CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 9 - MCQExams.com

The graph of y = g(x) in its domain is broken at
  • 1 point
  • 2 point
  • 3 point
  • None of these
Let $$g(x) = f(x) - 1$$. If $$f(x) + f(1 - x) = 2 \space \forall \space x \space \epsilon \space R$$, then $$g(x)$$ is symmetrical about
  • Origin
  • The line $$x = \frac{1} {2}$$
  • The point $$\left (1, 0 \right )$$
  • The point $$\left (\frac {1} {2}, 0 \right )$$
Which of the following is not true about $$h_1(x)$$?
  • It is periodic function with period $$\pi$$
  • Range is [0,1]
  • Domain if R
  • None of these
g(f(x)) is not defined if
  • $$a\space \epsilon ( 10, \infty) b\space \epsilon (5, \infty)$$
  • $$a\space \epsilon ( 4, 10) b\space \epsilon (5, \infty)$$
  • $$a\space \epsilon ( 10, \infty) b\space \epsilon (0, 1)$$
  • $$a\space \epsilon ( 4, 10) b\space \epsilon (1, 5)$$
Let $$f(x)$$ and $$g(x)$$ be differentiable for $$0\times  < 1$$ such that $$f(0)=0, g(0), f(1)=6$$. Let there exist a real number $$c$$ in $$(0,1)$$ such that $$f'(c)=2g'(c)$$, then the value of $$g(1)$$ must be 
  • $$1$$
  • $$3$$
  • $$2.5$$
  • $$-1$$
For set $$A, B$$ and $$C$$, let $$f:A\to B, g:B\to C$$ be functions such that $$gof $$ is surjective.
Then $$g$$ is surjective function.
  • True
  • False
Let $$A$$ be a finite set. Then, each injective function from $$A$$ into itself is not surjective.
  • True
  • False
For set $$A, B$$ and $$C$$, let $$f:A\to B, g:B\to C$$ be functions such that $$gof $$ is Injective.
Then $$f$$ is injective.
  • True
  • False
If g is the inverse of function $$f$$ and $$f'(x) = \frac{1}{1 + x}$$, then the value of g'(x) is equal to:
  • $$1 + x^7$$
  • $$\frac{1}{1 + [g(x)]^7}$$
  • $$1 + [g(x)]^7$$
  • $$7x^6$$
Computers use
  • decimal system
  • binary system
  • base 5 system
  • base 6 system
From $$ ] \dfrac{- \pi}{2} , \dfrac{- \pi}{2}[ $$ which of the following is one - one onto function defined in R
  • $$ f(x) = \tan x $$
  • $$ f(x) = \sin x $$
  • $$ f(x) = \cos x $$
  • $$ f(x) = e^{x} + r^{-x}$$
Which of the following in one -one function defined from R to R
  • $$ f(x) = |x| $$
  • $$ f(x) = \cos x $$
  • $$ f(x) =e^{x}$$
  • $$ f(x) = x^{2} $$
Subtraction is an operation on $$Z$$, which is 
  • commutative and associative
  • associative but not commutative
  • neither commutative and nor associative
  • commutative but not associative
Let $$ f: [ -1,1] \rightarrow [0,2]$$ be a linear function which is onto then f(x) is/are 
  • $$ 1 - x $$
  • $$ 1 + x $$
  • $$ x - 1$$
  • $$ x - 2 $$
If $$n \geq 2$$ then the number of surjections that can be defined from $$\{1, 2, 3, .......  n\}$$ onto $$\{1, 2\}$$ is
  • $$2n$$
  • $$^nP_2$$
  • $$2^n$$
  • $$2^{n}-2$$
Let $$f : X\rightarrow Y, f (x) = \sin x+ \cos x + 2\sqrt2$$ is invertible, then $$X\rightarrow Y$$ is$$/$$are
  • $$\left [\displaystyle \frac{\pi }{4},\frac{5\pi}{4} \right ]\rightarrow \left [ \sqrt{2},3\sqrt{2} \right ]$$
  • $$\left [ -\displaystyle \frac{\pi }{4},\frac{3\pi}{4} \right ]\rightarrow \left [ \sqrt{2},3\sqrt{2} \right ]$$
  • $$\left [\displaystyle \frac{\pi }{4},\frac{3\pi}{4} \right ]\rightarrow \left [ \sqrt{2},3\sqrt{2} \right ]$$
  • $$\left [ -\displaystyle \frac{3\pi }{4},\frac{\pi}{4} \right ]\rightarrow \left [ \sqrt{2},3\sqrt{2} \right ]$$
If $$f(x) = x^{2}$$, $$-1\leq x \leq 4$$ ,$$ g(x) = sec^{-1}x$$, $$x\geq 1$$ then
  • Domain of gof(x) is $$[1, 4] \cup {-1}$$
  • Domain of gof(x) is $$[1, 4] $$
  • Range of gof(x) is $$[0, sec^{-1}16]$$
  • Range of fog(x) is $$\left ( 0,\frac{\pi^{2}}{4} \right )$$
If $$f(x) = x^3- 4x + 1 \forall x > 0$$ and $$g(x)$$ is image of $$f(x)$$ with respect to line $$y = x$$ then
  • $$g(1) = 2$$
  • $$g'(1)=\displaystyle \frac{1}{8}$$
  • $$f''(g(1))=12$$
  • $$g''(1)=-\displaystyle \frac{3}{16}$$
Consider the function $$f (x)=\dfrac{x}{x-1}.$$ Which of the following statements are correct ?

