CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 1 - MCQExams.com

The sum of coefficient of integral powers of $$x$$ in the binomial expansion of $$(1-2\sqrt x)^{50}$$ is :
  • $$\dfrac {1}{2}(3^{50}+1)$$
  • $$\dfrac {1}{2}(3^{50})$$
  • $$\dfrac {1}{2}(3^{50}-1)$$
  • $$\dfrac {1}{2}(2^{50}+1)$$
If $$(2+\dfrac {x}{3})^{55}$$ is expanded in the ascending powers of $$x$$ and the coefficients of powers of $$x$$ in two consecutive terms of the expansion are equal, then these terms are :
  • $$7^{th}$$ and $$8^{th}$$
  • $$8^{th}$$ and $$9^{th}$$
  • $$28^{th}$$ and $$29^{th}$$
  • $$27^{th}$$ and $$28^{th}$$
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $$(x + a)^{n}$$ are $$A$$ and $$B$$ respectively, then the value of $$(x^{2} - a^{2})^{n}$$ is
  • $$A^{2} - B^{2}$$
  • $$A^{2} + B^{2}$$
  • $$4AB$$
  • None
$${ _{  }^{ 15 }{ C } }_{ 3 }+{ _{  }^{ 15 }{ C } }_{ 5 }+....+{ _{  }^{ 15 }{ C } }_{ 15 }$$ will be equal to
  • $${2}^{14}$$
  • $${2}^{14}-15$$
  • $${2}^{14}+15$$
  • $${2}^{14}-1$$
Find the sum of coefficient of middle terms of the expansion $$\left(3x-\dfrac{x^3}{6}\right)^7$$:
  • $$\dfrac {595}{48}$$
  • $$-\dfrac {595}{48}$$
  • $$-\dfrac {595}{24}$$
  • None of the above
The total number of terms in the expansion of $$(x+y)^{50}+(x-y)^{50}$$ is
  • $$51$$
  • $$26$$
  • $$102$$
  • $$25$$
In the expansion of $$(\sqrt 2+\sqrt [3]{5})^{20}$$ the number of rational terms will be:
  • $$3$$
  • $$10$$
  • $$4$$
  • $$8$$
What is $$\displaystyle\sum _{ r=0 }^{ n }{ C\left( n,r \right)  } $$ equal to?
  • $${2}^{n} - 1$$
  • $$n$$
  • $$n$$!
  • $${2}^{n}$$
The number of rational terms in the expansion of $$(9^{1/4} + 8^{1/6})^{1000}$$ is:
  • $$500$$
  • $$400$$
  • $$501$$
  • none of the above
The $$3rd$$ term of $${\left( {3x - \dfrac{{{y^3}}}{6}} \right)^4}$$ is
  • $$^4{C_2}{\left( {3x} \right)^2}{\left( { - \dfrac{{{y^3}}}{6}} \right)^2}$$
  • $$^4{C_2}{\left( {3x} \right)^2}{\left( { - \dfrac{{{y^2}}}{6}} \right)^2}$$
  • $$^4{C_2}{\left( {3x} \right)^3}{\left( { - \dfrac{{{y^3}}}{6}} \right)^2}$$
  • $$^4{C_2}{\left( {3x} \right)^3}{\left( { - \dfrac{{{y^2}}}{6}} \right)^2}$$
[AS 1] If $$A = \dfrac{1}{3} B \, and \, B = \dfrac{1}{2} C$$, then A : B : C = .. 
