CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 3 - MCQExams.com

Let $$S$$ be set of all rational numbers. The functions $$f:R\rightarrow R,\ g:R\rightarrow R$$ are defined as 
$$f(x)=\begin{cases}
0, & x \in S \\ 
1, & x \notin S
\end{cases}$$
$$g(x)=\begin{cases}
-1 & x\in S \\ 
 0 & x\notin S
\end{cases}$$
then, $$(fog) (\pi)+(gof)(e)=$$
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
Set $$A$$ has $$n$$ elements. The number of functions that can be defined from $$A$$ into $$A$$ is:
  • $$n^2$$
  • $$n!$$
  • $$n^n$$
  • $$n$$
If $$n\geq 1$$ is any integer, $$\mathrm{d}(n)$$ denotes the number of positive factors of $$n$$, then for any prime number $$\mathrm{p},\ \mathrm{d}(\mathrm{d}(\mathrm{d}(\mathrm{p}^{7})))=$$
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
Let $$\displaystyle f\left( x \right)={ x }^{ 2 }-x+1,x\ge \left( \frac { 1 }{ 2 }  \right) $$ then the solution of the equation $$f(x)=f^{-1}(x)$$ is
  • $$x=1$$
  • $$x=2$$
  • $$\displaystyle x=\frac{1}{2}$$
  • None of these
lf $$f:[-6,6]\rightarrow \mathbb{R}$$ is defined by $$f(x)=x^{2}-3$$ for $$x\in \mathbb{R}$$ then
$$(fofof)(-1)+(fofof)(0)+(fofof)(1)=$$
  • $$f(4\sqrt{2})$$
  • $$f(3\sqrt{2})$$
  • $$f(2\sqrt{2})$$
  • $$f(\sqrt{2})$$
lf $$f$$ : $$R\rightarrow R$$ is defined by
$$f(x)=\left\{\begin{array}{l}x+4 & x<-4\\3x+2 & -4\leq x<4\\x-4 & x\geq 4\end{array}\right.$$
then the correct matching of list I to List II is. 
List - IList - II
$$\mathrm{A}) f(-5)+f(-4)=$$$$\mathrm{i}) 14$$
$$\mathrm{B}) f(|f(-8)|)=$$ii $$) 4$$
$$\mathrm{C}) f(f(-7)+f(3))=$$$$\mathrm{i}\mathrm{i}\mathrm{i})-11$$
$$\mathrm{D}) f(f(f(f(0)))+1=$$$$\mathrm{i}\mathrm{v})-1$$
v) $$1$$
vi) $$0$$
  • A-iii , B-vi , C-ii , D- v
  • A-iii , B-iv , C-ii , D- vi
  • A-iv , B-iii , C-ii , D- i
  • A-ii , B-vi , C-v , D- ii

lf $$g(f(x)) =|\sin \mathrm{x}|,f(g(x)) =(\sin\sqrt{\mathrm{x}})^{2}$$, then
  • $${f}({x})=\sin^{2} {x},{g}({x})=\sqrt{{x}}$$
  • $${f}({x})=\sin x,g({x})=|{x}|$$
  • $${f}({x})={x}^{2},{g}({x})=\sin\sqrt{{x}}$$
  • $${f}, {g}$$ cannot be determined

lf $$ f(x)=x-x^{2}+x^{3}-x^{4}+\ldots..\infty$$ when $$|x|<1$$, then the ascending order of the following is
a) $$f(1/2)$$
b) $$f^{-1}(1/2)$$
c) $$ f(-1/2)$$
d) $$f^{-1}(-1/2)$$
  • a, b, c, d
  • c, d, a, b
  • b, a, d, c
  • d, c, a, b
