CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 8 - MCQExams.com

$$\sum\limits_{k = 1}^{n - r} {{}^{n - k}\mathop C\nolimits_r  = {}^x\mathop C\nolimits_y } $$
  • $$x = n + 1 ; y = r$$
  • $$x = n ; y = r + 1$$
  • $$x = n ; y = r$$
  • $$x = n + 1 ; y= r + 1$$
If $${ \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }.......{ C }_{ n }{ x }^{ n }$$, then $${ C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+{ C }_{ 2 }^{ 2 }+{ C }_{ 3 }^{ 2 }+.......+{ C }_{ n }^{ 2 }$$ is equal to
  • $$\dfrac {n!}{n!n!}$$
  • $$\dfrac {{(2n)}!}{n!n!}$$
  • $$\dfrac {{(2n)}!}{n!}$$
  • $$None\ of\ these$$
Find the $$13^{th}$$ terms in the expansion of $$\left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}, x \neq 0$$.
  • $$18564$$
  • $$87328$$
  • $$17374$$
  • $$35546$$
The co-efficient of x in  the expansion of $$(1-2x^3 + 3x^5) \left( 1 + \dfrac{1}{x}\right)^8$$ is :
  • 56
  • 65
  • 154
  • 62
The third term from the end in the expansion of $$\left( \dfrac{3x}{5}-\dfrac{5}{2x}\right)^8$$ is
  • $$\dfrac{35451}{15x^4}$$
  • $$\dfrac{45455}{16x^4}$$
  • $$\dfrac{39372}{15x^4}$$
  • $$\dfrac{39375}{16x^4}$$
The co-efficient of $${x^5}$$ in the expansion of $${\left( {1 + x} \right)^{21}} + {\left( {1 + x} \right)^{22}} + ........ + {\left( {1 + x} \right)^{30}}$$ is: 
  • $$^{51}{C_5}$$
  • $$^{9}{C_5}$$
  • $$^{31}{C_6}{ - ^{21}}{C_6}$$
  • $$^{30}{C_5}{ + ^{20}}{C_5}$$
If $${S_n} = \sum\limits_{r = 0}^n {\dfrac{1}{{^n{C_r}}}\,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {\dfrac{r}{{^n{C_r}}},\,\,then\,\,\dfrac{{{t_n}}}{{{s_n}}} = } } $$ 
  • $$\dfrac{1}{2}n$$
  • $$\dfrac{{2n - 1}}{2}$$
  • $$n - 1$$
  • $$2n$$
In the expansion of $$(1+ax)^n$$, $$n\in N$$, then the coefficient of x and $$x^2$$ are $$8$$ and $$24$$ respectively. Then?
  • $$a=2, n=4$$
  • $$a=4, n=2$$
  • $$a=2, n=6$$
  • None of these
If $$\left| x \right| < 1$$ then the coefficient of $${x^n}$$ in expansion of $${\left( {1 + x + {x^2} + {x^3}....} \right)^2}$$ is
  • $$n$$
  • $$n - 1$$
  • $$n + 2$$
  • $$n + 1$$
$${c_0},{c_1},{c_2}$$ denotes coefficents expansion of $${(1 + x)^n}$$ , then $${c_1} + {c_1}{c_2} + {c_2}{c_3} + .......{c_{n - 1}}{c_n} = \frac{{(2n)!}}{{(n + 1)!(n - 1)!}}$$
  • True
  • False
The sum of rational term in the expansion of $${(3^{\frac{1}{5}}+2^{\frac{1}{3}})}^{15}$$ is
  • $$31$$
  • $$59$$
  • $$51$$
  • $$61$$
The middle term (s) in the expansion of $${ \left( 1+x \right)  }^{ 2n+1 }$$ is (are)
  • $$_{ }^{ 2n+1 }{ { C }_{ n } }{ X }^{ n } and _{ }^{ 2n+1 }{ { C }_{ n+1 } }{ X }^{ n+1 }$$
  • $$_{ }^{ 2n+1 }{ { C }_{ n } }{ X }^{ n+1 } and _{ }^{ 2n+1 }{ { C }_{ n+1 } }{ X }^{ n }$$
  • $$^{ 2n+1 }{ { C }_{ n+1 } }{ X }^{ n }$$
  • $$^{ 2n+1 }{ { C }_{ n+1 } }{ X }^{ n+1 }$$
If $$6^{th}$$ term in the expansion of $$\left[\dfrac{1}{x^{8/3}}+x^{2 }\log_{10}x\right]^{8}$$ is $$5600$$, then $$x$$ is equal to 
  • $$5$$
  • $$4$$
  • $$8$$
  • $$none\ of\ these$$
$$ \sum _{ r=0 }^{ n } \left( \dfrac {r+2}{r+1} \right) $$ $$.^nCr$$ is equal to :
  • $$ \dfrac {2^n (n+2)-1}{(n+1)}$$
  • $$ \dfrac {2^n (n+1)-1}{(n+1)}$$
  • $$ \dfrac {2^n (n+4)-1}{(n+1)}$$
  • $$ \dfrac {2^n (n+3)-1}{(n+1)}$$
The value of $$\displaystyle\sum^{10}_{r=0}$$ $$^{20}C_r$$ is equal to?