  • 1 : 3 : 6
  • 2 : 3 : 6
  • 3 : 2 : 6
  • 3 : 1 : 2
The expression $$^{n}\textrm{C}_{0}+4\ ^{n}\textrm{C}_{1}+4^{2}\ ^{n}\textrm{C}_{2}+..........+4^{n}\ ^{n}\textrm{C}_{n}$$, equals 
  • $$2^{2n}$$
  • $$2^{3n}$$
  • $$5^{n}$$
  • None of these
The number of zeroes at the end of $$(101)^{11}-1$$ is
  • $$8$$
  • $$4$$
  • $$1100$$
  • $$2$$
The middle terms in the expansion of $$(x^{2}-a^{2})^{5}$$ is
  • $$10x^{6}a^{4},\ -10x^{4}a^{6}$$
  • $$-10x^{6}a^{4},\ 10x^{4}a^{6}$$
  • $$10x^{6}a^{4},\ 10x^{4}a^{6}$$
  • $$-10x^{6}a^{4},\ -10x^{4}a^{6}$$
If sum of the coefficients in the expansion of $$(2+3cx+c^2x^2)^{12}$$ vanishes, then $$c$$ equals to
  • $$-1,2$$
  • $$1,2$$
  • $$1,-2$$
  • $$-1,-2$$
$$^{100}C_{0}-^{100}C_{2}+^{100}C_{4}+^{100}C_{8}-........+^{100}C_{100}=$$__
  • $$2^{-49}$$
  • $$-2$$
  • $$2^{50}$$
  • $$-2^{50}$$
If the sum of binomial coefficient in the expansion $$(1+x)^{n}$$ is $$256$$, then $$n$$ is
  • $$6$$
  • $$7$$
  • $$8$$
  • $$9$$
The coefficient of $${ x }^{ 3 }$$ in the polynomial $$(x-1) (x-2) (x-4)$$ is
  • 1
  • -1
  • 7
  • None of these
If the constants term in the expansions of $$(\sqrt{x}-\dfrac{k}{x^2})^{10}$$ is $$405$$, then what can be the value of $$k$$ ?
  • $$\pm 2$$
  • $$\pm 3$$
  • $$\pm 5$$
  • $$\pm 9$$
The middle term in $${\left( {{x^2} + \dfrac{1}{{{x^2}}} + 2} \right)^n}$$ is 
  • $$\dfrac{{n!}}{{{{\left( {\left( {\frac{n}{2}} \right)!} \right)}^2}}}$$
  • $$\dfrac{{\left( {2n} \right)!}}{{{{\left( {\left( {\frac{n}{2}} \right)!} \right)}^2}}}$$
  • $$\dfrac{{1.3.5....\left( {2n + 1} \right)}}{{n!}}{2^n}$$
  • $$\dfrac{{\left( {2n} \right)!}}{{{{\left( {n!} \right)}^2}}}$$
The term containing $$ x^3 $$ in the expansion of $$ (x - 2y)^7 $$ is
  • $$3$$rd
  • $$4$$th
  • $$5$$th
  • $$6$$th
The greatest coefficient in the expansion of $$\left ( 1+X \right )^{2n+2}$$ is
  • $$\frac{\left ( 2n \right )!}{\left ( n! \right )^{2}}$$
  • $$\frac{\left ( 2n+2 \right )!}{\left (( n+1 \right )!)^{2}}$$
  • $$\frac{\left ( 2n+2 \right )!}{n!\left (n+1 \right )!}$$
  • $$\frac{\left ( 2n \right )!}{n!\left (n+1 \right )!}$$
In the expansion of $$ (2+\dfrac{x}{3})^n $$, coefficients of $$ x^7 $$ and $$ x^8 $$ are equal. Then $$ n = $$
  • $$49$$
  • $$50$$
  • $$55$$
  • $$56$$
The middle term in the expansion of $$ \left (\displaystyle \frac{x^{\frac{3}{2}}}{\sqrt{a}} - \frac{y^{\frac{5}{2}}}{b^{\frac{3}{2}}} \right )^8 $$ is 
  • $$ ^8C_4. \dfrac{x^6.y^{10}}{a^2b^6} $$
  • $$ ^8C_4. \dfrac{x^6.y^{8}}{a^2b^4} $$
  • $$ ^8C_5. \dfrac{x^5.y^{10}}{a^2b^5} $$
