystyle t_{n}= \sum_{r= 0}^{n}\frac{r}{{C_{r}}^{n}} then \displaystyle \frac{t_{n}}{s_{n}} equals

  • \displaystyle \frac{n}{2}
  • \displaystyle\frac{n\left ( n+1 \right )}{2}
  • \displaystyle\frac{n+1}{2}
  • None of these
If\displaystyle\left ( 1+x \right )^{n}= \sum_{r= 0}^{n}C_{r}x^{r}, then the value of C_{0}-C_{2}+C_{4}-C_{6}+C_{8}-C_{10}+... equals
  • \displaystyle2^{\tfrac{n}{2}}\cos \frac{nx}{4}
  • C_{1}-C_{3}+C_{5}-C_{7}+....
  • C_{0}+C_{4}+C_{8}+C_{12}+....
  • \displaystyle2^{\tfrac{n}{2}}\sin \frac{nx}{4}
If \left ( 1+x \right )^{n}= \sum_{r= 0}^{n}C_{r}x^{r} then the value of 3C_{1}+7C_{2}+11C^{3}+....+\left ( 4n-1 \right )C_{n} is
  • \left ( 4n-1 \right )2^{n}
  • \left ( 2n-1 \right )2^{n}
  • \left ( 2n-1 \right )2^{n}+1
  • \left ( 4n-1 \right )2^{n}-1
If {C_{r}}^{13} denoted by C_{r} then value of c_{1}+c_{5}+c_{7}+c_{9}+c_{11} is equal to
  • 2^{12}-287
  • 2^{12}-165
  • 2^{12}-C_{3}
  • 2^{12}-C_{2}-C_{13}
The Coefficient of x^{53} in \sum_{m= 0}^{100}{C_{m}}^{100}\left ( x-3 \right )^{100-m}2^{m} is
  • =-\:^{100}C_{53}
  • =-\:^{101}C_{53}
  • =\:^{101}C_{47}
  • =-\:^{100}C_{47}
The number of terms with integral coefficient in the expansion of \left ( 17^{\dfrac 13} +35^{\dfrac 12}\right )^{300} is
  • 50
  • 100
  • 150
  • 51
If A is the sum of the odd terms and B the sum of even terms in the expansion of { \left( x+a \right)  }^{ n }, then { A }^{ 2 }-{ B }^{ 2 }=
  • { \left( { x }^{ 2 }+{ a }^{ 2 } \right)  }^{ n }
  • { \left( { x }^{ 2 }-{ a }^{ 2 } \right)  }^{ n }
  • 2{ \left( { x }^{ 2 }-{ a }^{ 2 } \right)  }^{ n }
  • None of these
The sum of the series \displaystyle\sum_{r=0}^{n}\left ( ^{n+1}C_{r} \right ) equals
  • \left ( n+1 \right )2^{2n-1}
  • \displaystyle \frac{1}{2}\frac{2n!}{n!n!}
  • \displaystyle 2^{2n-1}\left ( n+1 \right )-\frac{1}{2}\frac{2n!}{n!n!}
  • 2^{n+1}-1
If x+y= 1 then \displaystyle\sum_{r= 0}^{n} r^{n}C_{r}x^{r}y^{n-r} equals
  • 1
  • n
  • nx
  • ny
Number of rational term is the expansion of \left ( 7^{1/3}+11^{1/9} \right )^{729}
  • 81
  • 82
  • 730
  • None of these
The values of x in the expansion \displaystyle \left ( x+x^{log_{10}x} \right )^{5} , if the third term in the expansion is 10,00,000
  • \displaystyle 10
  • \displaystyle 10^{2}
  • \displaystyle 10^{3}
  • None of these
If \displaystyle\left ( 1+x \right )^{n}=\sum_{r=0}^{n}C_{r}x^{r} and \sum { \sum _{ 0\le i<j\le n }{ { C }_{ i }\times { C }_{ j } }  }  represent the products of the C_{i}'s taken two at a time, then its value equals
  • \displaystyle2^{2n-1}-\frac{\left (2n\right)!}{\left (n!\right)^{2}}
  • \displaystyle2^{2n-1}+\frac{2n!}{n!n!}
  • \displaystyle2^{2n-1}-\frac{2n!}{2\cdot n!n!}
  • None of these
Sum of the coefficients of the terms of degree m in the expansion of
{ (1+x) }^{ n }{ (1+y) }^{ n }{ (1+z) }^{ n } is
  • { ({ _{ }^{ n }{ C } }_{ m }) }^{ 3 }
  • 3({ _{ }^{ n }{ C } }_{ m })
  • { _{ }^{ n }{ C } }_{ 3m }
  • { _{ }^{ 3n }{ C } }_{ m }
The number of irrational terms in the expansion of { ({ 5 }^{\tfrac 16 }+{ 2 }^{ \tfrac 18 }) }^{ 100 } is

  • 96
  • 97
  • 98
  • 99
Value of the expression { C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+{ C }_{ 2 }^{ 2 }+.....+(n+1){ C }_{ n }^{ 2 } is
  • (2n+1)({ _{ }^{ 2n }{ C } }_{ n })
  • (2n-1)({ _{ }^{ 2n }{ C } }_{ n })
  • \left( \cfrac { n }{ 2 } +1 \right) ({ _{ }^{ 2n }{ C } }_{ n })
  • \left( \cfrac { n }{ 2 } +1 \right) ({ _{ }^{ 2n-1 }{ C } }_{ n })
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