CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 6 - MCQExams.com

The sum of all the coefficients in the binomial expansion of $$(x^2 + x^3)^{319}$$ is
  • 1
  • $$(2)^{318}$$
  • $$(2)^{319}$$
  • 0
$$^{10}C_1+^{10}C_3+^{10}C_5+^{10}C_7+^{10}C_9=$$
  • $$2^9$$
  • $$2^{10}$$
  • $$2^{10}-1$$
  • None of these
The value of $$3{\;}^nC_0-8{\;}^nC_1+13 {\;}^nC_2-18 {\;}^nC_3+.....+n$$ terms is
  • $$0$$
  • $$3^n$$
  • $$5^n$$
  • none of these
The sum of the coefficients in the expansion of $$(x + 2y + z)^{10}$$ is
  • $$2^{10}$$
  • $$4^{10}$$
  • $$3^{10}$$
  • $$1$$
If the $$9^{th}$$ term in the expansion of $$\left(\displaystyle \frac{1}{x^2} + \frac{x}{2} log_2 x \right) ^{12}$$ is equal to 495, then the value of x can be
  • 1
  • An integer greater than 1
  • a fraction
  • an irrational number
In the expansion of $$\displaystyle \left [ 7^{1/3}+11^{1/9} \right ]^{6561}$$, the number of terms free from radicals is
  • $$730$$
  • $$729$$
  • $$725$$
  • $$750$$
In the expression of $$\displaystyle \left ( 3-\sqrt{\frac{17}{4}+3\sqrt{2}} \right )^{15}$$, the 11th term is a
  • positive integer
  • positive irrational number
  • negative integer
  • negative irrational number
The total number of distinct terms in the expansion of $$\displaystyle \left ( x+a \right )^{100}+\left ( x-a \right )^{100}$$ after simplification is
  • $$50$$
  • $$202$$
  • $$51$$
  • $$none\ of\ these$$
If $$C_0 , C_1 , C_2 .... C_n$$ denote the binomial coefficients in the expansion of $$(1 + x)^n$$ , then the value of $$\displaystyle \sum_{r=0}^n(r+1)C_r$$ is
  • $$n 2^n$$
  • $$(n+1)2^{n-1}$$
  • $$(n+2)2^{n-1}$$
  • $$(n+2)2^{n-2}$$
If $$(1+x-2x^2)^8= a_0 + a_1x + a_2x^2 +....+ a_{16}x^{16}$$ then the sum $$a_1+a_3+a_5+ .... +a_{15}$$ is equal to
  • $$-2^7$$
  • $$2^7$$
  • $$2^8$$
  • none of these
The number of terms which are free from radical signs in the expansion of $$(y^{\frac 15} + x^{\frac 1{10}})^{55}$$ are
  • 5
  • 6
  • 7
  • none of these
If $$(2x+3x^2)^6 = a_0 + a_1x + a_2x^2+.....+ a_{12} x^{12}$$, then values of $$a_0$$ and $$a_6$$ are
  • $$0, 6$$
  • $$0, 2^6$$
  • $$1, 6$$
  • $$0$$
If sum of all the coefficients in the expansion of $$(x^{3/2} + x^{1/3})^n$$ is 128, then the coefficient of $$x^5$$ is
  • 35
  • 45
  • 7
  • none of these
If the second term of the expansion $$\displaystyle \left [ a^{1/13}+\frac{a}{\sqrt{a^{-1}}} \right ]^{n}\: \: is\: \: 14a^{5/2}$$, then the value of $$\displaystyle \frac{^{n}{C}_{3}}{^{n}{C}_{2}}$$ is
  • $$4$$
  • $$3$$
  • $$12$$
  • $$6$$
The number of real negative terms in the bionomial expansion of $$ (1+ix)^{4n-2} ,n \in N,x>0 $$ is
  • $$n$$
  • $$n+1$$
  • $$n-1$$
  • $$2n$$
Find $$7^{th}$$ term of $$\displaystyle \left ( \frac{4x}{5}-\frac{5}{2x} \right )^{9}$$
  • $$\dfrac{10050}{x^{3}}$$
  • $$\dfrac{10500}{x^{3}}$$
  • $$\dfrac{1050}{x^{3}}$$
  • $$\dfrac{1000}{x^{3}}$$
Find $$28^{th}$$ term of $$\left ( 5x+8y \right )^{30}$$
  • $$^{30}\textrm{C}_{27}\left ( 5x \right )^{27}\left ( 8y \right )^{3}$$
  • $$^{30}\textrm{C}_{28}\left ( 5x \right )^{2}\left ( 8y \right )^{28}$$
  • $$^{30}\textrm{C}_{27}\left ( 5x \right )^{3}\left ( 8y \right )^{27}$$
  • $$^{30}\textrm{C}_{28}\left ( 5x \right )^{28}\left ( 8y \right )^{2}$$
If $$p+q=1$$, then the value of $$\displaystyle \sum _{ r=0 }^{ 15 }{ { _{  }^{ 15 }{ C } }_{ r } } { p }^{ 15-r }{ q }^{ r }$$
  • $$1$$
  • $$15$$
  • $$18$$
  • $$20$$
The value of $$ \sum _{ r=1 }^{ 10 }{ { r }^{ 2 }.\cfrac { { _{  }^{ 26 }{ C } }_{ r } }{ { _{  }^{ 26 }{ C } }_{ r-1 } }  } $$ is-
  • $$1100$$
  • $$1300$$
  • $$900$$
  • $$1800$$
If $${ \left( { x }^{ 2 }+\cfrac { 1 }{ { x }}  \right)  }^{ n }$$ has exactly one middle term which is equal to $$\alpha.{x}^{3}$$ then the value of $$(\alpha+n)$$ is-         ($$n\in N$$)
  • $$18$$
  • $$21$$
  • $$24$$
  • $$26$$
The number of integral terms in $${(\sqrt {3}+\sqrt [ 8 ]{ 2 } )}^{64}$$ is-
  • $$8$$
  • $$7$$
  • $$9$$
  • $$6$$
Value of $$\displaystyle \sum_{k=1}^{\infty} \sum _{r=0}^{k} \frac{1}{3^{k}} (^{k}C_{r})$$ is
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{4}{3}$$
  • $$2$$
  • $$1$$
Which term is the constant term in the expansion of $$\left ( 2x\, -\, \frac{7}{3x} \right )^6$$ ?
