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CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 12 - MCQExams.com

If Cr=(100Cr),thenE=n+4r=0(1)rcrcr+1
  • (100C45)
  • (100C47)
  • (101C50)
  • (100C51)
If the rth term in the expansion of \left(\dfrac{x}{3}-\dfrac{2}{x^{2}}\right)^{10} contains x^{4} then r is equal to
  • 2
  • 1
  • 3
  • 5
If C_{0},C_{1},C_{2},...,C_{n} are the binomial coefficients and n  is odd, then
2C_{1} +2^{3}.C_{5}+...+2^{n}\ ^{n}C_{n} equals
  • \dfrac {3^{n}+(-1)^{n}}{2}
  • \dfrac {3^{n}-(-1)^{n}}{2}
  • \dfrac {3^{n}+1}{2}
  • \dfrac {3^{n}-1}{2}
\begin{array} { l } { \text { Assertion } ( \mathrm { A } ) : \text { The expansion of } ( 1 + x ) ^ { n } = } \\ { C _ { 0 } + C _ { 1 } x + C _ { 2 } x ^ { 2 } + \ldots + C _ { n } x ^ { n } } \\ { \text { Reason (R): If } x = - 1 , \text { then the above expansion is } } \\ { \text { zero } } \end{array}
  • \begin{array} { l } { \text { Both A and R are true and } \mathrm { R } \text { is the correct } } \\ { \text { explanation of } \mathrm { A } } \end{array}
  • \begin{array} { l } { \text { Both A and R are true and R is not the } } \\ { \text { correct explanation of } \mathrm { A } } \end{array}
  • A is true, but R is false
  • A is false, but R is true
If the ratio T_2 : T_3 in the expansion of (a+b)^n and T_3 : T_4 in the expansion of  (a+b)^{n+3} are equal , then n =
  • 3
  • 4
  • 5
  • 6
The coefficient  of {x^5} in the expansion of {\left( {1 + {x^2}} \right)^5}{\left( {1 - x} \right)^4} is 
  • {4.^6}{C_4}
  • {2.^6}{C_4}
  • {2.^6}{C_2}
  • {4.^6}{C_2}
In the binomial expansion of ( a - b ) ^ { n } n > 0 and the sum 5 ^ { t h } and 6 ^ { \text { th } } terms is zero, then \frac { a } { b } equal to 
  • \frac { 5 } { n - 4 }
  • \frac { 6 } { n - 5 }
  • \frac { n - 5 } { 6 }
  • \frac { n - 4 } { 5 }
The co-efficient of x in the expansion of \left( 1 - 2 x ^ { 3 } + 3 x ^ { 5 } \right) \left( 1 + \dfrac { 1 } { x } \right) ^ { 8 } is 
  • 56
  • 65
  • 154
  • 62
If the sum of odd terms and the sum of even terms in (x+a)^{n} are p and q respectively then p^{2}+q^{2}=
  • \dfrac{(x+a)^{2n}-(x-a)^{2n}}{2}
  • (x+a)^{2n}-(x-a)^{2n}
  • \dfrac{(x+a)^{2n}+(x-a)^{2n}}{2}
  • (x+a)^{2n}+(x-a)^{2n}
The sum of the series
\dfrac {2\left(\dfrac {n}{2}\right)!\left(\dfrac {n}{2}\right)!}{(n!)}[C^{2}_{0}-2C^{2}_{1}+3C^{2}_{2}...+(-1)^{n}(n+1)C^{2}_{n}]
where n is an even positive integer, is equal to
  • 0
  • (-1)^{\dfrac {n}{2}}(n+1)
  • (-1)^{\dfrac {n}{2}}(n+2)
  • (-1)^{n}n
Integral part of (8+3\sqrt{7})^{n} is 
  • an even number
  • an odd number
  • an even or odd number depending upon the value of n
  • nothing can be said
If A and b are coefficients of ( 1 + x ) ^ { 2 } and ( 1 + x ) ^ { 2 n - 1 } respectively, then
  • d = B
  • A = 2 B
  • 2 A = B
  • A + B = 0
The total number of terms in the expansion of ( x + a ) ^ { 200 } + ( x - a ) ^ { 200 } after simplification is
  • 101
  • 102
  • 201
  • 202
The sum of the co-efficients of all odd degree terms in the expansion of { \left( x+\sqrt { { x }^{ 3 }-1 }  \right)  }^{ 5 }+{ \left( x-\sqrt { { x }^{ 3 }-1 }  \right)  }^{ 5 }
(x> 1) is:
  • 2
  • -1
  • 0
  • 1
Sum of the coefficient of integral powers of x in {\left(1-2\sqrt{x}\right)}^{50} is 
  • \dfrac{{3}^{50}+1}{2}
  • \dfrac{{3}^{50}}{2}
  • {2}^{49}-1
  • {2}^{49}+1
Find the middle terms(s) in the expansion of \left ( 3x-\dfrac{2}{x^{2}} \right )^{15}.
