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

Let $$\displaystyle a_{1},a_{2},a_{3},....,a_{11}$$ be real numbers satisfying $$\displaystyle a_{1}=15,27-2a_{2}> 0\:$$ and $$\: \: a_{k}=2a_{k-1}-a_{k-2}$$ for $$k = 3, 4, ......., 11$$.
 If $$\displaystyle \frac{a_{1}^{2}+a_{2}^{2}+...+a_{11}^{2}}{11}=90 $$, then the value of $$\displaystyle \frac{a_{1}+a_{2}+...+a_{11}}{11} $$ is equal to
  • 1
  • 5
  • 9
  • 0
If $$\sum_{r=1}^{n}(r)(r+1)(2r+3)=an^4+bn^3+cn^2+dn+e$$, then
  • $$a+c=b+d$$
  • $$e=0$$
  • $$a, b-\dfrac23, c-1$$ are in A.P.
  • $$\dfrac{c}a$$ is an integers
$$\sum_{k=1}^{n}\left ( \sum_{m=1}^{k} m^{2} \right )=an^{4}+bn^{3}+cn^{2}+dn+e$$ then 
  • $$a=\dfrac{1}{12}$$
  • $$b=\dfrac{1}{16}$$
  • $$d=\dfrac{1}{6}$$
  • $$e=0$$
Evaluate $$\sum_{r=1}^{n}\left [ \sum_{k=1}^{r}k \right ] \left [ \log_{1/2}\sqrt{(4x-x^{2})} \right ]^{r}$$. Find $$x$$ for which summation is a finite number as $$n\rightarrow \infty $$
  • $$x\in \left ( 0,2+\displaystyle \frac{\sqrt{15}}{2} \right )$$
  • $$x\in \left ( 0,2-\displaystyle \frac{\sqrt{15}}{2} \right )$$
  • $$x\in \left ( 0,-2+\displaystyle \frac{\sqrt{15}}{2} \right )$$
  • $$x\in \left ( 0,-2-\displaystyle \frac{\sqrt{15}}{2} \right )$$
It is known that $$\sum_{r=1}^{\infty }\frac{1}{\left ( 2r-1 \right )^{2}}=\frac{\pi ^{2}}{8}$$.  Then $$\sum_{r=1}^{\infty }\frac{1}{r^{2}}$$ is equal to
  • $$\dfrac{\pi ^{2}}{24}$$
  • $$\dfrac{\pi ^{2}}{3}$$
  • $$\dfrac{\pi ^{2}}{6}$$
  • none of these
if $$\displaystyle \frac{1}{1^2}\, +\, \displaystyle \frac{1}{2^2}\, +\, \displaystyle \frac{1}{3^2}$$ + .......... upto $$\infty\, =\, \displaystyle \frac{\pi^2}{6}$$, then $$\displaystyle \frac{1}{1^2}\, +\, \displaystyle \frac{1}{3^2}\, +\, \displaystyle \frac{1}{5^2}$$ + ........... = .......... .
  • $$\pi^2 / 8$$
  • $$\pi^2 / 12$$
  • $$\pi^2 / 3$$
  • $$\pi^2 / 9$$
Let $$\displaystyle \left \{ a_{n} \right \}\: and\: \left \{ b_{n} \right \}$$ are two sequences given by $$\displaystyle a_{n}=\left ( x \right )^{1/2^{n}}+\left ( y \right )^{1/2^{n}}\: \: and\: \: b_{n}=\left ( x \right )^{1/2^{n}}-\left ( y \right )^{1/2^{n}}$$ for all n $$\displaystyle \epsilon $$ N. The value of $$\displaystyle a_{1}a_{2}a_{3}...a_{n}$$ is equal to
  • x - y
  • $$\displaystyle \frac{x+y}{b_{n}}$$
  • $$\displaystyle \frac{x-y}{b_{n}}$$
  • $$\displaystyle \frac{xy}{b_{n}}$$
If a number sequence begins $$1, 3, 4, 6, 7, 9, 10, 12 . . .$$, which of the following numbers does NOT appear in the sequence?
