CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 5 - MCQExams.com

In the expansion of $$ \displaystyle \left ( \sqrt{2} + \sqrt[5]{3}\right )^{120} $$ the number of irrational terms is
  • $$12$$
  • $$13$$
  • $$108$$
  • $$54$$
If the fourth term in the expansion of $$\displaystyle { \left( \sqrt { \frac { 1 }{ { x }^{ \log { x } +1 } }  } +{ x }^{ 1/12 } \right)  }^{ 6 }$$ is equal to $$200$$ and $$x>1$$, then $$x$$ is equal to
  • $${ 10 }^{ \sqrt { 2 }  }$$
  • $$10$$
  • $${10}^{4}$$
  • None of these
If $$A$$ and $$B$$ are coefficients of $${x}^{n}$$ in the expansions of $${ (1+x) }^{ 2n }$$ and $${ (1+x) }^{ 2n-1 }$$ respectively, then $$\dfrac AB$$ is equal to
  • $$1$$
  • $$2$$
  • $$\dfrac 12$$
  • $$\dfrac 1n$$
The expression $${ C }_{ 0 }+2{ C }_{ 1 }+3{ C }_{ 2 }+......+(n+1){ C }_{ n }$$ is equal to
  • $${ 2 }^{ n-1 }$$
  • $$n({ 2 }^{ n-1 })$$
  • $$n({ 2 }^{ n-1 })+{ 2 }^{ n }$$
  • $$(n+1){ 2 }^{ n }$$
If $$f(n)=\sum _{ k=1 }^{ n }{ \sum _{ j=k }^{ n }{ ({ _{  }^{ n }{ C } }_{ j } } ) } ({ _{  }^{ j }{ C } }_{ k })$$, find $$f(n)$$.
  • $${ 3 }^{ n }-{ 2 }^{ n }$$
  • $${ 3 }^{ n }+{ 2 }^{ n }$$
  • $${ 3 }^{ n-1 }-{ 2 }^{ n -1}$$
  • $${ 3 }^{ n+1 }-{ 2 }^{ n+1 }$$
If the third term in the expansion $${ \left( x+{ x }^{ \log _{ 5 }{ x }  } \right)  }^{ 5 }$$ is $$2$$, then $$x$$ equals
  • $$1/5,5$$
  • $$1/5,1/\sqrt {5}$$
  • $$\sqrt{5},5$$
  • $$1/\sqrt{5},5$$
If $${ C }_{ r }=\begin{pmatrix} n \\ r \end{pmatrix}$$, then sum of series $${ \dfrac{{C }_{ 0 }^{ 2 }}{1}}+\dfrac{{ C }_{ 1 }^{ 2 }}{2}+\dfrac{{ C }_{ 2 }^{ 2 }}{3}+....$$ upto $$(n+1)$$ terms is
  • $$\cfrac { 1 }{ n+1 } \begin{pmatrix} 2n+1 \\ n+1 \end{pmatrix}$$
  • $$\cfrac { 1 }{ 2(n+1) } \begin{pmatrix} 2n+1 \\ n+1 \end{pmatrix}$$
  • $$\cfrac { 1 }{ n+1 } \begin{pmatrix} 2n+1 \\ n \end{pmatrix}$$
  • $$\cfrac { (2n+1)! }{ { (n+1)! }^{ 2 } } $$
Value of $$S=1\times 2\times 3\times 4+2\times 3\times 4\times 5+.......+n(n-1)(n+2)(n+3)$$ is
  • $$\cfrac { 1 }{ 5 } n(n+1)(n+2)(n+3)(n+4)$$
  • $$\cfrac { 1 }{ 5! } ({ _{ }^{ n+3 }{ C } }_{ 5 })$$
  • $$\cfrac { 1 }{ 5 }{ _{ }^{ n+4 }{ C } }_{ 4 }$$
  • None of these
Given that the $$4^{th}$$ term in the expansion of $${ \left( 2+\cfrac { 3x }{ 8 }  \right)  }^{ 10 }$$ has the maximum numerical value, then $$x$$ can lie in the interval(s)
  • $$\left( 2,\cfrac { 64 }{ 21 } \right) $$
  • $$\left( -\cfrac { 60 }{ 23 } ,-2 \right) $$