  • \dfrac {-6435\times 3^7\times 2^7}{x^6}, \dfrac {6437\times 3^7 \times 2^8}{x^9}
  • \dfrac {-6435\times 3^8\times 2^7}{x^6}, \dfrac {6437\times 3^7 \times 2^8}{x^9}
  • \dfrac {-6435\times 3^8\times 2^7}{x^6}, \dfrac {6437\times 3^7 \times 2^7}{x^9}
  • \dfrac {-6435\times 3^8\times 2^7}{x^6}, \dfrac {6437\times 3^8 \times 2^8}{x^9}
Sum of coefficients of { x }^{ 2r }, r= 1,2,3,....... in (1+x{ ) }^{ n } is
  • ({ 2 }^{ n-1 }-1)
  • ({ 2 }^{ n-1 }+1)
  • ({ 2 }^{ n-2 }+1)
  • ({ 2 }^{ n-2 }-1)
The sum of coefficients of integral powers of x in the binomial expansion of (1-2\sqrt x)^{50} is
  • \frac{1}{2}(3^{50}-1)
  • \frac{1}{2}(2^{50}+1)
  • \frac{1}{2}(3^{50}+1)
  • \frac{1}{2}(3^{50})
Coefficient of x^{n} in expansion of \dfrac{(1+2x)^{2}}{(1-x)^{3}} is
  • 2n
  • \dfrac{3}{2}(3n^{2}+n)
  • n^{2}+n-1
  • None\ of\ these
Coefficient of x^{r} in the expansion of (1-2x)^{-1/2} is
  • \dfrac{(2r)!}{(r!)^{2}}
  • \dfrac{(2r)!}{2^{r}(r!)^{2}}
  • \dfrac{(2r)!}{(r!){2^{2r}}}
  • \dfrac{(2r)!}{2^{r}(r+1)!(r+1)}
The coefficient of x^{99} in (x+1)(x+3)(x+5).....(x+199) is
  • 1+2+3+...+99
  • 1+3+5+...+199
  • 1.3.5............199
  • None\ of\ these
The sum of coefficient of integral powers of x in the binomial expansions {\left( {1 - 2\sqrt x } \right)^{50}} is:
  • \frac{1}{2}\left( {{3^{50}} - 1} \right)
  • \frac{1}{2}\left( {{2^{50}} + 1} \right)
  • \frac{1}{2}\left( {{3^{50}} + 1} \right)
  • \frac{1}{2}\left( {{3^{50}}} \right)
The middle term in the expansion of {\left(3x-\dfrac{{x}^{3}}{6}\right)}^{9} is 
  • \dfrac{21}{16}{x}^{19}
  • \dfrac{-21}{16}{x}^{19}
  • \dfrac{21}{16}{x}^{-19}
  • \dfrac{-21}{16}{x}^{-19}
The sum of the co-efficient of all odd degree terms in the expansion of \left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) 
  • 0
  • 1
  • 2
  • - 1
Sum of the series )^{100}C_1)^2 + 2(^{100}C_2)^2 + 3(^{100}C_3)^2+.......+100(^{100}C_{100})^2 equals
  • \frac{2^{99}[1.3.5 ... ... (199)]}{99!}
  • 100. ^{200}C_{100}
  • 50. ^{200}C_{100}
  • 100. ^{199}C_{99}
The sum of the series ^{2020}C_0- ^{2020}C_1+^{2020}C_2-^{2020}C_3+.....+^{2020}C_{1010} is 
  • \dfrac{1}{2}^{2020}C_{1010}
  • ^{2020}C_{1010}
  • Zero
  • \dfrac{-1}{2}^{2020}C_{1010}
The value of \dfrac{1}{12!}+\dfrac{1}{10!2!}+\dfrac{1}{8!4!}+...+\dfrac{1}{12!}
  • \dfrac{{2}^{12}}{12!}
  • \dfrac{{2}^{11}}{12!}
  • \dfrac{{2}^{11}}{11!}
  • None of these
