CBSE Questions for Class 11 Engineering Maths Sequences And Series Quiz 8 - MCQExams.com

The value of $$S=\sum _{ k=1 }^{ 6 }{ \left( \sin { \dfrac { 2\pi k }{ 7 }  } -i\cos { \dfrac { 2\pi k }{ 7 }  }  \right)  }$$ ?
  • $$-1$$
  • $$0$$
  • $$-i$$
  • $$i$$

 The value of
$$\frac{1}{1!} + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \frac{1+2+3+4}{4!}+
\cods$$ upto infinite terms is equal to 
  • (a)$$\frac{3e}{2}$$
  • (b)$$3e -1$$
  • (c)$$2e +1$$
  • (d)$$\frac{5e}{2}$$
Find sum $${1}^{2}+{3}^{2}+{5}^{2}+...+{(2n-1)}^{2}$$
  • $$\dfrac{n(2n+1)(2n+1)}{3}$$.
  • $$\dfrac{n(2n-1)(2n+1)}{3}$$.
  • $$\dfrac{2n(2n-1)(2n+1)}{3}$$.
  • None of these
The sum to the first n terms of the series  $$\dfrac{1}{2} + \dfrac{3}{4} + \dfrac{7}{8} + \dfrac{{15}}{{16}} + {\rm{ }} \ldots $$ is $$9 + {2^{ - 10}}$$.  The value of n is
  • $$10$$
  • $$9$$
  • $$8$$
  • none of these
If $$\sum\limits_{i = 1}^{20} {{{\sin }^{ - 1}}{\text{ }}{x_i} = 10\pi } then\;\sum\limits_{i = 1}^{20} {{x_i}} \;$$ is equal to 
  • 20
  • 10
  • 0
  • 1
What is the sum of the series $$\dfrac{1}{{{3^2} - 4}} + \dfrac{1}{{{7^2} - 4}} + \dfrac{1}{{{{11}^2} - 4}} + ..... + \dfrac{1}{{{{39}^2} - 4}}$$
  • $$\dfrac{{10}}{{41}}$$
  • $$\dfrac{{40}}{{41}}$$
  • $$\dfrac{{20}}{{41}}$$
  • $$\dfrac{{30}}{{41}}$$
Number of identical terms in the sequence $$2, 5, 8, 11,$$___ upto $$100$$ terms and $$3, 5, 7, 9, 11$$____ upto $$100$$ terms are
  • $$17$$
  • $$33$$
  • $$50$$
  • $$147$$
The value of the expression $$\left(\dfrac{1}{2^2 -1}\right) + \left(\dfrac{1}{4^2 -1}\right) + \left(\dfrac{1}{6^2 -1}\right) + ... + \left(\dfrac{1}{20^2 -1}\right)$$ is
  • $$\dfrac{9}{19}$$
  • $$\dfrac{10}{19}$$
  • $$\dfrac{10}{21}$$
  • $$\dfrac{11}{21}$$
$$67,84,95,.,133,158$$
  • $$116$$
  • $$129$$
  • $$108$$
  • $$111$$
Let $$S_n=\dfrac{1}{1+2008n}+\dfrac{1}{2+2008n}+...+\dfrac{1}{2009n}$$. Then $$\displaystyle\lim_{n\rightarrow\infty}S_n$$ equals?
  • $$log\left(\dfrac{1}{2008}\right)$$
  • $$log\left(1+\dfrac{1}{2008}\right)$$
  • $$log\left(\dfrac{1}{2009}\right)$$
  • $$log\left(1+\dfrac{1}{2009}\right)$$
Find the value of ?. 
1052324_a17d200592ec4c5eb92acd3b598dee6f.png
  • $$125$$
  • $$25$$
  • $$625$$
  • $$156$$
If $$\frac{a_2a_3}{a_1a_4}=\frac{a_2+a_3}{a_1+a_4}=3\left(\frac{a_2-a_3}{a_1-a_4}\right)$$ then $$a_1, a_2$$ and $$a_3$$ are in which progression?