Let $$f$$ be an injective function with domain $$\{x, y, z\}$$and range $$\{1,2,3\}$$ such that exactly one of the follwowing statements is correct and the remaining are false :
$${f}({x})=1,{f}({y})\neq 1$$,
$${f}({z})\neq 2$$, 
then the value of $${f}^{-1}(1)$$ is
  • $$x$$
  • $$y$$
  • $$z$$
  • none
If $$ f : R \rightarrow R$$ is defined by $$f(x)=2x-2,$$  then $$(f\circ f) (x) + 2 =$$
  • $$f(x)$$
  • $$2f(x)$$
  • $$3f(x)$$
  • $$-f(x)$$
If $$f(x) =\displaystyle \frac{x}{\sqrt{1-x^2}}, g(x)=\frac{x}{\sqrt{1+x^2}}$$ then $$(f\circ g)(x) =$$
  • $$\displaystyle \frac{x}{\sqrt{1-x^2}}$$
  • $$\displaystyle \frac{x}{\sqrt{1+x^2}}$$
  • $$\displaystyle \frac{1-x^2}{\sqrt{1-x^2}}$$
  • $$x$$
If $$f(x)=\log  x,  g(x) = x^3$$ then $$f[g(a)]+f[g(b)]= $$
  • $$f[g(a)+g(b)]$$
  • $$f[g(ab)]$$
  • $$g[f(ab)]$$
  • $$g[f(a)+f(b)]$$
If $$X=\left \{ 1, 2, 3, 4, 5 \right \}$$ and $$Y=\left \{ 1, 3, 5, 7, 9 \right \}$$, determine which of the following sets represent a relation and also a mapping.
  • $$R_{1}=\left \{ (x, y):y=x+2, x\in X, y\in Y \right \}$$
  • $$R_{2}=\left \{ (1, 1),(2, 1),(3, 3),(4, 3),(5, 5) \right \}$$
  • $$R_{3}=\left \{ (1, 1),(1, 3),(3, 5),(3, 7),(5, 7) \right \}$$
  • $$R_{4}=\left \{ (1, 3),(2, 5),(4, 7),(5, 9),(3, 1) \right \}$$
If $$f:R \rightarrow R$$ and $$g : R \rightarrow R$$ are defined by $$f(x)=2x+3$$ and $$g(x)=x^2+7$$, then the values of $$x$$ such that $$g(f(x)) =8$$ are:
  • $$1, 2$$
  • $$-1, 2$$
  • $$-1, -2$$
  • $$1, -2$$
If $$f(x) = \dfrac{2x+5}{x^{2} + x + 5}$$, then $$f\left [ f(- 1 ) \right ]$$ is equal to
  • $$\dfrac{149}{155}$$
  • $$\dfrac{155}{147}$$
  • $$\dfrac{155}{149}$$
  • $$\dfrac{147}{155}$$
Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A into B is :
  • 144
  • 12
  • 24
  • 64
If $$ABC\sim PQR$$, then AB:PQ =
  • $$AC:PB$$
  • $$AC:PR$$
  • $$AB:PR$$
  • $$AC:RQ$$
If $$f : R \rightarrow R$$ and $$g :R \rightarrow R$$ are defined by $$f(x) = x -[x]$$ and $$g(x) = [x]$$ for $$x \in R$$, where $$[x]$$ is the greatest integer not exceeding $$x$$, then for every $$x \in R, f(g(x)) =$$
  • $$x$$
  • $$0$$
  • $$f(x)$$
  • $$g(x)$$
If $$y=f(x) = \dfrac{2x-1}{x-2}$$, then $$f(y)=$$
  • $$x$$
  • $$y$$
  • $$2y-1$$
  • $$y-2$$
If $$f(g(x))$$ is one-one function, then
  • g(x) must be one-one
  • f(x) must be one-one
  • f(x) may not be one-one
  • g(x) may not be one-one
Which of the following functions are one-one?