If { \left( 1+x+2{ x }^{ 2 } \right)  }^{ 20 }=0+{ a }_{ 1 }x+{ a }_{ 2 }{ x }^{ 2 }+......+{ a }_{ 40 }{ x }^{ 40 } then { a }_{ 1 }+{ a }_{ 3 }+{ a }_{ 5\quad  }+......+{ a }_{ 37\quad  } equals -
  • { 2 }^{ 19}({ 2 }^{ 20 }-21)
  • { 2 }^{ 20 }({ 2 }^{ 19 }-19)
  • { 2 }^{ 19 }({ 2 }^{ 20 }+21)
  • None of these
The greatest terms of the expansion (2x+5y)^{13} when x=10, y=2 is?
  • ^{13}C_5\cdot 20^8\cdot 10^5
  • ^{13}C_6\cdot 20^7\cdot 10^4
  • ^{13}C_4\cdot 20^9\cdot 10^4
  • None of these
The coefficient of x^9 in (x - 1) (x - 4) (x - 9)........(x - 100) is
  • -235
  • 235
  • 385
  • None of these
Find the value of \dfrac{1}{\left(n-1\right)!}+\dfrac{1}{\left(n-3\right)!3!}+\dfrac{1}{\left(n-5\right)!5!}+...
  • \dfrac{{2}^{n-1}}{(n-1)!}
  • \dfrac{{2}^{n}}{n!}
  • \dfrac{{2}^{n-1}}{n!}
  • None of these
The constant term in the expansion of { (1+x) }^{ n }{ (1+\frac { 1 }{ x } ) }^{ n } is 
  • { C }_{ 0 }^{ 2 }+2{ C }_{ 1 }^{ 2 }+3{ C }_{ 2 }^{ 2 }+......+(n+1){ C }_{ n }^{ 2 }
  • { ({ C }_{ 0 }+{ C }_{ 1 }+.....+{ C }_{ n }) }^{ 2 }
  • { C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+.....+{ C }_{ n }^{ 2 }
  • None of these
The largest coefficient in the expansion of \left(4+3x\right)^{25} is 
  • ^{25}C_{11}3^{25}\left(\dfrac{4}{3}\right)^{14}
  • ^{25}C_{11}4^{25}\left(\dfrac{3}{4}\right)^{11}
  • ^{25}C_{14}4^{14}3^{11}
  • ^{25}C_{14}4^{11}.3^{14}
The sum of the co-efficient of all odd degree terms in the expansion of \left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , ( x > 1 ) is : 
  • - 1
  • 1
  • 0
  • 2
If the last term in the binomial expansion of \left(2^{1/3}-\dfrac {1}{\sqrt {2}}\right)^{n} is \left(\dfrac {1}{3^{5/3}}\right)^{\log_{3}8}, then the 5^{th} terms form the beginning is:
  • 210
  • 420
  • 103
  • None\ of\ these
Coefficient of x^{25} in (1+x+x^{2}+x^{3}+....+x^{10})^{7} is
  • ^{31}C_{15}-7.^{20}C_{14}
  • ^{31}C_{14}-7.^{20}C_{14}
  • 31
  • None\ of\ these
The coefficient of x^{8} in (1+2x^{2}-x^{3})^{9} is 
  • 1680
  • 2140
  • 2520
  • 2730
The coefficients of x^{10} in the expansion of (1+x)^{15}+(1+x)^{16}+(1+x)^{17}+....+(1+x)^{30} is 
  • ^{31}C_{10}-^{15}C_{10}
  • ^{31}C_{11}-^{15}C_{11}
  • ^{30}C_{10}-^{15}C_{10}
  • ^{31}C_{10}-^{14}C_{11}
The sum of the coefficient in the expansion of (a+2b+c)^{11} is-
  • 4^{11}
  • 32
  • 31
  • None\ of\ these
State true or false.
