CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 2 - MCQExams.com


The coefficient of $$\displaystyle \frac{1}{{x}}$$ in the expansion of $$(1+\displaystyle x)^{{n}}(1+\frac{1}{{x}})^{{n}}$$ is :
  • $$\displaystyle \frac{\mathrm{n}!}{(\mathrm{n}-1)!(\mathrm{n}+1)!}$$
  • $$\displaystyle \frac{2\mathrm{n}!}{(\mathrm{n}-1)!(\mathrm{n}+1)!}$$
  • $$\displaystyle \frac{\mathrm{n}!}{(2\mathrm{n}-1)!(2\mathrm{n}+1)!}$$
  • 2n!(2n−1)!(2n+1)!
Coefficient of $$x$$ in the expansion of $$(1-2\displaystyle {x}^{3}+3{x}^{5})(1+\frac{1}{{x}})^{8}$$ is
  • $$154$$
  • $$164$$
  • $$146$$
  • $$156$$
If $$a$$ is the coefficient of the middle term in the expansion of $$(1+x)^{2n}$$ and $$b, c$$ are the coefficients of the two middle terms in the expansion of $$(1+x)^{2n-1}$$ then 
  • $$a +b = c$$
  • $$a = b + c$$
  • $$a=b=c$$
  • $$b = a + c$$
The middle term in the expansion of $$(1-3{x}+3{x}^{2}-{x}^{3})^{2{n}}$$ is
  • $$6{n}_{{C}_{3{n}}}(-{x})^{3{n}}$$
  • $$6{n}_{{C}_{2{n}}}(-{x})^{2{n}+1}$$
  • $$4{n}_{{C}_{3{n}}}(-{x})^{3{n}}$$
  • $$6{n}_{{C}_{3{n}}}(-{x})^{3{n}-1}$$
Sum of the coefficients in the expansion of $$ (5x-4y)^n $$ where $$n$$ is a positive integer is
  • $$1$$
  • $$ 9^n $$
  • $$ (-1)^n $$
  • $$ 5^n $$
If $$n$$ is a positive integer, then the coefficient of $$ x^n $$ in the expansion of $$ \dfrac{(1+2x)^n}{1-x} $$ is
  • $$ n.3^n $$
  • $$ (n-1)3^n $$
  • $$ (n+1)3^n $$
  • $$ 3^n $$
The coefficient of $${x}^{p}$$ the expansion of $$(\displaystyle {x}^{2}+\frac{1}{{x}})^{{n}}$$, when it exists is
  • $$2n_{{C}_{\frac{4{n}+{p}}{3}}}$$
  • $$2n_{{C}_{\frac{2{n}+{p}}{3}}}$$
  • $$2n_{{C}_{\frac{{n}+{p}}{3}}}$$
  • $$n_{{C}_{\frac{{n}+{p}}{3}}}$$
If $$n$$ is a positive integer, then the coefficient of $$ x^n $$ in the expansion of $$ \dfrac{(1+x)^n}{1-x} $$ is

  • $$ (n+1)2^n $$
  • $$ 2^n $$
  • $$ 2^{n-1} $$
  • $$ n.2^n $$
The coefficient of the middle term in the binomial expansion in powers of $$x$$ of $$(1+\alpha x)^{4}$$ and of $$(1-\alpha x)^{6}$$ is same if $$\alpha=$$
  • -5/3
  • 3/5
  • -3/10
  • 10/3
The sum of the coefficients in the expansion of $$ (1-x)^{10} $$
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$ 2^{10} $$
In the expansion of $$(1+{x})^{\mathrm{n}}$$, the $$5^{{t}{h}}$$ term is $$4$$ times the $$4^{{t}{h}}$$ term and the $$4^{{t}{h}}$$ term is $$6$$ times the $$3^{{r}{d}}$$ term. than $$n =$$
  • 9
  • 10
  • 11
  • 12
Assertion (A) : The coefficient of $$x^{7}$$ in $$(\displaystyle \frac{x^{2}}{2}-\frac{2}{x})^{9}$$ is zero

