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

Let a1,a2,a3,....,a11 be real numbers satisfying a1=15,272a2>0 and ak=2ak1ak2 for k=3,4,.......,11.
 If a21+a22+...+a21111=90, then the value of a1+a2+...+a1111 is equal to
  • 1
  • 5
  • 9
  • 0
If nr=1(r)(r+1)(2r+3)=an4+bn3+cn2+dn+e, then
  • a+c=b+d
  • e=0
  • a,b23,c1 are in A.P.
  • ca is an integers
nk=1(km=1m2)=an4+bn3+cn2+dn+e then 
  • a=112
  • b=116
  • d=16
  • e=0
Evaluate nr=1[rk=1k][log1/2(4xx2)]r. Find x for which summation is a finite number as n
  • x(0,2+152)
  • x(0,2152)
  • x(0,2+152)
  • x(0,2152)
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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