CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 2 - MCQExams.com

If $$n (A) = 4$$ and $$n(B) = 6$$, then the number of surjections from $$A$$ to $$B$$ is
  • $$4^{6}$$
  • $$6^{4}$$
  • $$0$$
  • $$24$$
The number of injections that are possible from $$A$$ to itself is $$720,$$ then $$n (A) =$$
  • $$5$$
  • $$6$$
  • $$7$$
  • $$8$$
Let $$A=\{1,2,3\}, B =\{a, b, c\}$$ and If $$f=\{(1,a),(2,b),(3,c)\}, g=\{(1,b),(2,a),(3,b)\}, h=\{(1,b)(2,c),(3,a)\}$$ then
  • $$g$$ and $$h$$ are injections
  • $$f$$ and $$h$$ are injections
  • $$f$$ and $$g$$ injections
  • $$f,g$$ and $$h$$ are injections
The number of one-one functions that can be defined from $$A = \left \{ 1,2,3 \right \} $$ to $$  B = \left \{ a,e,i,o,u \right \}$$ is 
  • $$3^{5}$$
  • $$5^{3}$$
  • $${_{}}^{5}P_{3}$$
  • $$5!$$
The number of non-surjective mappings that can be defined from $$A = \left \{ 1,4,9,16 \right \}  $$ to$$  B=\left \{ 2,8,16,32,64 \right \}$$ is
  • $$1024$$
  • $$20$$
  • $$505$$
  • $$625$$
If $$ f:A\rightarrow B $$ is a constant function which is onto then $$B$$ is
  • a singleton set
  • a null set
  • an infinite set
  • a finite set
If $$ f:A\rightarrow B $$ is a bijection then $$ f^{-1} of = $$
  • $$fof^{-1}$$
  • $$f$$
  • $$f^{-1}$$
  • an identity
The number of injections possible from $$A=\{1,3,5,6\}$$ to $$B =\{2,8,11\}$$ is
  • $$8$$
  • $$64$$
  • $$2^{12}$$
  • $$0$$
The number of possible surjection from $$A=\{1,2,3,...n\}$$ to $$B = \{1,2\}$$ (where $$n \geq 2)$$ is $$62$$, then $$n=$$
  • $$5$$
  • $$6$$
  • $$7$$
  • $$8$$
If $$f:R\rightarrow R, g:R\rightarrow R$$ are defined by $$f(x)=x^{2}, g(x)=\cos x$$  then $$(gof)(x)=$$
  • $$\cos 2x$$
  • $$x^{2}\cos x$$
  • $$\cos x^{2}$$
  • $$\cos^{2} x^{2}$$
If $$f:R\rightarrow R $$ is defined by $$\displaystyle f(x)={\dfrac{2x+1}{3}}$$  then $$f^{-1}(x)=$$
  • $$\dfrac{3x-1}{2}$$
  • $$\dfrac{x-3}{2}$$
  • $$\dfrac{2x-1}{3}$$
  • $$\dfrac{x-4}{3}$$
Let $$f(x)=\dfrac{Kx}{x+1}(x\neq -1)$$ then the value of $$K$$ for which $$(fof)(x)=x$$ is
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$\sqrt{2}$$
$$f:\left ( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right )\rightarrow \left ( -\infty ,\infty  \right )$$ defined by $$f(x)=1+3x$$ is
  • one-one but not onto
  • onto but not one-one
  • neither one - one nor onto
  • bijective
If $$f:R\rightarrow R, g:R\rightarrow R$$ are defined by $$f(x)=4x-1,g(x)=x^{3}+2,$$ then $$(gof)\left(\dfrac{a+1}{4}\right)=$$ 