  • $$34$$
  • $$43$$
  • $$57$$
  • $$65$$
  • $$72$$
Sum of the series $$\displaystyle \sum_{r=1}^{88}\left ( -1 \right )^{r+1}\frac{1}{\sin ^{2}\left ( r+1 \right )^{\circ}-\sin ^{2}1^{\circ}}$$ is equal to
  • $$\displaystyle \frac{\cot 2^{\circ}}{\sin 2^{\circ}}$$
  • $$\displaystyle \frac{-\cot 2^{\circ}}{\sin 2^{\circ}}$$
  • $$\displaystyle \cot 2^{\circ}$$
  • $$\displaystyle \frac{\cot 2^{\circ}}{\sin ^{2}2^{\circ}}$$
The expression
$$\displaystyle \frac {2^2 + 1} {2^2 - 1} + \frac {3^2 + 1} {3^2 - 1} + \frac {4^2 + 1} {4^2 - 1} + ........... + \frac {(2011)^2 + 1} {(2011)^2 - 1} $$
lies in the interval
  • $$\displaystyle (2011, 2010 \frac {1} {2} )$$
  • $$\displaystyle \left( 2011 - \frac {1} {2011} , 2011 - \frac {1} {2012} \right)$$
  • $$\displaystyle (2011, 2011 \frac {1} {2} )$$
  • $$\displaystyle (2012, 2012 \frac {1} {2} )$$
Numbers can be classified into two categories,depending on their divisible conditions.
They are (i) Even numbers $$(2p) \vee p \epsilon N$$ (ii) odd numbers $$(2p + 1) \vee p \epsilon N$$
a.    $$a_1, a_2 ...... a_{2013}$$ are integers, not necessarily distinct.
$$x = (-1)^{a_1}+(-1)^{a_2}+.....+(-1)^{a_{1006}}$$
$$y = (-1)^{a_{1007}}+(-1)^{a_{1008}}+......+(-1)^{a_{2013}}$$

Then which of the following is true?
  • $$(-1)^x = 1; (-1)^y = 1$$
  • $$(-1)^x = 1; (-1)^y = -1$$
  • $$(-1)^x = -1; (-1)^y = 1$$
  • $$(-1)^x = -1; (-1)^y = -1$$
If j, k, and n are consecutive integers such that $$0 < j < k < n$$ and the units (ones) digit of the product jn is 9, what is the units digit of k ? 
  • 0
  • 1
  • 2
  • 3
  • 4
$$\displaystyle\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\cdots\cdots+\frac{1}{n(n+1)}=$$
  • $$\displaystyle\frac{1}{n+1}$$
  • $$\displaystyle\frac{n+2}{n(n+1)}$$
  • $$\displaystyle\frac{n+3}{n(n+1)}$$
  • $$\displaystyle\frac{n}{n+1}$$
It is given that $$\sum_{r = 1}^{\infty} \dfrac{1}{(2 r - 1)^2} = \dfrac{\pi^2}{8}$$, then $$\sum_{r = 1}^{\infty} \dfrac{1}{r^2}$$ is equal to
  • $$\dfrac{\pi^2}{24}$$
  • $$\dfrac{\pi^2}{3}$$
  • $$\dfrac{\pi^2}{6}$$
  • None of these
When $$x< 1$$, find the sum of the infinite series
$$\cfrac { 1 }{ \left( 1-x \right) \left( 1-{ x }^{ 3 } \right)  } +\cfrac { { x }^{ 2 } }{ \left( 1-{ x }^{ 3 } \right) \left( 1-{ x }^{ 5 } \right)  } +\cfrac { { x }^{ 4 } }{ \left( 1-{ x }^{ 5 } \right) \left( 1-{ x }^{ 7 } \right)  } +.....\quad $$
