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CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 10 - MCQExams.com

If the second term of the expansion [a1/13+aa1]n is 14a5/2, then the value of nC3nC2 is .
  • 4
  • 3
  • 12
  • 6
In the expansion of (1+x)n, The binomial coefficients of three consecutive terms are respectively 220, 495 and 795, the value 
  • 10
  • 11
  • 12
  • 13
If the last term in the binomial expansion of (21312)nis(1353)log38 , then the 5th term from the beginning  is
  • 210
  • 420
  • 105
  • none of these
The coefficient of xn in (1x+x22!x33!+...(1)nxnn!)2sequalto
  • (n)nn!
  • (1)n(2)nn!
  • 1(n!)2
  • 1(n!)2
If in the expansion of (1+x)20,the coefficients of rth and (r+4)th terms are equal,then r is equal to
  • 7
  • 8
  • 9
  • 10
if Pn denotes  the product of the binomial coefficients in the expansion of (1+x)n, then Pn+1Pn=?
  • n+1n
  • nnn
  • (n+1)nn!
  • none of these
The term independent of x in the expansion of  (x14)4(x+1x)3
  • 19/16
  • 0
  • 14
  • none of these
If (1+2x+3x2)10=a0+a1x+a2x2+ ...+a20x20, then a1 equals
  • 10
  • 20
  • 210
  • None of these
If the coefficients of (r5)th and (2r1)th terms in the expansion of (1+x)34 are equal, find r.
  • 21
  • 23
  • 14
  • 20
Let 2.^{20}C_0 +^{20} C_1 +^{20}C_2 + ..... +^{20}C_{20}. then sum of this series is 
  • 16. 2^{22}
  • 8. 2^{20}
  • 8. 2^{21}
  • 16. 2^{21}
The greatest term in the expansion of (2x + 3y)^{11} when x = 9 and y = 4 is :
  • (165)(10)^8(12)^3
  • (330) (18)^7 (12)^4
  • 462(18)^8 (12)^5
  • None of these
The coefficient of x^{18} in the product (1+x)(1-x)^{10}(1+x+x^2)^9 is?
  • -84
  • 84
  • 126
  • -126
If p is a real number and if the middle term in the expansion of \left(\dfrac{p}{2}+2\right)^{8} is 1120, find p
  • p=\pm 1
  • p=\pm 3
  • p=\pm 5
  • p=\pm 2
Find the 7^{th} term from the end in the expansion of \left(2x^{2}-\dfrac{3}{2x}\right)^{8}
  • 4033\ x^{9}
  • 4023\ x^{11}
  • 4032\ x^{10}
  • None of these
If the middle terms in the expansion of \left(x^2+\dfrac{1}{x}\right)^{2n} is 184756x^{10}, then what is the value of n
  • 10
  • 8
  • 5
  • 4
Find the coefficient of x^{10} in the expansion of \left(2x^{2}-\dfrac{1}{x}\right)^{20}
  • ^{20}C_{8}. 2^{8}
  • ^{20}C_{10}. 2^{10}
  • ^{20}C_{11}. 2^{11}
  • None of these
The coefficient of x^4 in the expansion of (1-2x)^5 is equal to  
  • 40
  • 320
  • -320
  • 80
Sum of last 30 coefficents in the binomial expansion of (1 + x)^{59} is 
  • 2^{29}
  • 2^{59}
  • 2^{58}
  • 2^{59} - 2^{29}
  • 2^{60}
Find the coefficient of x^{-15} in the expansion of \left(3x^{2}-\dfrac{a}{3x^{3}}\right)^{10}
  • -\dfrac {40}{21}a^6
  • -\dfrac {30}{23}a^8
  • -\dfrac {40}{27}a^7
  • None of these
If the sum of the coefficients in the expansions of (a^2 x^2 - 2ax + 1)^{51} is zero, then a is equal to 
  • 0
  • 1
  • -1
  • -2
  • 2
\dfrac{1}{9!} + \dfrac{1}{3! 7!} + \dfrac{1}{5! 5!} + \dfrac{1}{7! 3!} + \dfrac{1}{9!} is equal to 
  • \dfrac{2^9}{10!}
  • \dfrac{2^{10}}{8!}
  • \dfrac{2^{11}}{9!}
  • \dfrac{2^{10}}{7!}
  • \dfrac{2^{8}}{9!}
The arithmetic mean of ^nC_0 , \ ^nC_1, \ ^nC_2 ...., \ ^nC_n is 
  • \dfrac{2^n}{n + 1}
  • \dfrac{2^n}{n}
  • \dfrac{2^{n -1}}{n + 1}
  • \dfrac{2^{n - 1}}{n}
  • \dfrac{2^{n + 1}}{n}
The largest term in the expansion of (2 + 3x)^{25} where x = 2 is its
  • 13th term
  • 19th term
  • 20th term
  • 26th term
In the expansion of (1 + x)^{43}, the coefficients of the (2r+1)th and the (r + 2)th terms are equal, then the value of r, is
  • 14
  • 15
  • 16
  • 17
The total number of terms in the expansion of (x + a)^{100} + (x - a)^{100} after simplification,
  • 50
  • 51
  • 154
  • 202
The largest term in the expansion of \left(\frac{b}{2} + \frac{b}{2}\right)^{100} is 
  • b^{100}
  • \left(\frac{b}{2}\right)^{100}
  • ^{100}C_{50} \left(\frac{b}{2}\right)^{100}
