CBSE Questions for Class 11 Engineering Maths Sequences And Series Quiz 12 - MCQExams.com

 
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  • $$6$$
  • $$7$$
  • $$8$$
  • $$9$$
  • $$10$$
Find sum of series:
$$1.3.5+3.5.7+5.7.9.....$$ ?
  • $$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{8}+\frac{15}{8}$$
  • $$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{18}+\frac{15}{18}$$
  • $$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{20}+\frac{15}{8}$$
  • $$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{6}+\frac{1}{8}$$
$$\sin { \left( \dfrac  { 1 }{ \sqrt { 2 }  }  \right)  } +\sin { \left( \dfrac  { \sqrt { 2 } -1 }{ \sqrt { 6 }  }  \right)  } +\dots +\sin { \left( \dfrac  { \sqrt { n } -\sqrt { n-1 }  }{ \sqrt { n(n+1) }  }  \right)  } +\dots \infty =$$
  • $$\pi$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {3\pi}{2}$$
The sum of the series $$1+\dfrac{1}{1!}.\dfrac{1}{4}+\dfrac{1\cdot 3}{2!}\left(\dfrac{1}{4}\right)^{2}+\dfrac{1\cdot 3 \cdot 5}{3!} \left(\dfrac{1}{4}\right)^{3}+........$$ to $$\infty$$ is ?
  • $$\sqrt{2}$$
  • $$2$$
  • $$\dfrac{1}{\sqrt{2}}$$
  • $$\sqrt{3}$$
The value of $$(21{C}_{1}-10{C}_{1})+(21{C}_{2}-10{C}_{2})+(3{C}_{1}-10{C}^{3})+(21{C}_{4}-10{C}^{4})+.....(21{C}_{10}-10{C}_{10})$$ is
  • $${2}^{20}-{2}^{10}$$
  • $${2}^{21}-{2}^{11}$$
  • $${2}^{21}-{2}^{10}$$
  • $${2}^{20}-{2}^{9}$$
If $${ a }_{ n }=\sum _{ r=0 }^{ n }{ \cfrac { 1 }{ { _{  }^{ n }{ C } }_{ r } }  } $$, the value of $$\sum _{ r=0 }^{ n }{ \cfrac { n-2r }{ { _{  }^{ n }{ C } }_{ r } }  } $$
  • $$\cfrac{n}{2}{a}_{n}$$
  • $$\cfrac{1}{2}{a}_{n}$$
  • $$n{a}_{n}$$
  • $$0$$
The value of $$\displaystyle\sum^{100}_{r=2}\dfrac{3^r(2-2r)}{(r+1)(r+2)}$$ is equal to?
  • $$\dfrac{1}{2}-\dfrac{3^{100}}{100(101)}$$
  • $$\dfrac{3}{2}-\dfrac{3^{101}}{101(102)}$$
  • $$\dfrac{3}{2}-\dfrac{3^{100}}{100(101)}$$
  • None of these
The sum of $$n$$ terms of the series $${1^3} + {3^3} + {5^3} +  \ldots $$ is:
  • $$3n^2(2n^2-1)$$
  • $$n^2(2n^2-1)$$
  • $$n^2(n^2-1)$$
  • $$n^2(2n^2-5)$$
If $$f(x)=\displaystyle\Pi^3_{i=1}(x-a_i)+\displaystyle\sum^3_{i=1}a_i-3x$$ where $$a_i < a_{i+1}$$ for $$i=1, 2,$$ then $$f(x)=0$$ has:
  • Only one distinct real root
  • Exactly two distinct real root
  • Exactly $$3$$ distinct real roots
  • $$3$$ equal real roots
The sum of $$ \dfrac { 7 }{ 2\times 3 } \left( \dfrac { 1 }{ 3 }  \right) +\dfrac { 9 }{ 3\times 4 } { \left( \dfrac { { 1 } }{ 3 }  \right)  }^{ 2 }+\dfrac { 11 }{ 4\times 5 } { \left( \dfrac { 1 }{ 3 }  \right)  }^{ 3 }+$$ upto 10 terms is equal  to
  • $$ \dfrac { 1 }{ 2 } -\dfrac { 1 }{ { 12\times }3^{ 10 } } $$
  • $$ \dfrac { 1 }{ 3 } -\dfrac { 1 }{ { 12\times }3^{ 10 } } $$
  • $$ \dfrac { 1 }{ 2 } -\dfrac { 1 }{ { 10\times }3^{ 10 } } $$
  • $$None\ of\ these$$
The sum of the series 
$$(2)^2+2(4)^{2}+3(6)^{2}+....$$ upto $$10$$ terms is
  • $$12100$$
  • $$11300$$
  • $$11200$$
  • $$12300$$
The value of $$\sum^{10}_{x=1}\sum^{r=x-1}_{r=0}(2^{x}-2^{r})$$ is
  • $$16392$$
  • $$16398$$
  • $$15462$$
  • $$15468$$
Observe the pattern carefully
$$11\times11=121$$
$$111\times111=12321$$
$$1111\times1111=\,?$$
  • $$12345321$$
  • $$123421$$
  • $$1234321$$
  • $$12468$$