  • $$\cfrac { 1 }{ x(1-x^2)\left( 1-{ x }^{ 3 } \right)  } $$
  • $$\cfrac { 1 }{ (1-x)\left( 1-{ x }^{ 2 } \right)  } $$
  • $$\cfrac { 1 }{ x(1-x)\left( 1-{ x }^{ 2 } \right)  } $$
  • $$\cfrac { x }{ x^2(1-x)\left( 1-{ x }^{ 3 } \right)  } $$
The sum of the series $$\sum _{ n=1 }^{ \infty  }{ \sin { \left( \cfrac { n!\pi  }{ 720 }  \right)  }  } $$ is
  • $$\sin { \left( \cfrac { \pi }{ 180 } \right) } +\sin { \left( \cfrac { \pi }{ 360 } \right) } +\sin { \left( \cfrac { \pi }{ 540 } \right) } $$
  • $$\sin { \left( \cfrac { \pi }{ 6 } \right) } +\sin { \left( \cfrac { \pi }{ 30 } \right) } +\sin { \left( \cfrac { \pi }{ 120 } \right) } +\sin { \left( \cfrac { \pi }{ 360 } \right) } $$
  • $$\sin { \left( \cfrac { \pi }{ 6 } \right) } +\sin { \left( \cfrac { \pi }{ 30 } \right) } +\sin { \left( \cfrac { \pi }{ 120 } \right) } +\sin { \left( \cfrac { \pi }{ 360 } \right) } +\sin { \left( \cfrac { \pi }{ 720 } \right) } $$
  • $$\sin { \left( \cfrac { \pi }{ 180 } \right) } +\sin { \left( \cfrac { \pi }{ 360 } \right) } $$
The value of $$1000\left[\dfrac {1}{1\times 2}+\dfrac {1}{2\times 3}+\dfrac {1}{3\times 4}+...+\dfrac {1}{999\times 1000}\right]$$ is equal to 
  • $$1000$$
  • $$999$$
  • $$1001$$
  • $$\dfrac{1}{999}$$
Let $$\{a_n\}$$ be a sequence of numbers satisfying the relation $$(3-a_{n+1})(6+a_n)=18$$ for all $$n\ge 0$$ and $$a_0=3$$. Then
$$\displaystyle \underset{n\rightarrow \infty}{lim}\dfrac{1}{2^{n+2}}\sum_{j=0}^{n}\dfrac{1}{a_j}$$
  • $$\dfrac{1}{18}$$
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{3}$$
Let $$A=16{-4}+2^{-4}+3^{-4}+4^{-4}+...$$ and $$B=1^{-4}+3^{-4}+5^{-4}+7^{-4}+...$$. The ratio $$\dfrac{A}{B}$$ in the lowest form is
  • $$\dfrac{16}{15}$$
  • $$\dfrac{15}{14}$$
  • $$\dfrac{15}{16}$$
  • $$\dfrac{13}{12}$$
The variance of the series
$$a,a+d,a+2d,.....a+(2n-1)d,a+2nd$$ is
  • $$\cfrac { n(n+1) }{ 2 } { d }^{ 2 }$$
  • $$\cfrac { n(n-1) }{ 6 } { d }^{ 2 }$$
  • $$\cfrac { n(n+1) }{ 6 } { d }^{ 2 }$$
  • $$\cfrac { n(n+1) }{ 3 } { d }^{ 2 }$$
Which of the following options will continue the given series?
$$29$$, $$34$$, $$32$$, $$37$$, $$35$$, ?