  • A.P.
  • G.P.
  • H.P.
  • None of these
$$\sum n =55 \implies n=5$$
  • True
  • False
The sum of all $$2$$-digit numbers divisible by $$5$$ is _________?
  • $$1035$$
  • $$1245$$
  • $$1230$$
  • $$945$$
If $$S_r=\left |
\begin{array}{111}
2r & x & n(n+1)\\
6r^2-1 & y & n^2(2n+3)\\
4r^3-2nr & z & n^3(n+1)\\
\end{array}
\right |
$$ then $$\sum_\limits{r=1}^n S_r$$
does not depends on
  • x
  • y
  • n
  • all of these
If $$ (1^2-t_1)+ (2^2-t_2) +.......+(n^2-1)$$, then $$t_n$$ is
  • $$\frac{n}{2}$$
  • $$n -1$$
  • $$n+1$$
  • $$n$$
Insert the missing number in the given series : $$ 0,4,18,48, ?, 180$$ 
  • $$58$$
  • $$68$$
  • $$84$$
  • $$100$$
Ten students of the physics department decided to go on a educational trip.They hired a mini bus for the trip, but the bus can only carry eight students at a time and each student goes at least once. Find the minimum number of trips the bus has to make so that each students can go for equal number of trips.
  • $$5$$
  • $$4$$
  • $$6$$
  • $$8$$
For a sequence $$\left\{ { a }_{ n } \right\} ,{ a }_{ 1 }=2$$ and $$\dfrac { { a }_{ n+1 } }{ { a }_{ n } } =\dfrac { 1 }{ 3 }$$. Then $$\sum _{ r=1 }^{ 20 }{ { a }_{ r } } $$ is
  • $$\dfrac {20}{2}[4+19\times 3]$$
  • $$3(1-\dfrac {1}{3^{20}})$$
  • $$2(1-3^{20})$$
  • $$(1-\dfrac {1}{3^{20}})$$
Number of rectangles in the grid shown which are not squares is?
  • $$160$$
  • $$162$$
  • $$170$$
  • $$185$$
$$1,2,1,4,3,8,9,5,27,16,?,?,?$$
  • $$7,36,64$$
  • $$7,25,64$$
  • $$9,25,64$$
  • $$7,25,49$$
If $$\underset{r = 1}{\overset{n}{\sum}} r (r + 1) (2r + 3) = an^4 + bn^3 + cn^2 + dn + e$$, then 
  • a + c = b + d
  • e = 0
  • $$a, b - \dfrac{2}{3}, \, c - 1$$ are in A. P.
  • $$\dfrac{c}{a}$$ is an integer
$$\displaystyle \sum^{n}_{r=0} (-1)^r \,{^nC_r}. \dfrac{(1 + r \ell n 10)}{(1 + \ell n 10^n)^r} =$$
  • $$0$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$none\ of\ these$$
Evaluate the definite integral:
$$\displaystyle\int_{0}^{\pi/2}\dfrac{\cos^{2}x}{1+3\sin^{2}x}\ dx$$
  • $$\pi /4$$
  • $$\pi /2$$
  • $$\pi /6$$
  • $$\pi /3 $$
 $$\cfrac { 7 }{ 5 } \left( 1+\cfrac { 1 }{ { 10 }^{ 2 } } +\cfrac { 1.3 }{ 1.2 } .\cfrac { 1 }{ { 10 }^{ 4 } } +\cfrac { 1.3.5 }{ 1.2.3 } .\cfrac { 1 }{ { 10 }^{ 6 } } +....\infty \right) =\sqrt { 2 } $$
  • True
  • False
The sum $$2 \times 5 + 5 \times 9 + 8 \times 13 +  \ldots 10$$ term is 
  • $$4500$$
  • $$4555$$
  • $$5454$$
  • None of these
 $$1+\cfrac{1}{4}+\cfrac{1.3}{4.8}+\cfrac{1.3.5}{4.8.12}+.....\infty=\sqrt{2}$$
  • True
  • False
A series is given as: $$4+7+10+13+16+.....$$ Find the sum of the series up to $$10$$ terms.