  • 2nd term
  • 3rd term
  • 4th term
  • 5th term
If the middle term in the expansion of $$\left (x^2+\dfrac {1}{x}\right )^n$$ is $$924x^6$$, then $$n=$$
  • $$10$$
  • $$12$$
  • $$14$$
  • $$none\ of\ these$$
The middle term in the expansion of $$(x + 4)^4$$ is:
  • $$96x^3$$
  • $$96x^2$$
  • $$-96x^2$$
  • none of the above
Find the term which has the exponent of x as 8 in the expansion of $$\displaystyle\left(x^{\displaystyle\frac{5}{2}}-\frac{3}{x^3\sqrt{x}}\right)^{10}$$
  • $$T_2$$
  • $$T_3$$
  • $$T_4$$
  • Does not exist
Find the coefficient of the term independent of x in the expansion of $$\displaystyle\left(6x^3-\frac{5}{x^6}\right)^{12}$$.
  • $$^{12}C_45^86^4$$
  • $$^{12}C_55^76^5$$
  • $$^{12}C_46^85^4$$
  • None of these
If the coefficients of 6th and 5th terms of expansion $$(1+x)^n$$ are in the ratio 7:5, then find the value of n.
  • $$11$$
  • $$12$$
  • $$10$$
  • $$9$$
If 'p' and 'q' are the coefficients of $$x^a$$ and $$x^b$$ respectively in $$(1+x)^{a+b}$$, then
  • $$2p=q$$
  • $$p+q=0$$
  • $$p=q$$
  • $$p=2q$$
If sum of the first 3 coefficients is $$16$$ in the expansion $$\displaystyle\left(x+\frac{1}{x^3}\right)^n$$, then find n.
  • $$10$$
  • $$8$$
  • $$5$$
  • $$4$$
$$\displaystyle\sum_{r=2}^{16}{^{16}C_r}=$$
  • $$2^{15}-15$$
  • $$2^{16}-16$$
  • $$2^{16}-17$$
  • $$2^{17}-17$$
The value of the sum $${ \left( _{  }^{ n }{ { C }_{ 1 } } \right)  }^{ 2 }+{ \left( _{  }^{ n }{ { C }_{ 2 } } \right)  }^{ 2 }+{ \left( _{  }^{ n }{ { C }_{ 3 } } \right)  }^{ 2 }+\dots +{ \left( _{  }^{ n }{ { C }_{ n } } \right)  }^{ 2 }$$ is
  • $${ \left( _{ }^{ 2n }{ { C }_{ n } } \right) }^{ 2 }$$
  • $$^{ 2n }{ { Cn }_{ n } }$$
  • $$^{ 2n }{ { C }_{ n } }+1$$
  • $$^{ 2n }{ { C }_{ n } }-1$$
If the magnitude of the coefficient of $$\displaystyle { x }^{ 7 }$$ in the expansion of $$\displaystyle { \left( { ax }^{ 2 }+\frac { 1 }{ bx }  \right)  }^{ 8 }$$, where a, b are positive numbers, is eual to the magnitude of the coefficient of $$\displaystyle { x }^{ -7 }$$ in the expansion of $$\displaystyle { \left( { ax }^{ 2 }-\frac { 1 }{ bx }  \right)  }^{ 8 }$$, then a and b are connected by the relation
  • $$\displaystyle ab=1$$
  • $$\displaystyle ab=2$$
  • $$\displaystyle { a }^{ 2 }b=1$$
  • $$\displaystyle { ab }^{ 2 }=2$$
The value of $$\displaystyle \sum_{r = 1}^{15} r^{2} \left (\dfrac {^{15}C_{r}}{^{15}C_{r - 1}}\right )$$ is equal to:
  • $$680$$
  • $$1085$$
  • $$560$$
  • $$1240$$
The middle term in the expansion of $$(1 + x)^{2n}$$ is
  • $$\dfrac {1.3.5...(2n - 1)}{n}x^{n}$$
  • $$\dfrac {1.3.5...(2n - 1)}{n!}2^{n - 1}x^{n}$$