  • $$\left( -\cfrac { 64 }{ 21 } ,-2 \right) $$
  • $$\left( 2,-\cfrac { 60 }{ 23 } \right) $$
If the third term in the expansion of
$${ \left( \cfrac { 1 }{ x } +{ x }^{ \log _{ 2 }{ x }  } \right)  }^{ 5 }$$ is $$40$$ then $$x$$ equals
  • $$\dfrac {1}{\sqrt{2}},2$$
  • $$\sqrt{2},4$$
  • $$\dfrac {1}{\sqrt{2}},4$$
  • $$\sqrt{2},\dfrac{1}{\sqrt{2}}$$
If in the expansion of $${ \left( \cfrac { 1 }{ x } +x\tan { x }  \right)  }^{ 5 }$$ the ratio to 4th term to the 2nd term is $$\cfrac { 2 }{ 27 } { \pi  }^{ 4 }$$, then the value of $$x$$ can be
  • $$\cfrac { -\pi }{ 6 } $$
  • $$\cfrac { -\pi }{ 3 } $$
  • $$\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \pi }{ 12 } $$
If the coefficients of 2nd, 3rd and the 4th terms in the expansion of $${ \left( 1+x \right)  }^{ n }$$ are in A.P, then value of $$n$$ is
  • $$5$$
  • $$7$$
  • $$11$$
  • $$14$$
If the middle term of $${ \left( x+\cfrac { 1 }{ x } \sin ^{ -1 }{ x }  \right)  }^{ 8 }$$ is equal to $$\dfrac {630}{16}$$, then value of $$x$$ is (are)
  • $$\dfrac {\pi}3$$
  • $$\dfrac {\pi}6$$
  • $$-\dfrac {\pi}6$$
  • $$-\dfrac {\pi}3$$
Let $${ S }_{ n }= \sum _{ r=0 }^{ n }{ { (-2) }^{ r } } \left( \cfrac { { _{  }^{ n }{ C } }_{ r } }{ { _{  }^{ r+2 }{ C } }_{ r } }  \right) $$, then
  • $${ S }_{ n }=\cfrac { 1 }{ n+1 } $$ if $$n$$ is odd
  • $${ S }_{ n }=\cfrac { 1 }{ n+2 } $$ if $$n$$ is odd
  • $${ S }_{ n }=\cfrac { 1 }{ n+1 } $$ if $$n$$ is even
  • $${ S }_{ n }=\cfrac { 1 }{ n+2 } $$ if $$n$$ is even
Let $${C}_{r}$$ stand for $${ _{  }^{ n }{ C } }_{ r }$$ and $$S(n,r)={C}_{0}-{C}_{1}+{C}_{2}-{C}_{3}+....+{ (-1) }^{ r }{ C }_{ r }$$
  • If $$S(n,r)=28$$ then $$n=9,r=2$$ or $$n=9,r=6$$
  • If $$S(n,r)=-15$$, then $$n=7,r=3$$
  • $$S(n,n)=0$$
  • none of these
Value of
$$\sum _{ k=1 }^{ \infty  } \sum _{ r=0 }^{ k }{ \cfrac { 1 }{ { 3 }^{ k } }  } ({ _{  }^{ k }{ C } }_{ r })$$ is
  • $$2$$
  • $$\cfrac{2}{3}$$
  • $$\cfrac{1}{3}$$
  • none of these
Value of $$\sum { \sum _{ 0\le r<s\le n }^{  }{ { (r+s)\left( { C }_{ r }+{ C }_{ s } \right)  }^{ 2 } }  } $$ is
  • $$n\left[ (n-1)({ _{ }^{ 2n }{ C } }_{ n })+{ 2 }^{ 2n } \right] $$
  • $$n\left[ (n+1)({ _{ }^{ 2n }{ C } }_{ n })+{ 2 }^{ 2n } \right] $$
  • $$n\left[ { 2 }^{ 2n }-n({ _{ }^{ 2n }{ C } }_{ n }) \right] $$
  • None of these
If $$A=^{ 2n }{ { C }_{ 0 } }.^{ 2n }{ { C }_{ 1 } }+^{ 2n }{ { C }_{ 1 } }^{ 2n-1 }{ { C }_{ 1 } }+^{ 2n }{ { C }_{ 2 } }^{ 2n-2 }{ { C }_{ 1 } }+...$$ then $$A$$ is