  • $$f:R\rightarrow R$$ given by $$ f(x)={ 2x }^{ 2 }+1$$ for all $$\quad x\in R$$
  • $$g:Z\rightarrow Z$$ given by $$ g(x)={ x }^{ 4 }$$ for all $$\quad x\in R$$
  • $$h:R\rightarrow R$$ given by $$ h(x)={ x }^{ 3 }+4$$ for all $$\quad x\in R$$
  • $$\phi :C\rightarrow C$$ given $$ \phi (z)={ 2z }^{ 6 }+4$$ for all $$\quad x\in R$$
A mapping function $$f:X\rightarrow Y$$ is one-one, if
  • $$f({ x }_{ 1 })\neq f({ x }_{ 2 })\ $$for all $$ { x }_{ 1 },{ x }_{ 2 }\in X$$
  • $$f({ x }_{ 1 })=f({ x }_{ 2 })\Rightarrow { x }_{ 1 }={ x }_{ 2 }$$ for all $${ x }_{ 1 },{ x }_{ 2 }\in X$$
  • $${ x }_{ 1 }={ x }_{ 2 }\Rightarrow f({ x }_{ 1 })=f({ x }_{ 2 })$$ for all $${ x }_{ 1 },{ x }_{ 2 }\in X$$
  • none of these
Let $$\displaystyle f:R\rightarrow A=\left \{ y: 0\leq y< \dfrac{\pi}{2} \right \}$$ be a function such that $$\displaystyle f(x)=\tan^{-1}(x^{2}+x+k),$$ where $$k$$ is a constant. The value of $$k$$ for which $$f$$ is an onto function is 
  • $$1$$
  • $$0$$
  • $$\displaystyle \frac{1}{4}$$
  • none of these
If $$f:R\rightarrow R$$ given by $$f(x)={ x }^{ 3 }+({ a+2)x }^{ 2 }+3ax+5$$ is one-one, then $$a$$ belongs to the interval
  • $$(-\infty ,1)$$
  • $$(1 ,\infty)$$
  • $$(1 ,4)$$
  • $$(4 ,\infty)$$
Let R be the relation in the set N given by $$=\left \{ (a, b):a=b-2, b >6 \right \}$$. Choose the correct answer.
  • $$(2, 4)\in R$$
  • $$(3, 8)\in R$$
  • $$(6, 8)\in R$$
  • $$(8, 7)\in R$$
Let $$\displaystyle f:\left \{ x,y,z \right \}\rightarrow \left \{ a,b,c \right \}$$ be a one-one function and only one of the conditions $$(i)f(x)\neq b, (ii)f(y)=b,(iii)f(z)\neq a$$ is true then the function $$f$$  is given by the set 
  • $$\displaystyle \left \{ (x,a),(y,b),(z,c)\right \}$$
  • $$\displaystyle \left \{ (x,a),(y,c),(z,b)\right \}$$
  • $$\displaystyle \left \{ (x,b),(y,a),(z,c)\right \}$$
  • $$\displaystyle \left \{ (x,c),(y,b),(z,a)\right \}$$
Which of the following function is one-one?
  • $$f:R\rightarrow R$$ given by$$ f(x)=|x-1|$$ for all $$x\in R$$
  • $$g:\left[ -\dfrac{\pi }{ 2 },\dfrac{ \pi }{ 2 } \right] \rightarrow R$$ given by $$g(x)=|sinx|$$ for all $$ x\in \left[ \dfrac{ -\pi }{ 2 },\dfrac{ \pi }{ 2 } \right] $$
  • $$h:\left[ \dfrac{ -\pi }{ 2 },\dfrac{ \pi }{ 2 } \right] \in R$$ given by $$ h(x)=sinx$$ for all $$ x\in \left[ \dfrac{ -\pi }{ 2 },\dfrac{ \pi }{ 2 } \right] $$
  • $$\phi :R\rightarrow R$$ given by $$f(x)={ x }^{ 2 }-4$$ for all $$ x\in R$$
If $$f$$ and $$g$$ are one-one functions from $$R\to R$$, then
  • $$f+g$$ is one-one
  • $$fg$$ is one-one
  • $$fog$$ is one-one
  • none of these
If $$f:R^+\rightarrow A$$, Where $$A=\{x:-5<x<\infty\}$$ be defined by $$f(x)=x^2-5$$. Then $$f^{-1}(7)=$$
  • only $$-2{\sqrt3}$$