  • $$36$$
  • $$39$$
  • $$40$$
  • $$42$$
The value of $$S=\sqrt { 1+\cfrac { 1 }{ { 1 }^{ 2 } } +\cfrac { 1 }{ { 2 }^{ 2 } }  } +\sqrt { 1++\cfrac { 1 }{ { 2 }^{ 2 } } +\cfrac { 1 }{ { 3 }^{ 2 } }  } +...+\sqrt { 1++\cfrac { 1 }{ { (2014) }^{ 2 } } +\cfrac { 1 }{ { (2015) }^{ 2 } }  } $$ is
  • $$2015$$
  • $$2015-\cfrac { 1 }{ 2015 } $$
  • $$2016-\cfrac { 1 }{ 2016 } $$
  • $$2014-\cfrac { 1 }{ 2014 } $$
If $$ s_n = \displaystyle \sum_{r=0}^n \dfrac {1}{^nC_r}$$ and $$ t_n = \displaystyle \sum_{r=0}^n \dfrac {r}{^nC_r} , $$ then $$ \dfrac {t_n}{s_n} $$ is equal to - 
  • $$ \dfrac {n}{2} $$
  • $$ \dfrac {n}{2} -1 $$
  • $$ n - 1 $$
  • $$\dfrac{2n - 1}{2}$$
Solve the given series:
$$\dfrac {1.2^2+2.3^2+3.4^2+...n(n+1)^2}{1.2+2^2.3+3^2.4+...n^2(n+1)}$$
  • $$(3n+1)(3n+5)$$
  • $$\dfrac {n}{2n+1}$$
  • $$\dfrac {3n+1}{3n+5}$$
  • $$\dfrac {3n+5}{3n+1}$$
If $$\displaystyle \sum _{ r =1 }^{n}{ t_r = \frac{n(n+1)(n+2)(n+3)}{8} }$$, then $$\displaystyle \sum _{r = 1}^{n}{ \frac{1}{t_r}} $$ equals
  • $$\displaystyle - \left( \frac{1}{(n+1)(n+2)} - \frac{1}{2} \right) $$
  • $$\displaystyle \left( \frac{1}{(n+1)(n+2)} - \frac{1}{2} \right) $$
  • $$\displaystyle \left( \frac{1}{(n+1)(n+2)} + \frac{1}{2} \right) $$
  • $$\displaystyle \left( \frac{1}{(n-1)(n-2)} + \frac{1}{2} \right) $$
The value of $$\sum _{ n=1 }^{ 9999 }{ \cfrac { 1 }{ \left( \sqrt { n } +\sqrt { n+1 }  \right) \left( \sqrt [ 4 ]{ n } +\sqrt [ 4 ]{ n+1 }  \right)  }  } $$ is
  • $$9$$
  • $$99$$
  • $$999$$
  • $$9999$$
Let $$(1 + x)^{m} = C_{0} + C_{1}x + C_{2}x^{2} + C_{3}x^{3} + .... + C_{m}x^{m}$$, where $$C_{r} = ^{m}C_{r}$$ and $$A = C_{1}C_{3} + C_{2}C_{4} + C_{3}C_{5} + C_{4}C_{6} + .... + C_{m - 2}C_{m}$$, then
  • $$A\geq ^{2m}C_{m - 2}$$
  • $$A < ^{2m}C_{m - 2}$$
  • $$A < C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + .... C_{m}^{2}$$
  • $$A > C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + ....C_{m}^{2}$$
In a certain code language, DIPLOMA is written as FERHQIC, then what is the code for PENCILS in the language? 