  • $$\dfrac {1.3.5...(2n - 1)}{n!}x^{n}$$
  • $$\dfrac {1.3.5...(2n - 1)}{n!}2^{n}x^{n}$$
The number of irrational terms in the binomial expansion of $$(3^{1/5}+7^{1/3})^{100}$$ is
  • $$90$$
  • $$88$$
  • $$94$$
  • $$95$$
If in the expansion of $$(a-2b)^n$$, the sum of the $$5th$$ and $$6th$$ term is zero, then the value of $$\dfrac{a}{b}$$ is
  • $$\dfrac{n-4}{5}$$
  • $$\dfrac{2(n-4)}{5}$$
  • $$\dfrac{5}{n-4}$$
  • $$\dfrac{5}{2n-4}$$
The middle term in the expansion of $$\left (x - \dfrac {1}{x}\right )^{18}$$ is
  • $$^{18}C_{9}$$
  • $$-^{18}C_{9}$$
  • $$^{18}C_{10}$$
  • $$-^{18}C_{10}$$
If $${ \left( 1+x+{ x }^{ 2 }+{ x }^{ 3 } \right)  }^{ 5 }=\displaystyle\sum _{ k=0 }^{ 15 }{ { a }_{ k }{ x }^{ k } } $$ then $$\displaystyle\sum _{ k=0 }^{ 7 }{ { a }_{ 2k } } $$ is equal to
  • $$128$$
  • $$256$$
  • $$512$$
  • $$1024$$
Find the sum of coefficients in the expansion of $$\left(1-\dfrac 2x+\dfrac {4}{x^2}\right)^n$$ given that the number of terms are $$28$$
  • $$27$$
  • $$81$$
  • $$243$$
  • $$729$$
$$(x+1)^n=C_0+C_1x^1+C_2x^2....C_nx^n$$.
Find the value of $$C_0+C_1+C_2.....+C_n$$
  • $$n!$$
  • $$(n!)^2$$
  • $$2^n$$
  • $$n!-2^n$$
Let $$n \in N$$, such that $$(1+x+x^2)^n=a_0+a_1x+a_2x^2...a_{2n}x^{2n}$$ The value of $$a_r$$ when $$(0\leq r\leq 2n)$$
  • $$a_{2n-r}$$
  • $$a_{n-r}$$
  • $$a_{2n}$$
  • $$n.a_{2n-1}$$
Find the number of integral terms in the expansion of $$(\sqrt 3 + \sqrt[8]{5})^{256}$$
  • $$33$$
  • $$34$$
  • $$35$$
  • $$32$$
Find the coefficient of $$x^n$$ in expansion of $$(1 + x) (1 - x)^n$$.
  • $$(n - 1)$$
  • $$(-1)^{n - 1} n$$
  • $$(-1)^{n - 1}(n - 1)^2$$
  • $$(-1)^n (1- n)$$
The general term in $$(x + y)^n$$ is given by
$$T_{r+1}=$$ 
  • $$^{n}C_{r+1}$$
  • $$^nC_{n-r-1}$$
  • $$^{n-1}C_r$$
  • none
If the middle term of $$\left(\dfrac 1x +x\sin x\right)^{10}$$ is equal to $$7\dfrac 78$$, then the principal value of $$x$$ is: 
  • $$60^0$$
  • $$30^0$$
  • $$45^0$$
  • $$90^0$$
If the value of
$$C_0+2 \cdot C_1 + 3 \cdot C_2+........+(n+1)\cdot C_n=576$$, then n is ______.
  • $$7$$
  • $$5$$
  • $$6$$
  • $$9$$
The value of $$a_0 ^2-a_1 ^2+a_2 ^2....a_{2n} ^2$$ is
  • $$a_n$$
  • $$2a_n$$
  • $$3a_n$$
  • $$\dfrac {n+1}2 (a_0+a_1+a_2...a_{2n})$$

The first three terms in the expansion of $$(1+a)^n$$ are $$t_1=1,t_2=-18,t_3=144$$.

Use the general term to determine a and n.
  • $$a=-3,n=9$$
  • $$a=-2,n=9$$
  • $$a=-3,n=8$$
  • $$a=-2,n=8$$
The middle term of expansion of $$\left (\dfrac {10}{x} + \dfrac {x}{10}\right )^{10}$$
  • $$^{7}C_{5}$$
  • $$^{8}C_{5}$$
  • $$^{9}C_{5}$$
  • $$^{10}C_{5}$$
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