Reason (R) : $$r$$ in $$t_{r+1}$$ that contains coefficient of $$x^{7}$$ is not positive integer
  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true and R is not the correct explanation of A
  • A is true, but R is false
  • A is false, but R is true
lf the $$5^{\mathrm{t}\mathrm{h}}$$ term of $$(\displaystyle \frac{\mathrm{q}}{2\mathrm{x}}-\mathrm{p}\mathrm{x})^{8}$$ is 1120 and $$\mathrm{p}+\mathrm{q}=5,\ \mathrm{p}>\mathrm{q}$$ then $$\mathrm{p}=$$
  • 3
  • 6
  • 4
  • 7
lf $$(1+\mathrm{k}{\mathrm{x}})^{10}=\mathrm{a}_{0}+\mathrm{a}_{1}\mathrm{x}+\mathrm{a}_{2}\mathrm{x}^{2}+\ldots.+\mathrm{a}_{10}\mathrm{x}^{10}$$ and $$\displaystyle \mathrm{a}_{2}+\frac{7}{5}\mathrm{a}_{1}+1=0$$ then $$\mathrm{k}=$$
  • -1/5, -1/9
  • 1/5, 1/9
  • -1/5, -1/7
  • 1/5, -1/9

$$^{(n+1)}{{C}_{1}}+^{({n}+1)}{{C}_{2}}+^{({n}+1)}{{C}_{3}}+\ldots..+^{({n}+1)}{{C}_{{n}}}=$$
  • $$2(2^{{n}}+1)$$
  • $$2(2^{{n}}-1)$$
  • $$2^{{n}+1}$$
  • $$(2^{{n}+1}-1)$$
The ratio of $$(r + 1) ^{th}$$ and $$(r - 1)^ {th}$$ terms in the expansion of $$({a}-b)^{n}$$ is
  • $$\displaystyle \frac{(n-r+2)(n-r+1)}{r(r-1)}\cdot\frac{b^{2}}{\mathrm{a}^{2}}$$
  • $$\displaystyle \frac{(n-r+2)(n-r+1)}{r(r-1)}.\frac{\mathrm{a}^{2}}{b^{2}}$$
  • $$(\displaystyle \frac{n-r+2}{r})\cdot\frac{b}{\mathrm{a}}$$
  • $$(\displaystyle \frac{n-r+1}{r-1})\cdot\frac{b}{\mathrm{a}}$$
If the $$2^{nd}, 3^{rd}$$ and $$4^{th}$$ terms in the expansion of $$(a+b)^n$$ are $$135, 30$$ and $$\dfrac {10}{3}$$ respectively, then the value of $$n$$ is:
  • $$5$$
  • $$6$$
  • $$7$$
  • $$8$$
Assertion (A) : The coefficient of $$x^{-2}$$ in the expansion of $$(x^{2}+\displaystyle \frac{1}{x})^{5}$$ is equal to $${ ^{ 5 }C_{ 4 } }$$
Reason (R) : The value of r for the above expansion is 3.
  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true and R is not the correct explanation of A
  • A is true, but R is false
  • A is false, but R is true
If the term containing $$x^3$$ in $$(1-\dfrac {x}{n})^n$$ is $${\dfrac{7}{8}}$$ when $$ x = -2$$ and $$n$$ is a positive integer, then $$n =$$
  • 7
  • 8
  • 9
  • 10
In the expansion of $$\left[2^{\log_2 {\sqrt{9^{x-1}+7}}}+\displaystyle \frac{1}{2^{\frac{1}{5}\log_2 \displaystyle {(3^{x-1}+1})}}\right]^{7}$$, 6th term is $$84$$. Then $$x =$$
  • $$1$$
  • $$2$$
  • $$1$$ or $$2$$
  • $$2$$ or $$4$$

$$\displaystyle \sum_{{r}=0}^{{n}}({r}-4){C}_{{r}=}$$
  • $$(n-8)2^n$$
  • $$(n-8)2^{n-1}$$
  • $$(n-8)2^{n-4}$$
  • $$0$$

The sum of the coefficients of middle terms in the expansion of $$(1+{x})^{2{n}-1}$$
  • $$(2{n})!$$
  • $$\displaystyle \frac{(2{n})!}{{n}!}$$
  • $$\displaystyle \frac{(2{n})!}{({n}!)^{2}}$$
  • $$\displaystyle \frac{(2n-1)!}{{n}!}$$
$$^{(2n+1)}C_{0}-^{(2n+1)}C_{1}+^{(2n+1)}C_{2}-....^{2n+1}C_{2n}=$$
  • $$1$$
  • $$2^{2{n}}$$
  • $$-1$$
  • $$0$$