  • None of these
Which of the following expansion will have term containing x^2
  • \displaystyle(x^{-1/5} + 2x^{3/5})^{25}
  • \displaystyle(x^{3/5} + 2x^{-1/5})^{24}
  • \displaystyle (x^{3/5} - 2x^{-1/5})^{23}
  • \displaystyle (x^{3/5} + 2x^{-1/5})^{22}
If \displaystyle ( 1+ 2x + 3x^2)^{10} = a_0 + a_1x + a_2x^2 + \dots + a_{20}x^{20} then a_1equals
  • 10
  • 20
  • 210
  • 420
The coefficient if x^6 in the expansion of  ( 3x^2 - \frac {1}{3x})^9 is
  • 378
  • 756
  • 189
  • 567
If the second term in the expansion \displaystyle \lgroup ^{13}\sqrt{a} + \frac{a}{\sqrt{a^{-1}}} \rgroup^n is \displaystyle 14a^{5/2}, then the value of \displaystyle ^nC_3/^nC_2 is
  • 4
  • 3
  • 12
  • 6
If the fourth term of \displaystyle \left(\sqrt{_x\left(\frac{1}{1 + \log_{10} x}\right)} + ^{12}\sqrt{x} \right)^6 is equal to 200 and x > 1, then x is equal to 
  • \displaystyle 10\sqrt{2}
  • \displaystyle 10
  • \displaystyle 10^4
  • None of these
The sum of rational term in (\sqrt 2+\sqrt [3]{3}+\sqrt[6]{5})^{10} is equal to 
  • 12632
  • 1260
  • 1236
  • none of these
The number of distinct terms in the expansion of \left( x+\dfrac{1}{x}+x^2+\dfrac{1}{x^2}\right)^{15} is/are ( with respect to different power of x )
  • 255
  • 61
  • 127
  • none of these
The general term in the expansion of (x + a)^n
  • \,^nC_r\,x^{n-r} .a^r
  • \,^nC_r\,x^{r} .a^r
  • \,^nC_{n-r}\,x^{n-r} .a^r
  • \,^nC_{n-r}\,x^{r} .a^{n-r}
The 7th term in the expansion of \bigg(\dfrac{1}{2}+a\bigg)^8 is :
  • \,^8C_7\,\bigg(\dfrac{1}{2}\bigg)(a)^7
  • \,^8C_7\,\bigg(\dfrac{1}{2}\bigg)^7.a
  • \,^8C_6\,\bigg(\dfrac{1}{2}\bigg)^2(a)^6
  • \,^8C_6\,\bigg(\dfrac{1}{2}\bigg)^6(a)^2
Find the middle term of the expansion of \left ( 3x+\frac{1}{2x} \right )^7
  • ^{7}C_{4}\dfrac{2^5 }{16 x^2}
  • ^{7}C_{3}\dfrac{2^7 }{16 x}
  • ^{7}C_{4}\dfrac{2^7 }{16 x}
  • ^{-7}C_{3}\dfrac{2^7 }{16 x}
The sum of the coefficients of even powers of x in the expansion of
(1+x+x^2+x^3)^5 is
  • 512
  • 256
  • 128
  • 64
The coefficient of {x}^{5} in the expansion of (1+x)^{21}+(1+x)^{22}+\ldots\ldots..+(1+x)^{30} is 
  • ^{51}{{C}_{5}}
  • ^{9}{{C}_{5}}
  • ^{31}{{C}_{6}-}^{21}{{C}_{6}}
  • ^{30}{{C}_{5}+}^{20}{{C}_{5}}
The coefficient of middle term in the expansion of (1+{x})^{40} is
  • \displaystyle \frac{1.3.5\cdots 39}{20!}.2^{20}
  • \displaystyle \frac{1.3.5\cdots 39}{20!}
  • {\dfrac{40}{20}!}
  • 40! 20^{20}
If S be the sum of the coefficients in the expansion of (ax+by+-cz)^{n} where a, b, c are lengths of the sides of a triangle, then \lim_{n\rightarrow \infty }\dfrac{S}{(S^{1/n}+1)^{n}} is
  • 1
  • 0
  • e^{\left(\dfrac{ab}{c}\right)}
  • e^{\displaystyle \left(\frac{a+b+c}{a+b+c-1}\right)}
Match the elements of List I with List II

 List I List II
A) lf \lambda be the number of terms which are integers, in the expansion of
(5^{\frac16}+7^{\frac19})^{1824}, then \lambda is divisible by
P) 2
B) lf \lambda be the number of terms which are rational in the expansion of
(5^{\frac16}+2^{\frac18})^{100}, then 
\lambda is divisible by
Q) 3
C) lf \lambda be the number of terms which are irrational in the expansion of
(3^{\frac14}+4^{\frac13})^{99}, then 
\lambda is divisible by
R) 7
 S) 13 
 T) 17
The correct option which matches all the elements correctly, is :
  • (A) - P,Q,T (B) - P (C) - R,S
  • (A) - P,T (B) - Q (C) - S,T
  • (A) - P,S,T (B) - S,Q (C) - Q
  • (A) - P,S (B) - P,A (C) - P,R
Arrange the values of n in ascending order
A : If the term independent of x in the expansion of \left(\displaystyle \sqrt{x}-\frac{n}{x^{2}}\right)^{10} is 405
B : If the fourth term in the expansion of \left(\displaystyle \frac{1}{n}+n^{\log_{n}10}\right)^{5} is 1000( n< 10 )
C : In the  binomial expansion of (1+x)^{n} the coefficients of  5^{\mathrm{t}\mathrm{h}},\ 6^{\mathrm{t}\mathrm{h}} and 7^{\mathrm{t}\mathrm{h}} terms are in A.P.

