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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:RR, g:RR are defined as 
f(x)={0,xS1,xS
g(x)={1xS0xS
then, (fog)(π)+(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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