Sum of the series 
$$(1\times 2015)+(2 \times 2014)+(3\times 2013)......+(2015\times 1)$$ is equal to-
  • $$336\ \times 2015 \times 2016$$
  • $$336\ \times 2015 \times 2017$$
  • $$336\ \times 2016\times 201$$
  • $$None$$
It is known  that $$\sum\limits_{r = 1}^\infty  {\frac{1}{{{{(2r - 1)}^2}}} = \frac{{{\pi ^2}}}{8},} $$ then $$\sum\limits_{r = 1}^\infty  {\frac{1}{{{r^2}}}} $$ is equal to 
  • $$\frac{{{\pi ^2}}}{{24}}$$
  • $$\frac{{{\pi ^2}}}{{3}}$$
  • $$\frac{{{\pi ^2}}}{{6}}$$
  • none of these
Find the sum of the infinite series $$\frac{1}{9}+\frac{1}{81}+\frac{1}{729}+....\infty$$
  • $$\frac{1}{8}$$
  • $$\frac{1}{4}$$
  • $$\frac{1}{5}$$
  • $$\frac{2}{3}$$
The sum  $$\frac{3}{{1.2}},\frac{3}{{1.2}},\frac{1}{2},\frac{4}{{2.3}}{\left( {\frac{1}{2}} \right)^2} + \frac{5}{{3.4}}{\left( {\frac{1}{2}} \right)^2}$$
  • $$1 - \frac{1}{{\left( {n + 1} \right){2^n}}}$$
  • $$1 - \frac{1}{{n{{.2}^{n - 1}}}}$$
  • $$1 = \frac{1}{{\left( {n + 1} \right){2^n}}}$$
  • $$\frac{1}{{\left( {n - 1} \right){2^{n - 1}}}}$$
If f(x)= $$x+\frac{1}{2x+\frac{1}{\frac{1}{2x+...\infty }}}$$ ; 
then the value of f (2011). f'(2011) is :
  • 0
  • 1
  • 2011
  • 2010
$$1 + \frac { n } { 2 } + \frac { n ( n - 1 ) } { 2.4 } + \frac { n ( n - 1 ) ( n - 2 ) } { 2.4 .6 } + \dots \ldots \ldots \infty =$$ ?
  • $$\left( \frac { 2 } { 3 } \right) ^ { n }$$
  • $$\left( \frac { 3 } { 2 } \right) ^ { n }$$
  • $$\left( \frac { 3 } { 4 } \right) ^ { n }$$
  • $$\left( \frac { 4 } { 3 } \right) ^ { n }$$
Find the missing number, if same rule is followed in all the three figures.
1180459_2279085d5b404909b0098a54eab4e625.JPG
  • 12
  • 16
  • 14
  • 9
Choose the CORRECT options:-

  • $$\sum\limits_{k = 1}^{360} {\left( {\frac{1}{{k\sqrt {k + 1} + \left( {k + 1} \right)\sqrt k }}} \right)} $$ is the ratio of two relative prime positive integers $$m$$ nad $$n$$. The value of $$(m+n)$$ is equal to $$37$$.
  • If $$\frac{{5\pi }}{2} < x < 3\pi $$, then the value of the expression $$\frac{{\sqrt {1 - \sin x} + \sqrt {1 + \sin x} }}{{\sqrt {1 - \sin x} - \sqrt {1 + \sin x} }}$$ is $$ - \tan \frac{x}{2}$$.
  • The exact value of $$\frac{{96\sin 80^\circ \sin 65^\circ \sin 35^\circ }}{{\sin 20^\circ + \sin 50^\circ + \sin 110^\circ }}$$ is equal to $$24$$.
  • The sum $$\sum\limits_{n = 1}^\infty {\left( {\frac{n}{{{n^4} + 4}}} \right)} $$ is equal to $$3/16$$.
$$\sum\limits_{r = 1}^n {{{\sin }^{ - 1}}\left( {\frac{{\sqrt r  - \sqrt {r - 1} }}{{\sqrt {r\left( {r + 1} \right)} }}} \right)} $$ is equal to 
  • $${\tan ^{ - 1}}\left( {\sqrt n } \right) - \frac{\pi }{4}$$
  • $${\tan ^{ - 1}}\left( {\sqrt n + 1} \right) - \frac{\pi }{4}$$
  • $${\tan ^{ - 1}}\left( {\sqrt n } \right)$$
  • $${\tan ^{ - 1}}\left( {\sqrt n + 1} \right)$$
Direction: In a given question, one number is missing in the series. you have to understand the pattern of the series and insert the number 
JN    28   27    GP
CE   12   45    TU
LR     ?     ?     MS
  • $$34, 36$$
  • $$35, 35$$
  • $$30, 32$$
  • $$30, 41$$
Let $$A$$ be the sum of the first $$20$$ terms and $$B$$ be the sum of the first $$40$$ terms of the series $$1$$ + $$2.2^2$$ + $$3^2$$ + $$2.4^2$$ + $$5^2+2.6^2$$ +......... Find the value of $$A$$.