  • $$43$$
  • $$4a^3-1$$
  • $$a^{3}+2$$
  • $$64a^3 - 8a^{2}-1$$
The function $$f:(0,\infty )\rightarrow (-\infty ,\infty )$$ is defined by $$ f(x)=\log_{3} x $$ then $$ f^{-1}(x)=$$
  • $$3^{x}$$
  • $$3^{-x}$$
  • $$-3^{x}$$
  • $$-3x^{-x}$$
If $$f:R\rightarrow R,f(x)=3x-2$$ then $$ (fof)(x)+2=$$
  • $$f(x)$$
  • $$2f(x)$$
  • $$3f(x)$$
  • $$-f(x)$$
If $$f(x)=2x+1$$ and $$g(x)=x^{2}+1$$ then $$ (go(fof))(2)=$$
  • $$112$$
  • $$122$$
  • $$12$$
  • $$124$$
If $$f(x)=\dfrac{1}{x}, g(x)=\sqrt{x}$$  and $$ (go\sqrt{f})(16)=$$
  • $$2$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$4$$
If $$f(x)=x, g(x)=2x^{2}+1$$ and $$h(x)=x+1$$  then  $$(hogof)(x)$$ is equal to
  • $$x^{2}+2$$
  • $$2x^{2}+1$$
  • $$x^{2}+1$$
  • $$2(x^{2}+1)$$
If $$f(x)=\dfrac{e^{x}+e^{-x}}{2}$$, then the inverse of $$f(x)$$ is
  • $$\log_{e}(x+\sqrt{x^{2}+1})$$
  • $$\log_{e}\sqrt{x^{2}-1}$$
  • $$\log_{e}\left(\dfrac {x+\sqrt{x^{2}-1}}{2}\right)$$
  • $$\log_{e}(x+\sqrt{x^{2}-1})$$
If $$f:(-\infty ,\infty )\rightarrow (-\infty ,\infty )$$ is defined by $$f(x)=5x-6$$, then $$f^{-1}(x)=$$
  • $$\dfrac{x+5}{6}$$
  • $$\dfrac{x-5}{6}$$
  • $$\dfrac{x-6}{5}$$
  • $$\dfrac{x+6}{5}$$
If $$f(x)=\dfrac{5x+6}{7x+9}$$ then $$f^{-1}(x)=$$
  • $$\dfrac{y+6}{7y+9}$$
  • $$\dfrac{7y+9}{5y+6}$$
  • $$\dfrac{9y-6}{-7y+9}$$
  • $$\dfrac{9y-6}{-7y+5}$$
If $$f$$ from $$R$$ into $$R$$ is defined by $$f(x)=x^{3}-1$$, then $$f^{-1}\left \{ -2,0,7 \right \}=$$
  • $$\left \{ -1,1,2 \right \}$$
  • $$\left \{ 0,1,2 \right \}$$
  • $$\left \{ \pm 1,\pm 2 \right \}$$
  • $$\left \{ 0,\pm 2 \right \}$$
If $$f(x)=3x-1$$ and $$g(x)=5x+6$$ then $$(g^{-1}of^{-1})(2)=$$
  • $$10$$
  • $$-1$$
  • $$11$$
  • $$12$$
If $$f(x)=e^{5x+13}$$  then $$f^{-1}(x)=$$
  • $$\dfrac{13-\log y}{5}$$
  • $$\dfrac{-13+\log y}{5}$$
  • $$\dfrac{5+\log y}{13}$$
  • $$\dfrac{5-\log y}{13}$$
If $$f:[1,\infty )\rightarrow [2,\infty) $$ is given by $$f(x)=x+\dfrac{1}{x}$$, then $$f^{-1}(x)=$$
  • $$\dfrac{x+\sqrt{x^{2}-4}}{2}$$
  • $$\dfrac{x}{1+x^{2}}$$
  • $$\dfrac{x-\sqrt{x^{2}-4}}{2}$$
  • $$x+\sqrt{x^{2}-4}$$
If $$f:\left \{ 1,2,3,..... \right \}\rightarrow \left \{ 0,\pm 1,\pm 2,..... \right \}$$ is defined by  $$f(n)=\begin{cases} \dfrac{n}{2} & \text{ if } n  \space is \space even \\-\left (\dfrac{n-1}{2} \right ) & \text{ if } n \space is \space  odd \end{cases}$$ then  $$f^{-1}(-100)$$ is
  • Function is not invertible.