  • $$ ^8C_1. \dfrac{x^5.y^{10}}{a^2b^5} $$
The middle term in the expansion of $$ \left ( x + \dfrac{1}{x} \right )^{10} $$ is
  • $$ ^{10}C_{4}.\dfrac{1}{x} $$
  • $$ ^{10}C_{5} $$
  • $$ ^{10}C_{5}.\dfrac{1}{x} $$
  • $$ ^{10}C_{6}.x $$
The middle term in the expansion of $$ (1+x)^{2n} $$ is
  • $$ ^{2n}C_{n} $$
  • $$ ^{2n}C_{n-1}.x^{n+1} $$
  • $$ ^{2n}C_{n-1}.x^{n-1} $$
  • $$ ^{2n}C_{n}.x^n $$
The middle term of $$ \left ( x - \dfrac{1}{x} \right )^{2n+1} $$ is
  • $$ ^{2n+1}C_n.x $$
  • $$ ^{2n+1}C_n $$
  • $$ (-1)^{n}.^{2n+1}C_{n} $$
  • $$ (-1)^{n}.^{2n+1}C_{n}.x $$
If the coefficients of $$ (r+2)^{th} $$ and $$ (2r+1) ^{th}$$ terms $$ (r \neq 1) $$ are equal in the expansion of $$ (1+x)^{43} $$, then $$ r = $$
  • $$12$$
  • $$13$$
  • $$14$$
  • $$15$$
Sum of the coefficients of $$ (1+x)^n $$ is always a
  • an integer
  • positive integer
  • negative integer
  • zero
The number of terms in the expansion of $$ (1+x)^{21} $$ is
  • $$20$$
  • $$21$$
  • $$22$$
  • $$24$$
The ratio of the $${ r }^{ th }$$ term and the $$({ r+1 })^{ th }$$ term in the expansion of $$ (1+ x)^n $$ is 
  • $$\displaystyle \frac{r}{(n-r+1)x} $$
  • $$\displaystyle \frac{1}{(n-r+1)x} $$
  • $$\displaystyle \frac{r}{(n-r+1)} $$
  • $$\displaystyle \frac{(n-r+1)x}{r} $$
In the binomial expansion of $$ (a-b)^n , n \geq{5} $$, the sum of
5th and 6th terms is zero then a/b equal to
  • $$ \frac{5}{n-4} $$
  • $$ \frac{6}{n-5} $$
  • $$ \frac{n-5}{6} $$
  • $$ \frac{n-4}{5} $$
If the middle term of $$ (1+x)^{2n} $$ is the greatest term then $$x$$ lies between
  • $$n - 1 < x < n$$
  • $$ \dfrac{n}{n + 1} < x < \dfrac{n+1}{n} $$
  • $$ n < x < n+1 $$
  • $$ \dfrac{n+1}{n} < x < \dfrac{n}{n+1} $$
In the expansion of $$ (a+b)^n $$, the ratio of the binomial coefficients of $${ 2 }^{ nd }$$ and $${ 3 }^{ rd }$$ terms is equal to the ratio of the binomial coefficients of $${ 5 }^{ th }$$ and $${ 4 }^{ th }$$ terms, then $$n = $$
  • $$4$$
  • $$5$$
  • $$6$$
  • $$7$$
The ratio of (r+1)th and rth terms in the expansion of $$ (1-x)^n $$ is
  • $$ \dfrac{-r}{(n-r+1)x} $$
  • $$ \dfrac{r}{(n-r+1)x} $$
  • $$ \dfrac{-(n-r+1)x}{r} $$
  • $$ \dfrac{(n-r+1)x}{r} $$
In the expansion of $$ \left ( a^2\sqrt{a} + \frac{\sqrt[3]{a}}{a} \right )^n $$ the binomial coefficient of 3rd term isThe 7th term is :
  • $$ 84a^3\sqrt{a} $$
  • $$ 84a^2\sqrt{a} $$
  • $$ 84 a^2 $$
  • $$ 84 a^3 $$