  • $$0$$
  • $$n.2^{2n}$$
  • $${ 2 }^{ 10 }-2$$
  • $$1$$
The coefficient of $${ x }^{ 9 }$$ in the expansion of $${ \left( { x }^{ 3 }+\cfrac { 1 }{ { 2 }^{ t } }  \right)  }^{ 11 }$$, where $$t=\log _{ \sqrt { 2 }  }{ { (x }^{ \tfrac 32 } } )$$,
  • $$-5$$
  • $$330$$
  • $$520$$
  • $$5+\log _{ \sqrt { 2 } }{ (3 } )$$
Value of $$\sum { \sum _{ 0\le i<j\le n }^{  }{ { \left( { C }_{ i }+{ C }_{ j } \right)  }^{ 2 } }  } $$ is
  • $${ 2 }^{ 2n }(n+1)({ _{ }^{ 2n }{ C } }_{ n })$$
  • $${ 2 }^{ 2n }+(n+1)({ _{ }^{ 2n }{ C } }_{ n })$$
  • $${ 2 }^{ 2n-1 }+(n+1)({ _{ }^{ 2n }{ C } }_{ n })$$
  • none of these
If $${x}^{2r}$$ occurs in $${ \left( x+\cfrac { 2 }{ { x }^{ 2 } }  \right)  }^{ n }$$, then $$n-2r$$ must be of the form
  • $$3k$$
  • $$3k-1$$
  • $$3k+1$$
  • $$4k\pm 1$$
Value(s) of $$x$$ for which the fourth term in the expansion of
$${ \left( { \sqrt { x }  }^{ 1/(\log _{ 2 }{ x+1 } ) }+{ x }^{ 1/2 } \right)  }^{ 6 }$$ is $$40$$ is (are)
  • $$1/8$$
  • $$2$$
  • $$1/16,2$$
  • $$1/8,4$$
If the middle term of $${ \left( x+\cfrac { 1 }{ x } \sin ^{ -1 }{ x }  \right)  }^{ 8 }$$ is $$\cfrac { 35{ \pi  }^{ 4 } }{ 8 } $$, then value of $$x$$ can be
  • $$\dfrac12$$
  • $$\dfrac{\sqrt {3}}2$$
  • $$\dfrac1{\sqrt {2}}$$
  • $$1$$
The number of irrational terms in the expansion of $${ \left( { 4 }^{ 1/5 }+{ 7 }^{ 1/10 } \right)  }^{ 45 }$$ is
  • 40
  • 5
  • 41
  • none of these
If the expansion of $${ \left( 1+x \right)  }^{ 50 }$$, the sum of coefficients of add powers of $$x$$ is
  • $${ 2 }^{ 50 }$$
  • $${ 2 }^{ 49 }$$
  • $$0$$
  • None of these
The coefficients of three consecutive terms in the expansion of $${ (1+x) }^{ n }$$ are in the ratio $$5:10:21$$. find $$n$$
  • $$5$$
  • $$4$$
  • $$7$$
  • $$8$$
The coefficient of $$\displaystyle x^{99}$$ in the polynomial $$\displaystyle \left ( x-1 \right )\left ( x-2 \right )\left ( x-3 \right )...\left ( x-100 \right )$$ is
  • $$1$$
  • $$-5050$$
  • $$5050$$
  • $$-1$$
Value of the expression
$$\quad { C }_{ 0 }+({ C }_{ 0 }+{ C }_{ 1 })+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 })+....+({ C }_{ 0 }+{ C }_{ 1 }+....+{ C }_{ n-1 })$$ is
  • $${2}^{n-1}$$
  • $$n{2}^{n-2}$$
  • $$n{2}^{n-1}$$
  • $$2n$$
If $${ S }_{ n }=\cfrac { 1 }{ n+1 } \sum _{ i=0 }^{ n }{ \begin{pmatrix} n \\ i \end{pmatrix} } $$, then $$2{ S }_{ n+1 }-{ S }_{ n }$$ equals
  • $$=2^{n}[\dfrac{3n-2}{(n+1)(n+2)}]$$
  • $$=2^{n}[\dfrac{3n+2}{(n+1)(n+2)}]$$