  • $$\dfrac{1}{2}(2^{20}+$$ $$^{20}C_{10})$$
  • $$\dfrac{1}{2}(2^{28}+$$ $$^{19}C_{10})$$
  • $$20(2^{18}+$$ $$^{19}C_{11})$$
  • $$10(2^{18}+$$ $$^{19}C_{11})$$
In the binomial expansion of $$(a-b)^{n}, n\ge 5$$, the sum of the $$5^{th}$$ and $$6^{th}$$ terms is zero, then $$a/b$$ is equals:
  • $$\dfrac{n-5}{6}$$
  • $$\dfrac{n-4}{5}$$
  • $$\dfrac{5}{n-4}$$
  • $$\dfrac{6}{n-5}$$
The first $$3$$ terms in the expansion of $$(1+ax)^{n}(n\neq 0)$$ are $$1, 6x$$ and $$16x^{2}$$. Then the value of $$a$$ and $$n$$ are respectively 
  • $$2$$ and $$9$$
  • $$3$$ and $$2$$
  • $$2/3$$ and $$9$$
  • $$3/2$$ and $$6$$
If $$f ( x ) + 2 f ( 1 - x ) = x ^ { 2 } + 2 , \forall x \in R$$, then find $$f ( x )$$
  • $$\dfrac{(x+3)^2}{3}$$
  • $$\dfrac{(x-3)^2}{3}$$
  • $$\dfrac{(x-2)^2}{4}$$
  • $$\dfrac{(x-2)^2}{3}$$
The number of non-zero terms in the expansion of $$(\sqrt{7}+1)^{75}-(\sqrt{7}-1)^{75}$$ is
  • $$36$$
  • $$37$$
  • $$38$$
  • $$39$$
The coefficient of $${x^5}$$ in the expansion of $${\left( {1 + {x^2}} \right)^5}{\left( {1 - x} \right)^4}$$ is   
  • $${4.^6}{C_4}$$
  • $${2.^6}{C_4}$$
  • $${2.^6}{C_2}$$
  • $${4.^6}{C_2}$$
The number of rational terms in the expansion of $$\quad { \left( { 3 }^{ \cfrac { 1 }{ 4 }  }+{ 7 }^{ \cfrac { 1 }{ 6 }  } \right)  }^{ 144 }$$ is
  • $$33$$
  • $$23$$
  • $$12$$
  • $$13$$
In the expansion of $$(y^{1/5}+x^{1/10})^{55}$$, the number of terms free of a radical sign is
  • $$5$$
  • $$6$$
  • $$50$$
  • $$56$$
For $$r=0,1,2,,....10$$ let $${A}_{r},{B}_{r}$$ and $${C}_{r}$$ denote respectively the coefficient of $${x}^{r}$$ in the expansions of $${(1+x)}^{10},{(1+x)}^{20}$$ and $${(1+x)}^{30}$$. Then $$\sum _{ r=1 }^{ 10 }{ { A }_{ r }\left( { B }_{ 10 }{ B }_{ r }-{ C }_{ 10 }{ A }_{ r } \right)  } $$ is equal to
  • $${B}_{10}-{C}_{10}$$
  • $${A}_{10}({B}_{10}^{2}-{C}_{10}{A}_{10})$$
  • $$0$$
  • $${C}_{10}-{B}_{10}$$
 The coefficient of the middle term in the expansion of $$\left(1+x\right)^{2n}$$ is equal to the sum of the coefficient of middle terms in the expansion of $$\left(1+x\right)^{2n-1}$$
the statement is true and false
  • True
  • False
The numerical value of middle terms in $${ \left( 1-\cfrac { 1 }{ x }  \right)  }^{ n }{(1-x)}^{n}$$ is
  • $${ _{ }^{ 2n }{ C } }_{ n }$$
  • $${ _{ }^{ n }{ C } }_{ n }$$
  • $$\left( { _{ }^{ 2n }{ C } }_{ n } \right) $$
  • $$\left( { _{ }^{ n }{ C } }_{ n } \right) $$
The value of