In the expansion of $$\displaystyle  \left ( \sqrt{a} + \frac{1}{\sqrt{(3a)}} \right )^n  $$, if the ratio of the binomial coefficient of the $$4^{th}$$ term to the binomial coefficient of the $$3^{rd}$$ term is $$ \dfrac{10}{3} $$, the $$5^{th}$$ term is
  • $$ \dfrac{88a}{\sqrt 3} $$
  • $$ \dfrac {88a}{3} $$
  • $$ 50a^2 $$
  • $$ 55a^2 $$
Sum of the coefficients of $$ (1 - x)^{25} $$ is
  • $$-1$$
  • $$1$$
  • $$0$$
  • $$ 2^{25} $$
The product of two middle terms in the expansion of $$ \left (\displaystyle \frac{3x^2}{2} - \frac{1}{3x} \right )^9 $$ is 
  • $$ (^9C_4)^2.\displaystyle \frac{x^9}{512} $$
  • $$ ^{-9}C_4.^9C_5.\displaystyle \frac{x^8}{512} $$
  • $$ -^{9}C_4.^9C_5.\displaystyle \frac{x^9}{512} $$
  • $$ ^9C_4.^9C_5.\displaystyle \frac{x^9}{256} $$
The $$3$$rd, $$4$$th and $$5$$th terms in the expansion of $$ (1+x)^n $$ are $$60, 160$$ and $$240$$ respectively, then $$ x = $$
  • $$2$$
  • $$4$$
  • $$5$$
  • $$6$$
$$ \displaystyle \frac{^{15}C_1}{^{15}C_0}+2.\frac{^{15}C_2}{^{15}C_1}+3.\frac{^{15}C_3}{^{15}C_2}+\ldots+15.\frac{^{15}C_{15}}{^{15}C_{14}} = $$
  • 105
  • 91
  • 120
  • 15
The number of terms in the expansion of $$ (1+5\sqrt{2}x)^9 + (1-5\sqrt{2}x)^9 $$ is :
  • $$5$$
  • $$10$$
  • $$18$$
  • $$20$$
A. $$ ^{2n}C_n = C_0^2 + C_1^2 + C_2^2 + C_3^2 + \dots \dots + C_n^2 $$

B. $$ ^{2n}C_n = $$ term independent of $$x$$ in $$ (1+x)^n \left(1+\frac{1}{x} \right)^n $$

C. $$ ^{2n}C_n = \dfrac{1.3.5.7 \ldots \ldots (2n-1)}{n!} $$ then
  • A, B are false, C is true
  • A is false, B and C are true
  • A and B are true; C is false
  • A, B, C are true
The total number of terms in the expansion of $$ (x+a)^{100}+(x-a)^{100} $$ after simplification is
  • $$202$$
  • $$51$$
  • $$101$$
  • $$50$$
$$ C_0^2+3.C_1^2+5.C_2^2 + \ldots\ldots +(2n+1).C_n^2 = $$
  • $$ (n+1)2^n $$
  • $$ (2n+1) ^{2n}C_n $$
  • $$ (n+1). ^{2n}C_n $$
  • $$ (2n-1) ^{2n}C_n $$
$$ ^nC_0 + ^nC_2 + ^nC_4 + \dots \dots + ^nC_{2[n/2]} $$, where [ ] denotes greatest integer
  • $$ 2^{2n-1} $$
  • $$ 2^{2n-1}-1 $$
  • $$ 2^{n-1} $$
  • $$ 2^{n-1}-1$$
$$ ^5C_0+2.^5C_1+2^2.^5C_2+2^3.^5C_3 +2^4.^5C_4+2^5.^5C_5 = $$
  • 32
  • 243
  • 64
  • 729
Evaluate the following:
$$ C_1+2C_2+3C_3+\ldots\dots+nC_n  $$
  • $$ n2^n $$
  • $$ n2^{n-1} $$
  • $$ (n+1)2^n $$
  • $$ (n+1)2^{n-1} $$
$$ (1+x)^{15}=a_0+a_1x+\ldots\ldots+a_{15}x^{15} \Rightarrow \sum_{r=1}^{15}r\frac{a_r}{a_{r-1}}= $$
  • 110
  • 115
  • 120
  • 135
If $$ n\geq2 $$ then $$ (a-1).C_1-(a-2).C_2+(a-3).C_3-\ldots\ldots(-1)^{n-1}(a-n).C_n= $$
  • $$0$$
  • $$ a-1 $$
  • $$a$$
  • $$ a+1 $$
0:0:1


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