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  • $$f$$ has the same domain and range
  • $$f$$ has its own inverse
  • $$f$$ is not injective
  • $$f$$ is neither odd nor even
Let $$\displaystyle \mathrm{f}:\mathrm{R}\rightarrow \left[0,\frac{\pi}{2}\right)$$ be defined by $$\mathrm{f}(\mathrm{x})=\mathrm{t}\mathrm{a}\mathrm{n}^{-1}(\mathrm{x}^{2}+\mathrm{x}+\mathrm{a})$$. Then the set of values of '$$\mathrm{a}$$' for which $$\mathrm{f}$$ is onto is
  • $$[0,\infty)$$
  • $$[2, 1]$$
  • $$ \displaystyle \left\{ \frac{1}{4}\right\}$$
  • $$ \displaystyle \left[ \frac{1}{4}, \infty\right)$$
Let $$\displaystyle {f}({x})=\frac{{a}{x}+{b}}{{c}{x}+{d}}$$, then $$fof(x)={x}$$, provided
  • $$d = -a$$
  • $$d = a$$
  • $$a = b = c = d = 1$$
  • $$a = b = 1$$
Which of the following functions is/are injective map(s) ?
  • $$f(x)=x^2+2, x \in (-\infty,\infty)$$
  • $$f(x)=|x+2|, x \in [-2,\infty)$$
  • $$f(x)=(x-4)(x-5), x \in (-\infty,\infty)$$
  • $$f(x)=\dfrac{4x^2 + 3x -5}{4+3x-5x^2}, x\in(-\infty, \infty)$$
Which of the following functions is not injective ?
  • $$f:R \rightarrow R, f(x)=2x+7$$
  • $$f:[0,\pi]\rightarrow[-1,1],f(x)=\cos x$$
  • $$f:\left [ -\dfrac{\pi}{2},\dfrac{\pi}{2} \right ]\rightarrow R, f(x)=2 \sin x +3$$
  • $$f:R\rightarrow [-1,1],f(x)=\sin x$$
Let $$f(x)=max\left\{1+\sin x,1,1-\cos x \right\}, x\in \left [ 0,2\pi  \right ]$$  and $$g(x)=max\left\{ 1,\left | x-1 \right |\right\},x\in R$$ , then
  • $$g(f(0))=1$$
  • $$g(f(1))=1$$
  • $$f(g(1))=1$$
  • $$f(g(0))=\sin 1$$
If $$f\left( x \right)=\sqrt { 3\left| x \right| -x-2 } $$ and $$g(x)=\sin(x)$$, then the domain of the definition of $$f\circ g\left( x \right) $$ is
  • $$\displaystyle \left\{ 2n\pi +\frac { \pi  }{ 2 }  \right\} $$
  • $$\displaystyle \left( 2n\pi +\frac { 7\pi  }{ 6 } ,2n\pi +\frac { 11\pi  }{ 6 }  \right) $$
  • $$\displaystyle \left\{ 2n\pi +\frac { 7\pi  }{ 6 }  \right\} $$
  • $$\displaystyle \left( 2n\pi +\frac { 7\pi  }{ 6 } ,2n\pi +\frac { 11\pi  }{ 6 }  \right) \bigcup _{ n,m\in I }^{  }{ \left( 2n\pi +\frac { \pi  }{ 2 }  \right)  } $$
Let $$\displaystyle a > 1$$ be a real number and $$\displaystyle f\left ( x \right ) = \log _{a} x^{2}$$ for $$\displaystyle x > 0$$. If $$\displaystyle f^{-1}$$ is the inverse function of $$f$$ and $$b$$ and $$c$$ are real numbers then $$\displaystyle f^{-1} \left ( b+c \right )$$ is equal to
  • $$\displaystyle f^{-1}\left ( b \right ) . f^{-1}\left ( c \right )$$
  • $$\displaystyle f^{-1}\left ( b \right ) + f^{-1}\left ( c \right )$$
  • $$\displaystyle \frac {1}{f\left ( b+c \right )}$$