$$\left( { _{  }^{ 7 }{ C } }_{ 0 }+{ _{  }^{ 7 }{ C } }_{ 1 } \right) +\left( { _{  }^{ 7 }{ C } }_{ 1 }+{ _{  }^{ 7 }{ C } }_{ 2 } \right) +.....\left( { _{  }^{ 7 }{ C } }_{ 6 }+{ _{  }^{ 7 }{ C } }_{ 7 } \right) $$ is
  • $${2}^{7}-1$$
  • $${2}^{8}-2$$
  • $${2}^{8}-1$$
  • $${2}^{8}$$
The number of rational terms in the expansion of $${ \left( \sqrt [ 4 ]{ 5 } +\sqrt [ 5 ]{ 4 }  \right)  }^{ 100 }$$ is
  • $$50$$
  • $$5$$
  • $$6$$
  • $$51$$
The $$r$$th term of series $$2\dfrac{1}{2} + 1\dfrac{7}{{13}} + 1\dfrac{1}{9} + \dfrac{{20}}{{23}} + .....$$ is
  • $$\dfrac{{20}}{{5r + 3}}$$
  • $$\dfrac{{20}}{{5r - 3}}$$
  • $$20\left( {5r + 3} \right)$$
  • $$\dfrac{{20}}{{5{r^2} + 3}}$$
$$^{m}C_r.^{n}C_0+{}^{m}C_{r-1}.{}^{n}C_{1}+{}^{m}C_{r-2}.^{n}C_2+..........+^{m}C_0.^{n}C_r={}^{m+n}Cr$$
  • True
  • False
If $$r$$th term is middle term in $${ \left( { x }^{ 2 }-\cfrac { 1 }{ 2x }  \right)  }^{ 20 }$$ then $$(r+3)$$th term is:
  • $$\cfrac { { _{ }^{ 20 }{ C } }_{ 7 }x }{ { 2 }^{ 13 } } $$
  • $$-\left( \cfrac { { _{ }^{ 20 }{ C } }_{ 5 }x }{ { 4 }^{ 13 } } \right) $$
  • $$-\left( \cfrac { { _{ }^{ 20 }{ C } }_{ 7 }x }{ { 2 }^{ 13 } } \right) $$
  • $$-\left( \cfrac { { _{ }^{ 20 }{ C } }_{ 14 }x }{ { 4 }^{ 13 } } \right) $$
The middle term in the expansion of $$(1-3x+3x^{2}-x^{3})^{6}$$ is
  • $$^{18}C_{10}\ {x}^{10}$$
  • $$^{18}C_{9}( {-x})^{9}$$
  • $$^{18}C_{9}\ {x}^{9}$$
  • $$None\ of\ these$$
The value of $$\displaystyle\frac { { C }_{ 0 } }{ 1.3 } -\frac { { C }_{ 1 } }{ 2.3 } +\frac { { C }_{ 2 } }{ 3.3 } -\frac { { C }_{ 3 } }{ 4.3 } +.........+{ \left( -1 \right)  }^{ n }\frac { { C }_{ n } }{ (n+1).3 } $$ is :
  • $$\displaystyle\frac { 3 }{ n+1 } $$
  • $$\displaystyle\frac { n+1 }{ 3 } $$
  • $$\displaystyle\frac { 1 }{ 3n+3 } $$
  • none of these
The sum of  the series $$^{20}{C_0}{ - ^{20}}{C_1}{ + ^{20}}{C_2}{ - ^{20}}{C_3}{ + _{......}}{ - _{......}}{ + ^{20}}{C_{10}}\,is - $$
  • $$\frac{1}{2}{20_{{C_{10}}}}$$
  • $$0$$
  • $${ - ^{20}}{C_{10}}$$
  • $$^{20}{C_{10}}$$
If the number of terms is the expansion $${\left( {1 - \frac{2}{x} + \frac{4}{{{x^2}}}} \right)^n},x \ne 0,$$ is $$28$$, then the sum of coefficients of all the terms in this expansion, is :
  • 2187
  • 243
  • 729
  • 64
The sum of coefficients of integral powers of $$x$$ in the binomial expansion of $$(1-2\sqrt x)^{50}$$ is : 
  • $$\dfrac{1}{2}(3^{50}-1)$$
  • $$\dfrac{1}{2}(2^{50}+1)$$
  • $$\dfrac{1}{2}(3^{50}+1)$$
  • $$\dfrac{1}{2}(3^{50})$$
If $$A$$ and $$B$$ are coefficients of $$x ^ { n }$$ in the expansion of $$( 1 + x ) ^ { 2 n }$$ and $$( 1 + x ) ^ { 2 n - 1 }$$ respectively, then