  • $$199$$
  • $$201$$
  • $$200$$
$$f:R\rightarrow R$$ is defined by $$f(x)=x^{2}+4$$ then $$f^{-1}(13)=$$
  • $$\left \{ -3,3 \right \}$$
  • $$\left \{ -2,2 \right \}$$
  • $$\left \{ -1,1 \right \}$$
  • Not invertible
If $$f(x)=2+x^{3}$$, then $$f^{-1}(x)$$ is equal to
  • $$\sqrt[3]{x}+2$$
  • $$\sqrt[3]{x}-2$$
  • $$\sqrt[3]{x-2}$$
  • $$\sqrt[3]{x+2}$$
The solution of $$8x\equiv 6(mod \  14) $$ is
  • $$\{8, 6\}$$
  • $$\{6,14\}$$
  • $$\{6,13\}$$
  • $$\{8,14,6\}$$
If $$f(x)=(1-x)^{1/2}$$ and $$g(x)= \ln(x)$$  then  the  domain  of $$(gof)(x)$$ is
  • $$(-\infty ,2)$$
  • $$(-1,1)$$
  • $$(-\infty ,1]$$
  • $$(-\infty ,1)$$
If $$f:R^{+}\rightarrow R$$ such that $$f(x)=\log_{5} x$$ then $$f^{-1}(x)=$$
  • $$\log_{x}10$$
  • $$5^{x}$$
  • $$3^{-x}$$
  • $$3^{1/x}$$
If $$f(x)=\dfrac{x+1}{x-1}(x\neq 1)$$ then $$fofofof(x)=$$
  • $$f(x)$$
  • $$2\left ( \dfrac{x+1}{x-1} \right )$$
  • $$\dfrac{x-1}{x+1}$$
  • $$x$$
If $$F(n)=(-1)^{k-1}(n-1), G(n)=n-F(n)$$ then $$ (GoG)(n)=$$ (where $$k$$ is odd)
  • $$1$$
  • $$n$$
  • $$2$$
  • $$n-1$$
If $$f:[1,\infty )\rightarrow B$$  defined  by the function $$ f(x)=x^{2}-2x+6$$ is a surjection, then $$B$$ is equals to
  • $$[1,\infty )$$
  • $$[5,\infty )$$
  • $$[6,\infty )$$
  • $$[2,\infty )$$
If $$f:R\rightarrow R^{+}$$ then $$\displaystyle f(x)=\left(\dfrac{1}{3}\right)^{x}$$, then $$f^{-1}(x)=$$
  • $$\displaystyle \left(\dfrac{1}{3}\right)^{-x}$$
  • $$3^{x}$$
  • $$\displaystyle \log_{1/3}$$$$ x$$
  • $$\displaystyle \log_{x}\left(\dfrac{1}{3}\right)$$
If $$ X =\{1, 2,3,4,5\} $$ and $$Y =\{1,3,5,7,9\}$$, determine which of the following sets represent a relation and also a mapping?
  • $$R_{1}= \{(x,y)$$:$$ y=x+2, x \in Y,y \in Y\}$$
  • $$R_{2}=\{(1,1), (1,3), (3,5), (4,7), (5,9)\}$$
  • $$R_{3}=\{(1,1), (2,3), (3,5), (3,7), (5,7)\}$$
  • $$R_{4}=\{(1,3), (2,5), (4,7), (5,9), (3,1)\}$$
If $$f(x)=\dfrac{x}{\sqrt{1+x^{2}}}$$ then $$fofof(x)=$$
  • $$\dfrac{x}{\sqrt{1+3x^{2}}}$$
  • $$\dfrac{x}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{2x}{\sqrt{1+2x^{2}}}$$
  • $$\dfrac{x}{\sqrt{1+x^{2}}}$$
If A $$=$${$$x : x^{2}-3x+2= 0$$}, and $$R$$ is a universal relation on $$A$$, then $$R$$ is
  • $$\{(1,1),(2, 2)\}$$
  • $$\{(1,1)\}$$
  • $$\phi $$
  • $$\{(1,1),(1, 2)(2,1),(2,2)\}$$
Assertion(A):  If $$X=\left \{ x:-1\leq x\leq 1 \right \}$$  and  $$f:X\rightarrow X$$ defined by $$f(x)=\sin \pi x; \forall x\in A$$ is not invertible function

Reason (R): For a function $$f$$ to have inverse, it should be a bijection