  • $$496$$
  • $$232$$
  • $$248$$
  • $$464$$
If $$1^{2}+2^{2}+3^{2}++(2003) ^{2}=(2003)(4007)(334)$$ and $$(1)(2003)+ (2)(2002)= (2003)(334)(x)$$ then $$x$$ equals
  • $$2005$$
  • $$2004$$
  • $$2003$$
  • $$2001$$
The number of the three digits numbers having only two consecutive digits identical is:
  • 153
  • 160
  • 180
  • 161
The sum of the first n terms of the series $$\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+$$ is equal to:
  • $$2^{-n}- n -1$$
  • $$1 - 2^{-n}$$
  • $$n + 2^{-n} - 1$$
  • $$2^{-n} - 1$$
$$1+(1+2)+(1+2+{2}^{2})+(1+2+{2}^{2}+{2}^{3})=....$$ upto $$n$$ terms $$=$$______
  • $${2}^{n+2}-n-4$$
  • $$2({2}^{n}-1)-n$$
  • $${2}^{n+1}-n$$
  • $${2}^{2n+1}-1$$
If $$24 + 37 = 7,12 + 18 = 3$$, then 54+21=?
  • 11
  • 9
  • 12
  • 3
In the following number series,one number is wrong.find out the ?

1,2,8,33,149,765,4626

  • $$33$$
  • $$8$$
  • $$149$$
  • $$0$$
If $$ '-'$$ denotes $$'\div ',\ '\div '$$ denotes $$'\times ',\ '+'$$ denotes' - 'and $$'\times '$$ denotes $$'+'$$, then find the value of $$116+9\div 52-4\times 5$$.
  • $$16$$
  • $$8$$
  • $$9$$
  • $$4$$
$$1+\dfrac{1}{2}(1+2)+\dfrac{1}{3}(1+2+3)+\dfrac{1}{4}(1+2+3+4)+.....$$ upto $$20$$ terms is
  • $$110$$
  • $$111$$
  • $$115$$
  • $$116$$
Find the missing term in the series given below. $$12,\ 13,\ 18,\ 19,\ 24,\ 25 $$?
  • $$27$$
  • $$30$$
  • $$29$$
  • $$26$$
Let a sequence whose $$n^{th}$$ term is $${a_n}$$ be define as $${a_1} = \frac{1}{2}$$ and $$(n - 1){a_{n - 1}} = (n + 1){a_n}$$ for $$n \ge 2$$ ; then 
  • $${a_n} = \frac{1}{{n(n + 1)}}$$
  • $${S_n} = 1 - \frac{1}{{n + 1}}$$
  • $$\mathop {\lim }\limits_{n \to \infty } {S_n} = 1$$
  • $${S_n} = \frac{1}{{n + 1}}$$
The sum of the series 1.2.3+2.3.4+...3.4.5.+...to n terms is 
  • $$n(n+1)(n+2)$$
  • $$(n+1)(n+2)(n+3)$$
  • $$\frac { 1 }{ 4 } n(n+1)(n+2)(n+3)$$
  • $$\frac { 1 }{ 4 } (n+1)(n+2)(n+3)$$
The sum of first $$9$$ terms of the series
$$\dfrac { { 1 }^{ 3 } }{ 1 } +\dfrac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +\dfrac { { 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 } }{ 1+3+5 } +\dots$$
  • $$96$$
  • $$142$$
  • $$192$$
  • $$71$$
If $$\left( { 1-y } \right) ^{ 35 }\left( 1+y \right) ^{ 45 }={ A }_{ 0 }+{ A }_{ 1 }Y+{ A }_{ 2 }{ Y }^{ 2 }+{ A }_{ 3 }{ Y }^{ 3 }+.....+{ A }_{ 80 }{ Y }^{ 80 },$$ then
  • $$\frac { { A }_{ 2 } }{ A_{ 1 } } <2$$
  • $${ A }_{ 1 }={ A }_{ 2 }$$
  • $$\frac { { A }_{ 2 } }{ { A }_{ 1 } } <1$$
  • $$1<\frac { { A }_{ 2 } }{ { A }_{ 1 } } <2$$
If $$\dfrac{1}{1^{2}}+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+........\infty =\dfrac{\pi^{2}}{6}$$ then $$\dfrac{1}{1^{2}}+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+......\infty$$