  • $$\displaystyle \frac {1}{f^{-1}\left ( b \right ) + f^{-1} \left ( c \right )}$$
Suppose $$f(x)=ax+b$$ and $$g(x)=bx+a$$, where $$a$$ and $$b$$ are positive integers. If  $$f\left ( g(50) \right )-g\left ( f(50) \right )=28$$ then the product $$(ab)$$ can have the value equal to
  • $$12$$
  • $$48$$
  • $$180$$
  • $$210$$
Which one of the following functions is not one-one?
  • $$f:(-1,\infty )\rightarrow R$$ given by $$ f(x)={ x }^{ 2 }+2x\quad $$
  • $$g:(2,\infty )\rightarrow R$$ given by $$g(x)={ e }^{ { x }^{ 3 }-3x+2 }\quad $$
  • $$h:R\rightarrow R$$ given by $$h(x)={ 2 }^{ { x }(x-1) }\quad $$
  • $$\phi :(-\infty ,0)\rightarrow R$$ given by $$\phi (x)=\cfrac { { x }^{ 2 } }{ { x }^{ 2 }+1 } $$
If $$f:R\rightarrow \left [\dfrac {\pi}{6}, \dfrac {\pi}{2}\right ), f(x)=\sin^{-1}\left (\dfrac {x^2-a}{x^2+1}\right )$$ is a onto function, then set of values of $$a$$ is
  • $$\left \{-\dfrac {1}{2}\right \}$$
  • $$\left [-\dfrac {1}{2}, -1\right )$$
  • $$(-1, \infty)$$
  • none of these
If $$f (x) = x + 2, g (x) = 2 x +3,$$ then find gof
  • $$2x -7 $$
  • $$7x + 2$$
  • $$2x + 7$$
  • $$7 + 2x$$
Which of the function defined below are one-one function(s)?
  • $$f(x)=x+1,(x\geq-1)$$
  • $$g(x)=x+\dfrac1x,(x\geq0)$$
  • $$h(x)=x^2+4x-5,(x>0)$$
  • $$f(x)=e^{-x},(x\geq0)$$
$$f(x)=x^3+3x^2+4x+b \sin x+c \cos x, \forall x\in R$$ is a one-one function, then the value of $$b^2+c^2$$ is
  • $$\geq 1$$
  • $$\geq 2$$
  • $$\leq 1$$
  • none of these
If $$f(x)=2x+|x|, g(x)=\dfrac {1}{3}(2x-|x|)$$ and $$h(x)=f(g(x))$$, then domain of $$\sin^{-1}\underset {\text {n times}}{\underbrace {(h(h(h(h.....h(x).....))))}}$$ is
  • $$[-1, 1]$$
  • $$\left [-1, -\dfrac {1}{2}\right ]\cup \left [\dfrac {1}{2}, 1\right ]$$
  • $$\left [-1, -\dfrac {1}{2}\right ]$$
  • $$\left [\dfrac {1}{2}, 1\right ]$$
Let $$f:{x, y, z}\rightarrow (a, b, c)$$ be a one-one function. It is known that only one of the following statements is true:
(i) $$f(x)\neq b$$
(ii)$$f(y)=b$$
(iii)$$f(z)\neq  a$$
  • $$f=\{(x, a), (y, b), (z, c)\}$$
  • $$f=\{(x, b), (y, a), (z, c)\}$$
  • $$f=\{(x, b), (y, c), (z, c)\}$$
  • $$f=\{(x, b), (y, c), (z, a)\}$$
The inverse of the function $$\displaystyle y= \frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}} $$ is
  • $$\displaystyle \log_{e}\left ( \frac{1+2x}{1-2x} \right )$$
  • $$\displaystyle \frac{1}{4}\log_{e}\left ( \frac{1-x}{1+x} \right )$$
  • $$\displaystyle \frac{1}{4}\log_{e}\left ( \frac{1+x}{1-x} \right )$$
  • an odd function
Let $$\displaystyle f(x)=\begin{cases}x^{2} & \mbox{if}  \quad0< x< 2\\2x-3 & \mbox{if}  \quad2\leq x< 3 \\ x+2 & \mbox{if}\quad  x\geq 3\end{cases}$$.