  • $$110$$
  • $$125$$
  • $$162$$
  • $$175$$
The unit's place digit in $$(1446)^{4n + 3}$$ is
  • $$n$$
  • $$0$$
  • $$6$$
  • None of these
The sum of the infinite series, $$ { 1 }^{ 2 }-\dfrac { { 2 }^{ 2 } }{ 5 } +\dfrac { { 3 }^{ 2 } }{ { 5 }^{ 2 } } -\dfrac { { 4 }^{ 2 } }{ { 5 }^{ 3 } } +\dfrac { { 5 }^{ 2 } }{ { 5 }^{ 4 } } -\dfrac { { 6 }^{ 2 } }{ { 5 }^{ 3 } } + ..........  $$ is
  • $$\dfrac { 1 }{ 2 }$$
  • $$ \dfrac { 25 }{ 24 }$$
  • $$ \dfrac { 25 }{ 54 }$$
  • $$ \dfrac { 125 }{ 252 }$$
The sum of infinity of the series $$\dfrac{1}{1} + \dfrac{1}{1 + 2} + \dfrac{1}{1+2+3}+$$______ is equal to:
  • $$2$$
  • $$\cfrac{5}{2}$$
  • $$3$$
  • None of these
Solve then inequality 
$$\dfrac {x-1}{x}\geq 2$$
  • $$x\leq 1$$
  • $$x\leq -1$$
  • $$x\leq0$$
  • $$x \in R$$
The sum of the series $$1+2.2+3.2^{2}+4.2^{3}+5.2^{4}+.+100.2^{99}$$ is  ?
  • $$99.2^{100} -1$$
  • $$100.2^{100} +1$$
  • $$99.2^[100}$$
  • $$99.2^{100}+1$$
If $${ S }_{ n }=\overset { n }{ \underset { r=1 }{ \Sigma  }  } { t }_{ r }=\dfrac { 1 }{ 6 } n\left( 2{ n }^{ 2 }+9n+13 \right) $$, then $$\overset { n }{ \underset { r=1 }{ \Sigma  }  } \sqrt { { t }_{ r } } $$ equals ?
  • $$\dfrac { 1 }{ 2 } n\left( n+1 \right) $$
  • $$\dfrac { 1 }{ 2 } n\left( n+2 \right) $$
  • $$\dfrac { 1 }{ 2 } n\left( n+3 \right) $$
  • $$\dfrac { 1 }{ 2 } n\left( n+5 \right) $$
If $$a_n=n(n!)$$, then $$\displaystyle\sum^{100}_{r=1} a_r$$ is equal?
  • $$101!$$
  • $$100!-1$$
  • $$101!-1$$
  • $$101!+1$$
If $$a_{1}=a_{2}=2,a_{n}=a_{n-1}-1(n > 2)$$ then $$a_{5}$$ is ?
  • $$1$$
  • $$-1$$
  • $$0`$$
  • $$-2$$
The sum of the series $$1+\dfrac{1}{4\times 2!}+\dfrac{1}{16\times 4!}+\dfrac{1}{64\times 6!}+....\infty$$ is?
  • $$\dfrac{e+1}{\sqrt{e}}$$
  • $$\dfrac{e-1}{\sqrt{e}}$$
  • $$\dfrac{e+1}{2\sqrt{e}}$$
  • $$\dfrac{e-1}{2\sqrt{e}}$$
If $${x}_{1},{x}_{2},. . . . .,{x}_{n}$$ are any real number and $$n$$ is any positive integer, then ?