  • only $$2{\sqrt3}$$
  • both options A and B
  • none
If the function $$f:[1,\infty)\rightarrow [1,\infty)$$ is defined by $$\displaystyle f(x)=3^{x(x-1)}$$ then $$f^{-1}(x) $$ is 
  • $$\displaystyle \left ( \frac{1}{2} \right )^{x(x-1)}$$
  • $$\displaystyle \frac{1}{2}\left ( 1+\sqrt{1+4\log_{3}x}\right)$$
  • $$\displaystyle \frac{1}{2}\left ( 1-\sqrt{1+4\log_{3}x}\right)$$
  • not defined
If the function $$f:R\rightarrow R$$ be such that $$\displaystyle f(x)=x-[x],$$ where $$[y]$$ denotes the greatest integer less than or equal to $$y$$, then $$f^{-1}(x)$$ is
  • $$\displaystyle \frac{1}{x-[x]}$$
  • $$\displaystyle [x]-x$$
  • not defined
  • none of these
If $$\displaystyle f:\left ( 3,4 \right )\rightarrow \left ( 0,1 \right )$$ is defined by $$\displaystyle f\left ( x \right )=x-\left [ x \right ]$$ where $$\displaystyle [x] $$denotes the greatest integer function then $$ f^{-1}(x)$$ is
  • $$\displaystyle \frac{1}{x-\left [ x \right ]}$$
  • $$\displaystyle [x]-x$$
  • $$\displaystyle x-3$$
  • $$\displaystyle x+3$$
Let $$\displaystyle f:(-\infty,1]\rightarrow (-\infty,1]$$ such that $$\displaystyle f(x)=x(2-x).$$ Then $$f^{-1}(x)$$ is
  • $$1+\sqrt{1-x}$$
  • $$1-\sqrt{1-x}$$
  • $$\sqrt{1-x}$$
  • none of these
If $$f(x)=ax+b$$ and $$g(x)=cx+d$$, then $$f(g(x))=g(f(x))$$ implies
  • $$f(a)=g(c)$$
  • $$f(b)=g(b)$$
  • $$f(d)=g(b)$$
  • $$f(c)=g(a)$$
If $$\displaystyle f(x)=\frac{1}{1-x},x\neq 0,1$$ then the graph of the function $$\displaystyle y=f\left \{ f(f(x)) \right \},x> 1,$$ is
  • a circle
  • an ellipse
  • a straight line
  • a pair of straight lines
If $$\displaystyle f(x)=3x-5$$ then $$f^{-1}(x)=$$
  • $$\displaystyle \frac{1}{3x-5}$$
  • $$\displaystyle \frac{x+5}{3}$$
  • does not exist because $$f$$ is not one-one
  • does not exist because $$f$$ is not onto
If $$f$$ and $$g$$ are two functions such that  $$\displaystyle \left ( fg \right )\left ( x \right )=\left ( gf \right )\left ( x \right )$$ for all $$x$$. Then $$f $$ and $$g$$ may be defined as
  • $$\displaystyle f\left ( x \right )=\sqrt{x}, g\left ( x \right )=\cos x$$
  • $$\displaystyle f\left ( x \right )=x^{3}, g\left ( x \right )=x+1$$
  • $$\displaystyle f\left ( x \right )=x-1, g\left ( x \right )=x^{2}+1$$
  • $$\displaystyle f\left ( x \right )=x^{m}, g\left ( x \right )=x^{n}$$ where $$m, n$$ are unequal integers
If $$\displaystyle f(x)=x^{n},n\in N$$ and $$(gof)(x)=ng(x)$$ then $$g(x)$$ can be 
  • $$n\:|x|$$
  • $$3.\sqrt[3]{x}$$
  • $$e^{x}$$
  • $$\log\:|x|$$
The composite mapping $$fog$$ of the map $$f: R\rightarrow R,f(x)=\sin x$$ and $$g: R\rightarrow R, g(x)=x^2$$ is
  • $$x^2 \sin x$$
  • $$(\sin x)^2$$
  • $$\sin x^2$$
  • $$\dfrac{ \sin x}{x^2}$$