Then 
  • $$\displaystyle f\left \{ f\left ( f\left ( \frac{3}{2} \right ) \right ) \right \}=f\left ( \frac{3}{2} \right )$$
  • $$\displaystyle 1+f\left \{ f\left ( f\left ( \frac{5}{2} \right ) \right ) \right \}=f\left ( \frac{5}{2} \right )$$
  • $$\displaystyle f\left \{ f(0) \right \}=f\left ( 1 \right )=1$$
  • none of these
The function $$f$$ is one to one and the sum of all the intercepts of the graph is $$5$$. The sum of all the intercept of the graph $$\displaystyle y = f^{-1} \left ( x \right )$$ is
  • $$5$$
  • $$\dfrac15$$
  • $$\dfrac25$$
  • $$-5$$
Suppose $$\displaystyle  f\left ( x \right )=\left ( x+1 \right )^{2}$$ for $$\displaystyle x\geq -1$$. If $$\displaystyle g(x)$$ is the function whose graph is the reflection of the graph of $$\displaystyle f(x)$$ with respect to the line $$y=x $$, then $$\displaystyle g(x)$$ is equal to
  • $$\displaystyle -\sqrt{x}-1$$ for $$\displaystyle x\geq 0$$
  • $$\displaystyle \frac{1}{\left ( x+1 \right )^{2}}$$ for $$\displaystyle x > -1$$
  • $$\displaystyle \sqrt{x+1}$$ for $$\displaystyle x > -1$$
  • $$\displaystyle \sqrt{x}-1$$ for $$\displaystyle x\geq 0$$
Let $$f\left( x \right) =\left\{ \begin{matrix} 1+|x|,\; x<-1 \\ \left[ x \right] ,\; x\ge -1 \end{matrix} \right. $$ where $$[\cdot]$$ denotes the greatest integer function. Then $$\displaystyle f\left \{f(-2.3) \right\}$$ is equal to 
  • $$4$$
  • $$2$$
  • $$-3$$
  • $$3$$
The inverse function of the function $$\displaystyle f(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$ is 
  • $$\displaystyle \frac{1}{2}\log\frac{1+x}{1-x}$$
  • $$\displaystyle \frac{1}{2}\log\frac{2+x}{2-x}$$
  • $$\displaystyle \frac{1}{2}\log\frac{1-x}{1+x}$$
  • none of these
If $$\displaystyle f \left ( x \right ) = px + q$$ and $$\displaystyle f \left ( f\left ( f\left ( x \right ) \right ) \right ) = 8x + 21$$, where $$p$$ and $$q$$ are real numbers, the $$ p + q$$ equals
  • $$3$$
  • $$5$$
  • $$7$$
  • $$11$$
$$K(x)$$ is a function such that $$K(f(x))=a+b+c+d$$,
Where,
$$a=\begin{cases}
0 & \text{ if f(x) is even}  \\ 
-1 & \text{ if f(x) is odd} \\ 
2 & \text{ if f(x) is neither even nor odd} 
\end{cases}$$
$$b=\begin{cases}
3 & \text{ if  f(x) is periodic} \\ 
4 & \text{  if  f(x) is  aperiodic}
\end{cases}$$
$$c=\begin{cases}
5 & \text{ if  f(x) is  one one} \\ 
6 & \text{  if  f(x) is many one}
\end{cases}$$
$$d=\begin{cases}
7 & \text{ if  f(x) is onto} \\ 
8 & \text{  if  f(x) is into}
\end{cases}$$ 
$$h:R\rightarrow R,h(x)=\left ( \displaystyle \frac{e^{2x}+e^{x}+1}{e^{2x}-e^{x}+1} \right )$$ 

On the basis of above information, answer the following questions.$$K(\phi(x)) $$
  • $$15$$
  • $$16$$
  • $$17$$
  • $$18$$
The total number of injective mappings from a set with $$m$$ elements to a set with $$n$$ elements, $$m \leq n $$ is 