The number of irrational terms in the expansion $${ \left( \sqrt [ 4 ]{ 3 } +\sqrt [ 3 ]{ 7 }  \right)  }^{ 36 }$$ is
  • $$30$$
  • $$33$$
  • $$31$$
  • $$29$$

$$\displaystyle \sum_{{r}=0}^{{n}}{r}.{C}_{{r}=}^{2}$$
  • $$\dfrac{n}{2}\dfrac{(2n)!}{(n!)^{2}}$$
  • $$\dfrac{(2n)!}{(n!)^{2}}$$
  • $$(2n)!$$
  • $$\displaystyle \frac{\mathrm{n}(2\mathrm{n})!}{2}$$
The sum of the coefficients of the first $$10$$ terms in the expansion of $$(1-x)^{-3}$$
  • $$220$$
  • $$286$$
  • $$120$$
  • $$150$$
$$\displaystyle \sum_{{r}=2}^{{n}}{ ^{ 5r-3 }C_{ r } }=$$
  • $$(5{n}+6).2^{{n}-1}-2{n}+2$$
  • $$(5{n}+6).2^{{n}-1}-2{n}+3$$
  • $$(5{n}-6)2^{{n}-1}-2{n}+2$$
  • $$(5{n}-6)2^{{n}-1}-2{n}+3$$

$$S_{1}=^{m}{c}_{1}+(m+1)_{C_{2}+}(m+2)_{C_{3}+\ldots.+}(m+n-1)_{C_{n}}$$

$$S_{2}=^{n}{c}_{1}+(n+1)_{C_{2}+}(n+2)_{C_{3}+\ldots.+}(m+n-1)_{C_{n}}$$
  • $${S}_{1}+{S}_{2}=0$$
  • $${S}_{1}-{S}_{2}=0$$
  • $$S_1+S_2=2^n$$
  • $${S}_{1}+{S}_{2}=2^{{n}}-1$$

If $$\mathrm{C}_{0}+2\mathrm{C}_{1}+4\mathrm{C}_{2}+\ldots.+2^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}=243$$, then n =
  • 3
  • 4
  • 5
  • 6

The sum $$\displaystyle \sum_{\lrcorner 0<\leq}\sum_{{j}\leq 10}{C}_{{j}}({C}_{{i}})=$$
  • $$2^{10}$$
  • $$2^{10}-1$$
  • $$3^{10}-1$$
  • $$3^{10}$$
The number of non zero terms in the expansion of $$(1+3\sqrt{2}x)^{9}+(1-3\sqrt{2}x)^{9}$$ is
  • $$9$$
  • $$0$$
  • $$5$$
  • $$10$$
If $$x + y = 1$$, then $$\displaystyle \sum_{r=0}^{n}r^{n}C_{r}x^{r}.y^{n-r}=$$
  • $$1$$
  • $$n$$
  • $$nx$$
  • $$ny$$
If $$x + y = 1$$ then $$\displaystyle \sum_{r=0}^{n}r^{2}$$  $$^{n}C_{r}x^{r}y^{n-r}$$
  • $$nxy$$
  • $$nx(x + yn)$$
  • $$nx(nx + y)$$
  • $$nx$$

The number of non zero terms in $$(x+\mathrm{a})^{75}+(x-\mathrm{a})^{75}$$
  • $$38$$
  • $$76$$
  • $$34$$
  • $$32$$
If the fourth term in the expansion of 