Let $$S$$ be a set containing $$n$$ elements. Then the total number of binary operations on $$S$$ is
  • $$n^n$$
  • $${2^n}^{2}$$
  • $$n^{n^2}$$
  • $$n^2$$
Let $$f: R\rightarrow R$$ be defined by $$f(x)=3x-4$$ then $$f^{-1}(x)$$ is
  • $$\dfrac{1}{3}(x+4)$$
  • $$\dfrac{1}{3}(x-4)$$
  • $$3x+4$$
  • not defined
Let $$\displaystyle f(x)=\frac{ax}{x+1}$$, where $$\displaystyle x\neq -1$$. Then for what value of $$\displaystyle a$$ is $$\displaystyle f( f(x))=x$$ always true
  • $$\displaystyle \sqrt{2}$$
  • $$\displaystyle -\sqrt{2}$$
  • $$1$$
  • $$-1$$
A function $$y=f\left ( x \right )$$ is invertible only when
  • $$y=f\left ( x \right )$$ is monotonic increasing
  • $$y=f\left ( x \right )$$ is bijective
  • $$y=f\left ( x \right )$$ is monotonic decreasing
  • invertible
Let $$ f:R \rightarrow R$$ and $$g:R \rightarrow R$$ be defined by $$f(x)=x^2+2x-3,g(x)=3x-4$$ then $$(gof) (x)=$$
  • $$3x^2+6x-13$$
  • $$3x^2-6x-13$$
  • $$3x^2+6x+13$$
  • $$-3x^2+6x-13$$
If $$\displaystyle f:[1,+\infty ]\rightarrow [2,+\infty )$$ is given by $$f(x)=x+\dfrac{1}{x}$$  then $$f^{-1}(x)$$ equals
  • $$\displaystyle \frac{x+\sqrt{x^{2}+4}}{2}$$
  • $$\displaystyle \frac{x}{1+x^{2}}$$
  • $$\displaystyle \frac{x+\sqrt{x^{2}-4}}{2}$$
  • $$\displaystyle 1+\sqrt{x^{2}-4}$$
$$f:R\rightarrow R$$ is a function defined by $$f(x)=10x-7$$. If $$g=f^{-1}$$, then $$g(x)$$ is equals 
  • $$\dfrac{1}{10x-7}$$
  • $$\dfrac{1}{10x+7}$$
  • $$\dfrac{x+7}{10}$$
  • $$\dfrac{x-7}{10}$$
Set $$A$$ has $$3$$ elements and set $$B$$ has $$4$$ elements. The number of injections that can be defined from $$A$$ to $$B$$ is
  • $$144$$
  • $$12$$
  • $$24$$
  • $$64$$
If $$f:R\rightarrow R$$ such that $$f(x)=log_3 x$$ then $$f^{-1} x$$ is equal to
  • $$log x^3$$
  • $$3^x$$
  • $$3^{-x}$$
  • $$3^{1/x}$$
The inverse of $$\displaystyle f\left ( x \right )=\frac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$$ is
  • $$\displaystyle \frac{1}{6}\log_{10}\left ( \frac{1+x}{1-x} \right )$$
  • $$\displaystyle \frac{1}{6}\log_{10}\left ( \frac{x}{1-x} \right )$$
  • $$\displaystyle \frac{1}{6}\log_{e}\left ( \frac{1+x}{1-x} \right )$$
  • $$\displaystyle \frac{1}{6}\log_{e}\left ( \frac{1-x}{1+x} \right )$$
If $$\displaystyle X= \left \{ 1,2,3,4,5 \right \}$$ and $$\displaystyle Y= \left \{ 1,3,5,7,9 \right \}$$ then which of the following sets are relation from $$X$$ to $$Y$$
  • $$\displaystyle R_{1}= \left \{ (x,a):a= x+2,x\in X,a\in Y \right \} $$
  • $$\displaystyle R_{2}= \left \{ (1,1),(2,1),(3,3),(4,3),(5,5) \right \} $$
  • $$\displaystyle R_{3}= \left \{ (1,1),(1,3),(3,5),(3,7),(5,7) \right \} $$
  • $$\displaystyle R_{4}= \left \{ (1,3),(2,5),(4,7),(5,9),(3,1) \right \} $$
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