  • $$A = B$$
  • $$A = 2 B$$
  • $$2 A = B$$
  • $$A + B = 0$$
If$$(1+x)^n=C_0+C_1x+C_2x^2+......+C_nx^n,$$  then $$C_0+5C_1+9C_2+.....+(4n+1)C_n$$ is equal to 
  • $$n.2^n$$
  • $$(n+1)2^n$$
  • $$(2n+1)2^n$$
  • $$(4n+1)2^n$$
If the rth term in the expansion of $${\left( {{x \over 3} - {2 \over {{x^2}}}} \right)^{10}}$$ contains $${x^4}$$ then r is equal to 
  • 2
  • 3
  • 4
  • 5
If the sum of the binomial coefficients in the expansion of $$\left(x^{2}+\dfrac{2}{x^{3}}\right)^{n}$$ is $$243$$, the term independent of $$x$$ is equal to 
  • $$40$$
  • $$30$$
  • $$20$$
  • $$10$$
If $$x^{4}$$ occurs in the $$rth$$ term in the expansion of $$\left(x^{4}+\dfrac{1}{x^{3}}\right)^{15}$$, then $$r=$$
  • $$7$$
  • $$8$$
  • $$9$$
  • $$10$$
If $$(1-x-x^2)^{20}$$ = $$\sum _{ r=0 }^{ 40 }{ a_4,x^x },$$ then 
$$a_1+3a_3+5a_5+........+39a_{39}=$$
  • $$40$$
  • $$-40$$
  • $$80$$
  • $$-80$$
whether the sum of the coefficients in the expansion of $${\left(1+x-3{x}^{2}\right)}^{2163}$$ is $$-6$$
  • True
  • False
If the constant term in the expansion of $$\left(x^{2}-\dfrac{1}{x}\right)^{n}$$ is $$15$$ then the value of $$n$$ is
  • $$6$$
  • $$9$$
  • $$12$$
  • $$15$$
In the expansion of $$(1+2x+3x^{2})^{10},$$ coefficient of $$x^{4}$$ is not divisible by
  • $$12$$
  • $$7$$
  • $$11$$
  • $$5$$
Coefficient of $$\alpha $$ in the expansion of $$(\alpha +p)^{m-1}+(\alpha +p)^{m-2}(\alpha +q)^{m-3}(\alpha +q)^{2}+....(\alpha +q)^{m-1}$$ where $$\alpha \neq -q$$ and $$p\neq q$$ is: 
  • $$\frac{^{m}C_{1}(p^{1}-q^{1})}{p-q}$$
  • $$\frac{^{m}C_{1}(p^{m-1}-q^{m-1})}{p-q}$$
  • $$\frac{^{m}C_{1}(p^{1}+q^{1})}{p-q}$$
  • $$\frac{^{m}C_{1}(p^{m-1}+q^{m-1})}{p-q}$$
If the $$r^{th}$$ and the $$(r+1)^{th}$$ terms in the expansion of $$(p+q)^{n}$$ are equal, then $$\dfrac{(n+1)q}{r(p+q)}$$ is
  • $$1/2$$
  • $$1/4$$
  • $$1$$
  • $$0$$
Number of distinct terms in the expansion of $$(x+y-z)^{16}$$ is 
  • $$816$$
  • $$152$$
  • $$153$$
  • $$136$$
Find the coefficient of $$x^{11}$$ in the expansion of $$\left(x^{3}-\dfrac{2}{x^{2}}\right)^{12}$$
  • $$-25344$$
  • $$-25250$$
  • $$-25000$$
  • $$-25750$$
The number of terms in the expansion of $$(1+x)^{101}(1+x^{2}-x)^{100}$$ in power of $$x$$ is:
  • $$302$$
  • $$301$$
  • $$202$$
  • $$101$$
If $$(1+x+x^2)^8=a_0+a_1x+.....a_{16}x^{16}$$ then $$a_1-a_3+a_5-a_7+....... -a_{15}$$=
  • $$1$$
  • $$2$$
  • $$3$$
  • $$0$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 11 Engineering Maths Quiz Questions and Answers