  • RAPYHKU
  • RAPYKHU
  • RPAYKHU
  • RAPKYHU
If $$x\in R$$ and $$S=1-{ C }_{ 1 }\cfrac { 1+x }{ { \left( 1+nx \right)  }^{  } } +{ C }_{ 2 }\cfrac { 1+2x }{ { \left( 1+nx \right)  }^{ 2 } } -{ C }_{ 3 }\cfrac { 1+3x }{ { \left( 1+nx \right)  }^{ 3 } } +...upto\quad (n+1)$$ terms, then $$S$$
  • equal $${x}^{2}$$
  • equals $$1$$
  • equals $$0$$
  • is independent
The sum to infinite of the series
$$S=1+\cfrac { 2 }{ 3 } +\cfrac { 6 }{ { 3 }^{ 2 } } +\cfrac { 6 }{ { 3 }^{ 3 } } +\cfrac { 6 }{ { 3 }^{ 4 } } +.....\quad $$ is
  • $$4$$
  • $$3$$
  • $$2$$
  • $$6$$
The $$(n+1)^{th} $$ term from the end in $$(x - \frac{1}{x})^{3n}$$ is 
  • $$3nc_n.X^{-n}$$
  • $$(-1)^n. 3nc_n .X^{-n}$$
  • $$3nc_nX^n$$
  • $$(-1)^n. 3nc_n.X^n$$
If sum of the series $$\displaystyle \sum_{n = 0}^{\infty} r^{n} = S$$, for $$|r| < 1$$, then sum of the series $$\displaystyle \sum_{n = 0}^{\infty} r^{2n}$$
  • $$S^{2}$$
  • $$\dfrac {S^{2}}{2S + 1}$$
  • $$\dfrac {S^{2}}{S^{2} - 1}$$
  • None of these
The value of 
$$\displaystyle \sum _{ n=0 }^{ 1947 }{ \cfrac { 1 }{ { 2 }^{ n }+\sqrt { { 2 }^{ 1947 } }  }  } $$ is equal to
  • $$\cfrac { 487 }{ \sqrt { { 2 }^{ 1945 } } } $$
  • $$\cfrac { 1946 }{ \sqrt { { 2 }^{ 1947 } } } $$
  • $$\cfrac { 1947 }{ \sqrt { { 2 }^{ 1947 } } } $$
  • $$\cfrac { 1948 }{ \sqrt { { 2 }^{ 1947 } } } $$
$$\displaystyle \sum_{r = 0}^{n}{t^3 \left( \frac{^nC_r}{^nC_{r-1}} \right)^2 }$$ is equal to
  • $$\displaystyle \frac{n(n+1)(n+2)^2}{12} $$
  • $$\displaystyle \frac{n(n+1)^2 (n+2)}{12} $$
  • $$\displaystyle \frac{n(n+1)(n+2)}{12} $$
  • None of these
If $$x=\dfrac{1}{5}+\dfrac{1.3}{5.10}+\dfrac{1.3.5}{5.10.15}+.....\infty$$ then $$3x^2+6x=$$
  • 1
  • 2
  • 3
  • 4
The value of $$\displaystyle\sum^{n}_{i=1}\sum^{i}_{j=1}\sum^j_{k=1}1=220$$, then the value of n equals.
  • $$11$$
  • $$12$$
  • $$10$$
  • $$9$$
Let $${ T }_{ r }$$ and $${ S }_{ r }$$ be the $${ r }^{ th }$$ term and sum up to $${ r }^{ th }$$ term of a series respectively. If for an odd natural number $$n,{ S }_{ n }=n$$ and $${ T }_{ n }=\dfrac { { T }_{ n-1 } }{ { n }^{ 2 } }$$, then $${ T }_{ m }$$ ($$m$$ being even) is:
  • $$\dfrac { 2 }{ 1+{ m }^{ 2 } }$$
  • $$\dfrac { 2{ m }^{ 2 } }{ 1+{ m }^{ 2 } }$$
  • $$\dfrac { { \left( m+1 \right) }^{ 2 } }{ 2+{ \left( m+1 \right) }^{ 2 } }$$
  • $$\dfrac { 2{ \left( m+1 \right) }^{ 2 } }{ 1+{ \left( m+1 \right) }^{ 2 } }$$
Sum of the series $$\displaystyle\sum^n_{r=1}(r^2+1)r!$$ is?
  • $$(n+1)!$$
  • $$(n+2)!-1$$
  • $$n-(n+1)!$$
  • $$n-(n+2)!$$
If $$a_{1}, a_{2}, ......., a_n(n > 3)$$ are all unequal positive real numbers, and 

$$E = \dfrac{(1 + a_{1} + a_{1}^{2})(1 + a_{2} + a_{2}^{2})......(1 + a_{n} + a_{n}^{2})}{a_{1}, a_{2}, ......., a_{n}}$$
 then which of the following best describes E?