The general term for 3,7,13,21,31,43 ........ is n^2−(n−1),n=1,2,3,...
  • True
  • False
The coefficient of x^{3} in the expansion of (1+2x+3x^{2})^{10} is
  • Less than 200
  • Less than 400 but greater than 200
  • 1400
  • 1500
The coefficient of {x}^{n} in the expansion of \dfrac { 1 }{ \left( 1-x \right) \left( 1-2x \right) \left( 1-3x \right)  } is
  • \dfrac { 1 }{ 2 } \left( { 2 }^{ n+2 }-{ 3 }^{ n+3 }+1 \right)
  • \dfrac { 1 }{ 2 } \left( { 2 }^{ n+2 }-{ 2 }^{ n+3 }+1 \right)
  • \dfrac { 1 }{ 2 } \left( { 2 }^{ n+2 }-{ 3 }^{ n+2 }+1 \right)
  • None\ of\ these
The number of rational terms in the expansion of { \left( 1+\sqrt { 2 } +\sqrt [ 3 ]{ 3 }  \right)  }^{ 6 } is
  • 6
  • 7
  • 5
  • 8
The coefficient of x^4 in the expansion of (1+x+x^2+x^3)^{11} is 
  • 990
  • 495
  • 330
  • none of these
The coefficient of x^{24} in the expansion of 
(1 +3x + 6x^2 + 10x^3+ -----------+\infty)^{2/3} =
  • 300
  • 250
  • 25
  • 205
The co-efficient of x^{k} in expansion of 1+\left(1+x\right)+\left(1+x\right)^{2}++\left(1+x\right)^{n} is : \left(n>k\right)
  • ^{n}C_{k}
  • ^{n+1}C_{k}
  • ^{n+1}C_{k+1}
  • None\ of\ these
The number of terms in the expansion of { \left[ { a }^{ 3 }+\dfrac { 1 }{ { a }^{ 3 } } +1 \right]  }^{ 100 } is
  • 201
  • 300
  • 200
  • ^{ 100 }{ C }_{ 3 }
For x\in R, x\neq -1 if (1+x)^{2016}+x(1+x)^{2015}+x(1+x)^{2014}+.+x^{2016}=\sum _{ i=0 }^{ 2016 }{ { a }_{ i }{ x }^{ i } }, then  a_{17} is equal to 
  • \dfrac{2017!}{ 17!2000!}
  • \dfrac{2016!}{ 17!1999!}
  • \dfrac{2017!}{2000!}
  • \dfrac{2016!}{ 16!}
The middle term in the expansion of \left(1-\dfrac{1}{x}\right)^{n}\left(1-x\right)^{n} is
  • ^{2n}C_{n}
  • -^{2n}C_{n}
  • -^{2n}C_{n-1}
  • none\ of\ these
If the middle term in the expansion of ( 1 + x ) ^ { 2 n } is the greatest term, then x lies in the interval ___________________.
  • \left( \frac { n } { n + 1 } , \frac { n + 1 } { n } \right)
  • \left( \frac { n + 1 } { n } , \frac { n } { n + 1 } \right)
  • ( n - 2 , n )
  • ( n - 1 , n )
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