  • $$n\sum _{ i=1 }^{ n }{ { x }_{ i }^{ 2 }<{ \left( \sum _{ i=1 }^{ n }{ { x }_{ i } } \right) }^{ 2 } }$$
  • $$n\sum _{ i=1 }^{ n }{ { x }_{ i }^{ 2 }\ge { \left( \sum _{ i=1 }^{ n }{ { x }_{ i } } \right) }^{ 2 } }$$
  • $$n\sum _{ i=1 }^{ n }{ { x }_{ i }^{ 2 }\ge n{ \left( \sum _{ i=1 }^{ n }{ { x }_{ i } } \right) }^{ 2 } }$$
  • $$None\ of\ the\ above $$
Sum  of first n terms of the series $$\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{{15}}{{16}} + ....$$ is equal to 
  • $${2^n} - n - 1$$
  • $$1 - {2^{ - n}}$$
  • $$n + {2^{ - 2}} - 1$$
  • none of these
The $$nth$$ term of the series $$4, 14, 30, 52, 80, 114, ...$$ is
  • $$n^{2} + n + 2$$
  • $$3n^{2} + n$$
  • $$3n^{2} - 5n + 2$$
  • $$(n + 1)^{2}$$
The sum of $$(n+1)$$ terms of $$\frac { 1 }{ 1 } +\frac { 1 }{ 1+2 } +\frac { 1 }{ 1+2+3 } +.......is$$
  • $$\frac { n }{ n+1 } $$
  • $$\frac { 2n }{ n+1 } $$
  • $$\frac { 2 }{ n\left( n+1 \right) } $$
  • $$\frac { 2\left( n+1 \right) }{ n+2 } $$
The sum of the series $$\dfrac {5}{1 \cdot 2 \cdot 3}+\dfrac {7}{3 \cdot 4 \cdot 5}+\dfrac {9}{5 \cdot 6 \cdot 7}+....$$ is
  • $$\log \dfrac {8}{e}$$
  • $$\log \dfrac {e}{8}$$
  • $$\log2$$
  • $$\log \dfrac {1}{2}$$
If the sum of the series $$1+\dfrac{3}{x}+\dfrac{9}{x^{2}}+\dfrac{27}{x^{3}}+....$$ to $$\infty$$ is a finite number then 
  • $$x>3$$
  • $$x<-3$$
  • $$x<-3\ or\ x>3$$
  • $$None\ of\ these$$
The sum of the series $$1^{3}+3^{3}+5^{3}+....$$ upto $$20$$ terms is
  • $$319600$$
  • $$321760$$
  • $$306000$$
  • $$347500$$
The value of $$\sum_{n=1}^{\infty}\dfrac{1}{(3n-2)(3n+1)}$$ is equal to $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are relatively prime natural numbers. Then the value of $$(p^2+q^2)$$ is equal to
  • $$4$$
  • $$9$$
  • $$10$$
  • $$13$$
Select the missing number from the given responses ?
1152619_1c6c0bc365ab41c2a693e97a6343ff43.PNG
  • $$15$$
  • $$10$$
  • $$12$$
  • $$17$$
If $$abc=1$$, then the value of $$\dfrac{1}{1+a+b^{-1}}+\dfrac{1}{1+b+c^{-1}}+\dfrac{1}{1+c+a^{-1}}$$ is
  • $$abc$$
  • $$a$$
  • $$1$$
  • $$3$$
The sum $$1 + 3 + 7 + 15 + 31 +  \ldots $$ to $$n$$ term is 
  • $${2^n} - 2 - n$$
  • $${2^{n - 1}} - 1 - n$$
  • $${2^{n + 1}} - 2 - n$$
  • None of these
Find: $$6,25,62,123,(?),341$$
  • $$216$$
  • $$214$$
  • $$215$$
  • $$218$$
Sum of $$n$$ terms of the series $$8+88+888+.$$ equals
  • $$\dfrac {8}{18}[10^{n+1}-9n-10]$$
  • $$\dfrac {8}{18}[10^{n}-9n-10]$$
  • $$\dfrac {8}{18}[10^{n+1}+9n-10]$$
  • $$None\ of\ these$$
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