  • $$\displaystyle m^{n}$$
  • $$\displaystyle n^{m}$$
  • $$\displaystyle \frac{n!}{(n-m)!}$$
  • $$n!$$
If $$f:R\rightarrow R$$ and $$g:R\rightarrow R$$ are given by $$f(x)=|x|$$ and $$g(x)=[x]$$ for each $$x\in R,$$ then $$\left\{ x\in R:g\left( f\left( x \right) \right) \le f\left( g\left( x \right) \right)  \right\} =$$
  • $$Z\cup \left( -\infty ,0 \right) $$
  • $$\left( -\infty ,0 \right) $$
  • $$Z$$
  • $$R$$
Let $$\displaystyle f:\left[-\frac{\pi}{3},\frac{2\pi}{3}\right]\rightarrow [0,4]$$ be a function defined by $$\displaystyle f(x)=\sqrt 3 \sin x-\cos x+2$$ then $$\displaystyle f^{-1}(x)$$ equals
  • $$\displaystyle \frac{2\pi}{3}-\cos ^{-1}\left(\frac{x-2}{2}\right)$$
  • $$\displaystyle \sin ^{-1}\left ( \frac{x-2}{2} \right )+\frac{\pi}{3}$$
  • $$\displaystyle \sin ^{-1}\left ( \frac{x-2}{2} \right )$$
  • $$\displaystyle \sin ^{-1}\left ( \frac{x+2}{2} \right )+\frac{\pi}{6}$$
Let $$f$$ and $$g$$ be increasing and decreasing functions respectively from $$\displaystyle \left ( 0,\infty  \right )$$ to $$\left ( 0,\infty  \right )$$ and let $$h\left ( x \right )=f\left [ g\left ( x \right ) \right ]$$. If $$h\left ( 0 \right )=0$$ then $$ h\left ( x \right )-h\left ( 1 \right )$$ is
  • always zero
  • always negative
  • always positive
  • strictly increasing
  • None of these
If $$\displaystyle f\left ( x \right )= \frac{3x+2}{5x-3}$$ , then
  • $$\displaystyle f\left ( f{}'\left ( x \right ) \right )=f^{-1}\left ( x \right )$$
  • $$\displaystyle f^{-1}\left ( x \right )=f\left ( x \right )$$
  • $$\displaystyle f\left ( f{}'\left ( x \right ) \right )=f'\left ( x \right )$$
  • $$\displaystyle f\left ( f\left ( x \right ) \right )=-x^{2}$$
If $$\displaystyle f\left ( x \right )+f\left ( \dfrac1x \right )=0,\; f(e)=1; g\left ( x \right )=f^{-1}\left ( x \right )$$ then $$\displaystyle g{}'\left ( x \right )$$ equals
  • $$\displaystyle e^{x}$$
  • $$\displaystyle x $$
  • $$\displaystyle x^{2}$$
  • $$\displaystyle e^{-x}$$
If $$f:[1,\infty )\rightarrow [2,\infty )$$ is given by $$\displaystyle f\left( x \right)=x+\frac { 1 }{ x } ,$$ then $$f^{-1}(x)$$ equals
  • $$\displaystyle \frac { x+\sqrt { { x }^{ 2 }-4 }  }{ 2 } $$
  • $$\displaystyle \frac { x }{ 1+{ x }^{ 2 } } $$
  • $$\displaystyle \frac { x-\sqrt { { x }^{ 2 }-4 }  }{ 2 } $$
  • $$1+\sqrt{x^2-4}$$
Let $$\displaystyle f\left ( x \right )=\frac{ax^{2}+2x+1}{2x^{2}-2x+1}$$, the value of $$a$$ for which $$\displaystyle f:R\rightarrow \left [ -1,2 \right ]$$ is onto , is
  • $$\displaystyle \left [ 2,5 \right ]$$
  • $$\displaystyle \left [ -5,-2 \right ]$$
  • $$\displaystyle \left [ 0,5 \right ]$$
  • None of these.
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