$$\left(\sqrt{x^\dfrac{1}{\log x+1}}+x^\dfrac{1}{12}\right)^{6}$$ is equal to $$200$$ and $$x>1$$, then $$x$$ is
  • $$10$$
  • $$10^{-4}$$
  • $$1$$
  • $$-4$$
If in the expansion of $$(\displaystyle \frac{1}{x}+x\tan x)^{5}$$, the ratio of $$4^{th}$$ term to the $$2^{nd}$$ term is $$\displaystyle \frac{2}{27}\pi^{4}$$, then the value of $${x}$$ can be
  • $$\displaystyle \frac{-\pi}{6}$$
  • $$\displaystyle \frac{-\pi}{3}$$
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{\pi}{12}$$
In the expansion of $$(1+x)^{n}.(1+y)^{n}.(1+\mathrm{z})^{n}$$ the sum of the coefficients of the terms of degree $$r$$ is
  • $$(^{n}C_{r})^{3}$$
  • $$^{3n}C_{r}$$
  • $$3\times nC_{r}$$
  • $$nC_{3r}$$
The number of terms in the expansion of $$\left [ (a+4b)^{3}(a-4b)^{3} \right ]^{2}$$ are
  • $$6$$
  • $$7$$
  • $$8$$
  • $$32$$
The number of terms in the expansion of $$\left [ (a+4b)^{3}+(a-4b)^{3} \right ]^{2}$$ are
  • $$6$$
  • $$8$$
  • $$7$$
  • $$3$$
The sum of the coefficients of the middle terms of $$(1+{x})^{2{n}-1}$$ is
  • $$^{2n-1}C_{n}$$
  • $$^{2n-1}C_{n+1}$$
  • $$^{2n}C_{n-1}$$
  • $$^{2n}C_{n}$$
The coefficient of $$x^4$$ in $$\displaystyle \left ( \frac{x}{2} - \frac{3}{x^2} \right )^{10}$$ is
  • $$\displaystyle \frac{45}{64}$$
  • $$\displaystyle \frac{243}{128}$$
  • $$\displaystyle \frac{405}{256}$$
  • $$\displaystyle \frac{810}{512}$$
If the coefficients of $$r^{th}$$ term and $$(r+1)^{th}$$ term in the expansion of $$(1+x)^{20}$$ are in the ration 1 : 2, then $$r=$$
  • 6
  • 7
  • 8
  • 9
The coefficient of the $$8$$th term in the expansion of $$(1+x)^{10}$$ is
  • $$120$$
  • $$7$$
  • $$^{10}C_8$$
  • $$210$$
If $$T_r$$ denotes the rth term in the expansion of $$\displaystyle \left ( x+\frac{1}{y} \right)^{23}$$ then
  • $$T_{12}=T_{13}$$
  • $$x^2T_{13}=T_{12}$$
  • $$T_{12} = xy T_{13}$$
  • $$T_{12} + T_{12} = 25$$
The coefficient of $$x^3$$ in $$\displaystyle \left ( \sqrt{x^5}+ \frac{3}{\sqrt{x^3}} \right )^5$$ is
  • 0
  • 120
  • 420
  • 540
The coefficient of the middle term in the expansion of $$(1+x)^{2n}$$ is
  • $$^{2n}C_{n}$$
  • $$\displaystyle \frac{1.3.5\ldots..(2n-1)}{n!}2^{n}$$
  • $$2.6\ldots(4n-2)$$
  • $$2.4\ldots\ldots\ldots 2n$$

Assertion (A) : Number of the disimilar terms in the sum of expansion $$(x+a)^{102}+(x-a)^{102}$$ is $$206$$

Reason (R) : Number of terms in the expansion of $$(x+b)^{n}$$ is n + 1
  • Both A and R are individually true and R is the correct explanation of A.
  • Both A and R are individually true and R is not correct explanation of A.
  • A is true but R is false
  • A is false but R is true
In the binomial expansion of $$(a-b)^n, n \geq 5$$, the sum of $$5^{th}$$ and $$6^{th}$$ terms is zero, then $$\dfrac ab$$ equals
  • $$\displaystyle \frac{5}{n-4}$$
  • $$\displaystyle \frac{6}{n-5}$$
  • $$\displaystyle \frac{n-5}{6}$$
  • $$\displaystyle \frac{n-4}{5}$$
The total number of rational terms in the expansion of $$\left(7^{\frac 13} + 11^{\frac 19}\right)^{6561}$$ is
  • $$731$$
  • $$729$$
  • $$728$$
  • $$730$$
  • $$732$$
If the coefficient of $$x^7$$ in $$\displaystyle \left [ ax^2 + \left ( \dfrac{1}{bx} \right ) \right ]^{11}$$ equals the coefficient of $$x^{-7}$$ in $$\displaystyle \left [ ax - \left ( \dfrac{1}{bx^2} \right ) \right ]^{11}$$, then $$a$$ and $$b$$ satisfy the relation
  • $$a-b=1$$
  • $$a+b=1$$
  • $$\dfrac{a}{b}=1$$
  • $$ab=1$$
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