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

If the second term of the expansion $${ [{ a }^{ { 1 }/{ 13 } }+\frac { a }{ \sqrt { { a }^{ -1 } }  } ] }^{ n }$$ is $$14{ a }^{ { 5 }/{ 2 } }$$, then the value of $$\frac { ^{ n }{ C }_{ 3 } }{^{ n } { C }_{ 2 } } $$ is .
  • 4
  • 3
  • 12
  • 6
In the expansion of $$\left( 1+x \right) ^{ n }$$, 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 $$ (2{  }^{ \frac { 1 }{ 3 }  }-\frac { 1 }{ \sqrt { 2 }  } ){  }^{ n }\quad is\quad (\frac { 1 }{ { 3 }^{ \frac { 5 }{ 3 }  } } ){  }^{ log{  }_{ 3 }8 }$$ , then the 5th term from the beginning  is
  • 210
  • 420
  • 105
  • none of these
The coefficient of $$x^{n}$$ in $$(1-x+\frac { { x }^{ 2 } }{ 2! } -\frac { { x }^{ 3 } }{ 3! } +...-\frac { (-1){  }^{ n }x{  }^{ n } }{ n! } ){  }^{ 2 }s\quad equal\quad to\quad $$
  • $$\frac { (n){ }^{ n } }{ n! } $$
  • $$\frac { (-1)^n(2){ }^{ n } }{ n! } $$
  • $$\frac { 1 }{ (n!){ }^{ 2 } } $$
  • $$-\frac { 1 }{ (n!){ }^{ 2 } } $$
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 $${ P }_{ n }$$ denotes  the product of the binomial coefficients in the expansion of $$(1+x)^{n}$$, then $$\frac { { P }_{ n+1 } }{ { P }_{ n } } =?$$
  • $$\cfrac { n+1 }{ n } $$
  • $$\cfrac { n^{ n } }{ n } $$
  • $$\cfrac {(n+1)^n}{n!}$$
  • none of these
The term independent of x in the expansion of  $$\displaystyle \left ( x -\dfrac{1}{4} \right )^4\left ( x + \dfrac{1}{x} \right )^3$$
  • 19/16
  • 0
  • 14
  • none of these
If $$(1+2x+3x^2 )^{10} = a_0+a_1x+a_2x^2+\ ...\,+a_{20}x^{20}$$, then $$a_1$$ equals
  • $$10$$
  • $$20$$
  • $$210$$
  • None of these
If the coefficients of $$(r-5)^{th}$$ and $$(2r-1)^{th}$$ 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 $$a_1$$equals
  • 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
  • $$128$$
  • $$129$$
  • $$130$$
  • $$131$$
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