  • A,B,C
  • B,A,C
  • A,C,B
  • C,A,B
The value of the expression \displaystyle \frac{C_1}{C_0}+2\frac{C_2}{C_1}+3\frac{C_3}{C_2}+\ldots\ldots\ldots +n\frac{C_n}{C_{n-1}} is
  • \dfrac{(n+1)(n+2)}{2}
  • \dfrac{n(n+1)}{2}
  • \dfrac{n(n-1)}{2}
  • \dfrac{n(n+2)}{2}
The coefficient of x^r[0 \le r \le n-1] in the expression of (x + 2)^{n-1} + (x+2)^{n-2} .(x+1) + (x+2)^{n-3}. (x+1)^2+...+(x+1)^{n-1} is
  • ^nC_r(2^r - 1)
  • ^nC_r(2^{n-r} - 1)
  • ^nC_r(2^r + 1)
  • ^nC_r(2^{n-r} + 1)
\cfrac { { C }_{ 0 } }{ x } -\cfrac { { C }_{ 1 } }{ x+1 } +\cfrac { { C }_{ 2 } }{ x+2 } -......+{ \left( -1 \right)  }^{ n }\cfrac { { C }_{ n } }{ x+n } =_______ where { C }_{ r } stands for { _{  }^{ n }{ C } }_{ r }.
  • \cfrac { n! }{ (x+1)...(x+n) }
  • \cfrac { n! }{ x(x+1)...(x+n-1) }
  • \cfrac { n! }{ x(x+1)...(x+n) }
  • \cfrac { n-1! }{ x(x+1)...(x+n) }
If { \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }{x}+{ C }_{ 2 }{ x }^{ 2 }+\cdot \cdot \cdot \cdot \cdot +{ C }_{ n }{ x }^{ n }, then 
  • \cfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\cfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\cfrac { { C }_{ 3 } }{ { C }_{ 2 } } +\cdot \cdot \cdot \cdot +n\cfrac { { C }_{ n } }{ { C }_{ n-1 } } =\cfrac { n(n+1) }{ 2 }
  • { C }_{ n-1 } = n
  • \cfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\cfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\cfrac { { C }_{ 3 } }{ { C }_{ 2 } } +\cdot \cdot \cdot \cdot +n\cfrac { { C }_{ n } }{ { C }_{ n-1 } } =\cfrac { n(n-1) }{ 2 }
  • None of the above
If the (n+1) numbers a,b,c,d,... be all different and each of them a prime number, then the number of different factors (other than 1) of a^m.b.c.d.... is
  • m-2^n
  • (m+1)2^n
  • (m+1)2^n-1
  • None of these
Find the sum of the series
3.{ _{  }^{ n }{ C } }_{ 0 }-8.{ _{  }^{ n }{ C } }_{ 1 }+13._{  }^{ n }{ { C }_{ 2 } }-18.{ _{  }^{ n }{ { C }_{ 3 } } }+\ldots+(n+1)\quad terms
  • 0
  • -1
  • +1
  • None of these
if the coefficient of the middle term in the expansion of\displaystyle (1+x)^{2n+2}and p and the coefficients of middle terms in the expansion of\left ( 1+x \right )^{2n+1}are q and r,then
  • \displaystyle p+q=r
  • \displaystyle p+r=q
  • \displaystyle p=q+r
  • \displaystyle p+q+r=0
In the expansion of (5^{\tfrac 12} + 2^{\tfrac 18})^{1024}, the number of integral terms is
  • 128
  • 129
  • 130
  • 131
0:0:2


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