  • $$\dfrac{\pi^{2}}{8}$$
  • $$\dfrac{\pi^{2}}{12}$$
  • $$\dfrac{\pi^{2}}{3}$$
  • $$\dfrac{\pi^{2}}{2}$$
Sum of the series 
$$1+2.2+3.2^{2}+...+100.2^{10}=$$
  • $$100.2^{100}+1$$
  • $$99.2^{100}+1$$
  • $$99.2^{100}-1$$
  • $$100.2^{100}-1$$
Sum of series $$\displaystyle \sum^{n}_{r=1}(r^{2}+1)_{r!}$$ is
  • $$(n+1)!$$
  • $$(n+2)!-1$$
  • $$n(n+2)!$$
  • $$(n+1)!-1$$
$$\dfrac {1}{1.4}+\dfrac {1}{4.7}+\dfrac {1}{7.10}+...+\dfrac {1}{(3n-5)(3n-2)}$$
  • $$\dfrac {n-1}{3n+2}$$
  • $$\dfrac {n-1}{3n-2}$$
  • $$\dfrac {n-1}{3n-1}$$
  • $$\dfrac {n-1}{3n-4}$$
If $$x=\sqrt 1+\dfrac{1}{{1}^{2}}+\dfrac{1}{{2}^{2}}+\sqrt 1+\dfrac{1}{{2}^{2}}+\dfrac{1}{{3}^{2}}+\sqrt 1+\dfrac{1}{{2019}^{2}}+\dfrac{1}{{2020}^{2}}$$ then $$4040(2020-x)$$
  • $$1$$
  • $$2$$
  • $$-1$$
  • $$-2$$
The sum of the series $$1+\frac { 1.3 }{ 6 } +\frac { 1.3.5 }{ 6.8 } +...\infty $$ is 
  • $$4$$
  • $$0$$
  • $$\infty $$
  • $${ (4) }^{ \frac { 2 }{ 3 } }$$
The sum of the infinite series $$\cot { ^{ 1 } } \left( \dfrac { 7 }{ 4 } \right) +\cot { ^{ -1 }\left( \dfrac { 19 }{ 4 } \right) } +\cot { ^{ -1 }\left( \dfrac { 39 }{ 4 } \right) } +\cot { ^{ -1 }\left( \dfrac { 67 }{ 4 } \right) } +........\infty $$ is:
  • $$\\ \dfrac { \pi }{ 4 } -\cot { ^{ -1 } } (3)$$
  • $$\\ \dfrac { \pi }{ 4 } -\tan { ^{ -1 } } (3)$$
  • $$\\ \dfrac { \pi }{ 4 } +\cot { ^{ -1 } } (3)$$
  • $$\\ \dfrac { \pi }{ 4 } +\tan { ^{ -1 } } (3)$$
$$40280625, 732375, 16275, 465, 18.6, 1.24,?$$
  • $$0.248$$
  • $$0.336$$
  • $$0.424$$
  • $$0.512$$
  • $$0.639$$
Sum of the series  $$ S = 1^{2}-2^{2}+3^{2}-4^{2}+......-2002^{2}+2003^{2}$$ is
  • 2007006
  • 1005004
  • 2000506
  • none of these
In the sum $$3+33+333+3333+.........2015$$ terms the number formed by taking the last four digits in that order is 
  • $$6365$$
  • $$6255$$
  • $$6465$$
  • $$6565$$
Sum of the series $$\dfrac{1^2}{1!}+\dfrac{2^2}{2!}+\dfrac{3^2}{3!}+\dfrac{4^2}{4!}+.....$$(infinite terms) is?
  • e
  • $$2e$$
  • $$e^2$$
  • $$\infty$$
If $$a_{k}=\dfrac{1}{K(K+1)}$$ for $$K=1, 2, 3,.....n,$$ then $$\left(\sum_{k=1}^{n}{a_{k}}\right)^{2}=$$
  • $$\dfrac{n}{n+1}$$
  • $$\dfrac{n^{2}}{(n+1)^{2}}$$
  • $$\dfrac{n^{4}}{(n+1)^{4}}$$
  • $$\dfrac{n^{6}}{(n+1)^{6}}$$
$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\dfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\frac { { C }_{ 3 } }{ { C }_{ 2 } } +......+\dfrac { n{ C }_{ n } }{ { C }_{ n-1 } } $$ equals 
  • $${ n2 }^{ n-1 }$$
  • $$\dfrac { { n2 }^{ n-1 } }{ { 2 }^{ n }-1 } $$
  • $$\dfrac { n(n+1) }{ 2 } $$
  • $$\dfrac { (n+1)(n+2) }{ 2 } $$
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