  • $$=2^{n}[\dfrac{3n+2}{(n-1)(n+2)}]$$
  • 0
In the expansion of $$\displaystyle \left ( x^{3}+3.2^{-\log_{2}x^{3}} \right )^{11}$$
  • there appears a term with the power $$\displaystyle x^{2}$$
  • there does not appear a term with the power $$\displaystyle x^{2}$$
  • there appear a term with the power $$\displaystyle x^{-3}$$
  • the ratio of the co-efficient of $$\displaystyle x^{3}$$ to that of $$\displaystyle x^{-3}$$ is $$\displaystyle \frac{1}{3}$$
If $$\displaystyle \left ( 1+x+x^{2} \right )^{25}=a_{0}+a_{1}x+a_{2}x^{2}+\cdot \cdot \cdot +a_{50}\cdot x^{50}$$ then $$\displaystyle a_{0}+a_{2}+a_{4}+\cdot \cdot \cdot +a_{50}$$ is
  • even
  • odd & of the form $$3n$$.
  • odd & of the form $$(3n-1)$$
  • Odd & of the form $$(3n+1)$$
If $$\displaystyle \begin{pmatrix} p   \\ q  \end{pmatrix}$$ =0 for $$\displaystyle p< q$$ p,$$\displaystyle q\epsilon W$$ then $$\displaystyle \sum_{r=0}^{\infty}$$ $$\displaystyle \begin{pmatrix} n \\ 2r \end{pmatrix}$$
  • $$\displaystyle 2^{n}$$
  • $$\displaystyle 2^{n-1}$$
  • $$\displaystyle 2^{2n-1}$$
  • $$\displaystyle \hat{2n}C_{n}$$
Number of terms free from radical sign in the expansion of $$\displaystyle \left ( 1+3^{1/3}+7^{1/2} \right )^{10}$$ is
  • 4
  • 5
  • 6
  • 8
The sum of the coefficients of all the even powers of x in the expansion of $$\displaystyle \left ( 2x^{2}-3x+1 \right )^{11}$$ is
  • $$\displaystyle 2 . 6^{10}$$
  • $$\displaystyle 3 . 6^{10}$$
  • $$\displaystyle 6^{11}$$
  • None of the above
Number of rational terms in the expansion of $$\displaystyle \left ( \sqrt{2}+\sqrt[4]{3} \right )^{100}$$ is
  • $$25$$
  • $$26$$
  • $$27$$
  • $$28$$
The sum of the binomial coefficients of $$\displaystyle \left [2 x+\frac{1}{x} \right ]^{n}$$ is equal to  value of n is
  • 5
  • 6
  • 7
  • 8
$$(r + 1)^{th}$$ term in the expansion of $$(x + a)^n$$ will be
  • $$^nC_rx^na^{n-r}$$
  • $$^nC_rx^{n-r}a^{r}$$
  • $$^nC_rx^{n-r}a^{n}$$
  • $$^nC_rx^ra^{n-r}$$
If coefficients of $$x^n$$ in $$(1+x)^{101}(1-x+x^2)^{100}$$ is non-zero then $$n$$ cannot be of the form
  • $$3t+1$$
  • $$3t$$
  • $$3t+2$$
  • $$4t+1$$
$$\displaystyle \binom{n}{o}+\binom{n}{1}+\binom{n}{2}+.........+\binom{n}{n}=$$
  • $$\displaystyle 2^{n-1}$$
  • $$\displaystyle ^{2n}C_{n}$$
  • $$\displaystyle 2^n$$
  • $$\displaystyle 2^{n+1}$$
Find the $$\displaystyle 7^{th}$$ term of $$\displaystyle \left ( 3x^{2}-\frac{1}{3}\right)^{10}$$.