  • $$E \leq 2^{n}$$
  • $$E \geq 3^{n}$$
  • $$E > 3^{n}$$
  • $$E > 2^{n}$$
The sum of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + .......$$ up to low upon to n term is to the 

  • $$n - 1 + {2^{ - n}}$$
  • $${2^{ - n}} - 1$$
  • $$n - 2$$
  • None of these
The sume of the series $$1^3 - 2^3 + 3^3 - ........ + 9^3$$ =
  • 300
  • 125
  • 425
  • 0
The positive integer n for which $$2 \times {2^2} + 3 \times {2^3} + 4 \times {2^4} + ....... + n \times {2^n} = {2^{^{n + 10}}}$$ is______
  • $$510$$
  • $$511$$
  • $$512$$
  • $$513$$
The sum to infinite of the series
$$1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + ........$$
  • 2
  • 3
  • 4
  • 6
If $$ \displaystyle \lim _{ x\rightarrow 0^+ }{ x\left( \left[ \dfrac { 1 }{ x }  \right] +\left[ \dfrac { 5 }{ x }  \right] +\left[ \dfrac { 11 }{ x }  \right] +\left[ \dfrac { 19 }{ x }  \right] +\left[ \dfrac { 29 }{ x }  \right] +.......to\quad n\quad terms \right)  }=430$$ (where [.] denotes the greatest integer function), then $$n=$$
  • $$8$$
  • $$9$$
  • $$10$$
  • $$11$$
If $$\displaystyle\sum _{ n=1 }^{ 2013 }{ \tan { \left( \dfrac { \theta  }{ { 2 }^{ n } }  \right)  }  } \sec { \left( \dfrac { \theta  }{ { 2 }^{ n-1 } }  \right)  } =\left( \dfrac { \theta  }{ { 2 }^{ a } }  \right) -\left( \dfrac { \theta  }{ { 2 }^{ b } }  \right)$$ then $$(b+a)$$ equals 
  • $$2014$$
  • $$2012$$
  • $$2013$$
  • $$2019$$
If $$\sum^5_{n=1}\dfrac{1}{n(n+1)(n+2)(n+3)}=\dfrac{k}{3}$$, then k is equal to?
  • $$\dfrac{55}{336}$$
  • $$\dfrac{17}{105}$$
  • $$\dfrac{19}{112}$$
  • $$\dfrac{1}{6}$$
Sum to $$n$$ terms of the series$$\dfrac { 1 }{ 1.2.3.4 } +\dfrac { 1 }{ 2.3.4.5 } +\dfrac { 1 }{ 3.4.5.6 } +..........$$, is
  • $$\dfrac { { n }^{ 3 } }{ 3\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }$$
  • $$\dfrac { { n }^{ 3 }+{ 6n }^{ 2 }-3n }{ 6\left( n+2 \right) \left( n+3 \right) \left( n+4 \right) }$$
  • $$\dfrac { 15{ n }^{ 2 }+7n }{ 4n\left( n+1 \right) \left( n+5 \right) }$$
  • $$\dfrac { { n }^{ 3 }+6{ n }^{ 2 }+11n }{ 18\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }$$
The sum to infinity of the series:
$$\dfrac {3}{{1}^{3}}+\dfrac {5}{{1}^{3}+{2}^{3}}+\dfrac {7}{{1}^{3}+{2}^{3}+{3}^{3}}+..$$ is-
  • 5
    • 4
  • 6
  • 1
The sum of infinite series $$\dfrac{1.3}{2}+\dfrac{3.5}{2^2}+\dfrac{5.7}{2^3}+\dfrac{7.9}{2^4}+...\infty$$.
  • $$21$$
  • $$22$$
  • $$23$$
  • None
If $$\dfrac{x}{0.2} + \dfrac{x}{0.3} + \dfrac{x}{0.6} + \dfrac{x}{0.4} + \dfrac{x}{0.5} = 87$$, then the value of x is equal to 
  • 0
  • 4
  • 6
  • 1
0:0:1


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