  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true but R is not correct explanation of A
  • A is true but R is false
  • A is false but R is true
If $$f(x)=\displaystyle \dfrac{x}{\sqrt{1-x^{2}}},g(x)=\displaystyle \dfrac{x}{\sqrt{1+x^{2}}} $$, then $$(fog)(x)=$$       
  • $$x$$
  • $$\dfrac{x}{\sqrt{1+x^{2}}}$$
  • $$\sqrt{1+x^{2}}$$
  • $$2x$$
If $$f(x)=1+x+x^{2}+x^{3}+\ldots\ldots $$ for $$\left | x \right |<1$$  then $$f^{-1}(x)=$$
  • $$\dfrac{x-1}{x+1}$$
  • $$\dfrac{x+1}{x}$$
  • $$\dfrac{x}{x-1}$$
  • $$\dfrac{x-1}{x}$$
If the function is $$f:R\rightarrow R,  g:R\rightarrow R$$ are defined as $$f(x)=2x+3, g(x)=x^{2}+7$$  and  $$f[g(x)]=25$$  then  $$x=$$    
  • $$f(x)$$
  • $$\pm 2$$
  • $$\pm 3$$
  • $$\pm 4$$
If $$f(x)=\displaystyle \frac{2^{x}+2^{-x}}{2^{x}-2^{-x}}$$,  then  $$f^{-1}(x)=$$
  • $$\displaystyle\frac{1}{2}\log_{2}\left ( \frac{x-1}{x+1} \right )$$
  • $$\displaystyle\frac{1}{2}\log_{2}\left ( \frac{x+1}{x-1} \right )$$
  • $$\displaystyle\frac{1}{2}\log_{2}\left ( \frac{x+1}{x-2} \right )$$
  • $$\displaystyle\frac{1}{2}\log_{2}\left ( \frac{x-2}{x-1} \right )$$
If $$f(x)=\dfrac{x}{\sqrt{1-x^{2}}}$$, then $$ (fof)(x)=$$
  • $$\dfrac{x}{\sqrt{1-x^{2}}}$$
  • $$\dfrac{x}{\sqrt{1-2x^{2}}}$$
  • $$\dfrac{x}{\sqrt{1-3x^{2}}}$$
  • $$x$$
If $$f:R\rightarrow R$$ is defined by $$f(x)=x^{2}-10x+21 $$ then $$ f^{-1}(-3)$$ is
  • $$\left \{ -4,6 \right \}$$
  • $$\left \{ 4,6 \right \}$$
  • $$\left \{ -4, 4, 6 \right \}$$
  • Not Invertible
I: If $$f:A\rightarrow B$$ is a bijection only then does $$f$$ have an inverse function
II: The inverse function $$f:R^{+}\rightarrow R^{+}$$ defined by $$f(x)=x^{2}$$ is $$f^{-1}(x)=\sqrt{x}$$
  • only I is true
  • only II is true
  • both I and II are true
  • neither I nor II true
If $$f(x)=\sin^{-1}\left \{ 3-(x-6)^{4} \right \}^{1/3}$$  then $$ f^{-1}(x)=$$
  • $$6+\sqrt[4]{3+\sin^{3}x}$$
  • $$6+\sqrt[4]{3-\sin^{3}x}$$
  • $$6+\sqrt[4]{3+\sin x}$$
  • $$6+\sqrt[4]{3-\sin x}$$
Which of the following functions defined from $$(-\infty ,\infty )$$ to $$ (-\infty ,\infty )$$ is invertible ?
  • $$f(x) = \sin (2x+3)$$
  • $$f(x) = x^{2} + 4$$
  • $$ f (x) =x^{3}$$
  • $$f (x) = \cos x$$
lf $${f}\left({x}\right)=\sin^{2}{x}+\sin^{2}\left({x}+\displaystyle \dfrac{\pi}{3}\right)+ \cos x \cos \left({x}+\displaystyle \dfrac{\pi}{3}\right)$$ and $${g}\left(\displaystyle\dfrac{5}{4}\right)=1$$, $$g\left(1\right) = 0 $$ then $$\left({g}{o}{f}\right)\left({x}\right)=$$
  • $$1$$
  • $$0$$
  • $$\sin x$$
  • Data is insufficient
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