  • $$\displaystyle \frac{66}{3}x^{7}$$
  • $$\displaystyle \frac{70}{3}x^{7}$$
  • $$\displaystyle \frac{66}{3}x^{8}$$
  • $$\displaystyle \frac{70}{3}x^{8}$$
The value of $$\displaystyle 2\sum_{r=0}^{n}a_{2r-1}$$ is
  • $$\displaystyle 9^{n}-1$$
  • $$\displaystyle 9^{n}+1$$
  • $$\displaystyle 9^{n}-2$$
  • $$\displaystyle 9^{n}+2$$
If $${C}_{r}$$ stands for $$^{n}{C}_{r}$$, then the sum of the series $$\displaystyle\frac { 2\left( \frac { n }{ 2 }  \right) !\left( \frac { n }{ 2 }  \right) ! }{ n! } \left[ { C }_{ 0 }^{ 2 }-2{ C }_{ 1 }^{ 2 }+3{ C }_{ 2 }^{ 2 }-...+{ \left( -1 \right)  }^{ n }\left( n+1 \right) { C }_{ n }^{ 2 } \right] $$ where $$n$$ is an even positive integer, is
  • $$0$$
  • $${ \left( -1 \right)  }^{ n/2 }\left( n+1 \right) $$
  • $${ \left( -1 \right)  }^{ n/2 }\left( n+2 \right) $$
  • $${ \left( -1 \right)  }^{ n }n$$
Find the middle terms in the expansion of $$\displaystyle \left ( 2x^{2}-\frac{1}{x} \right )^{7}$$
  • $$\displaystyle -560x^{5},\:280x^{2}$$
  • $$\displaystyle -280x^{5},\:560x^{2}$$
  • $$\displaystyle 560x^{5},\:-280x^{2}$$
  • $$\displaystyle 280x^{5},\:-560x^{2}$$
The number of terms in the expansion of $$( 1 + 5\sqrt 2 x)^9 + ( 1 -5\sqrt 2 x)^9$$ is
  • 5
  • 7
  • 9
  • 10
In the expansion of $$(1 + x)^n$$, the sum of coefficients of odd powers of $$x$$ is
  • $$2^n+1$$
  • $$2^n-1$$
  • $$2^n$$
  • $$2^{n-1}$$
If $$(1+x)^n=C_0+C_1x+C_2x^2+.....+C_nx^2$$, then the value of $$C_0+C_2+C_4+ .....$$ is
  • $$2^{n-1}$$
  • $$2^n-1$$
  • $$2^n$$
  • $$2^{n-1}-1$$
Which term in the expansion of $$\left (\displaystyle \frac {x}{3}-\frac {2}{x^2}\right )^{10}$$ contains $$x^4$$?
  • 1
  • 3
  • 4
  • 5
If $$C_0, C_1, C_2, ..............C_n$$ are binomial coefficients then $$\displaystyle \frac {1}{n!0!}+\frac {1}{(n-1)!1!}+\frac {1}{(n-2)!2!}+ ....+\frac {1}{0!n!}$$ is equal to
  • $$2^n$$
  • $$\displaystyle \frac {2^{n-1}}{n!}$$
  • $$\displaystyle \frac {2^n}{n!}$$
  • none of these
If the sum of the coefficients in the expansion of $$(1 -3x + 10x^2)^n$$ is $$a$$ and if the sum of the coefficients in the expansion of $$(1 + x^2)^n$$ is $$b$$, then
  • $$a = 3b$$
  • $$a = b^3$$
  • $$b=a^3$$
  • none of these
$$\binom{n}{0}+2\binom{n}{1}+2^2\binom{n}{2}+ .... +2^n\binom{n}{n}$$ is equal to
  • $$2^n$$
  • $$0$$
  • $$3^n$$
  • none of these
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 11 Engineering Maths Quiz Questions and Answers