CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 8 - MCQExams.com

Let $$f\left( x \right) = {x^2}$$ and $$g\left( x \right) = {2^x}$$. Then the solution of the equation $$fog\left( x \right) = gof\left( x \right)$$ is
  • $$R$$
  • $$\left\{ {0} \right\}$$
  • $$\left\{ {0,\,2} \right\}$$
  • none
Let $$g(x)=1+x-[x]\quad $$ and $$f(x)=\begin{cases} -1\quad if\quad x<0 \\ 0\quad \quad if\quad x=0 \\ 1\quad \quad if\quad x>0 \end{cases}$$ , then $$\forall \:x,fog(x)$$ equals 
  • $$x$$
  • $$1$$
  • $$f(x)$$
  • $$g(x)$$
If $$f ( x ) = \left( a - x ^ { n } \right) ^ { 1 / n }$$ where $$a > 0$$ and $$n$$ is a positive integer then $$( f o f ) ( x )$$ is 
  • $$f ( x )$$
  • $$x$$
  • $$0$$
  • $$1$$
The inverse of the function $$f(x)=\dfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}+2$$ is given by 
  • $$\log { e{ \left( \dfrac { x-1 }{ x+1 } \right) }^{ -2 } }$$
  • $$\log { e{ \left( \dfrac { x-2 }{ x+1 } \right) }^{ 1/2 } }$$
  • $$\log { e{ \left( \dfrac { x }{ 2-x } \right) }^{ 1/2 } }$$
  • $$\log { e{ \left( \dfrac { x-1 }{ 3-x } \right) }^{ 1/2 } }$$
$$f:R \rightarrow R$$ such that $$f(x)=\ell n(x+\sqrt {x^{2}+1})$$. Another function $$g(x)$$ is defined such that $$gof(x)=x\ \forall\ x \in\ R$$. Then $$g(2)$$ is -
  • $$\dfrac {e^{2}+e^{-2}}{2}$$
  • $$e^{2}$$
  • $$\dfrac {e^{2}-e^{-2}}{2}$$
  • $$e^{-2}$$
Let $$f:R\rightarrow R$$ is a function satisfying $$f(2-x)=f(2+x)$$ and $$f(20-x)=f(x)\forall x\in R$$
If $$f(0)=5$$ then the minimum possible no. of values of $$x$$ satisfying $$f(x)=5$$ for $$x=[0.,70]$$, is
  • $$21$$
  • $$12$$
  • $$11$$
  • $$22$$
All values of a for which f : R $$ \to R$$ defined by f(x)= $${x^3} + a{x^2} + 3x + 100$$ is a one one functions, are
  • $$( - \infty , - 2)$$
  • $$( - \infty ,4)$$
  • $$(4, - 4)$$
  • $$( - 3,3)$$
Let $$A = \{ 1,2,3 \}$$ . Which of the following functions on A is invertible?
  • $$f = \{ ( 1,1 ) , ( 2,1 ) , ( 3,1 ) \}$$
  • $$f = \{ ( 1,2 ) , ( 2,3 ) , ( 3,1 ) \}$$
  • $$f = \{ ( 1,2 ) , ( 2,3 ) , ( 3,2 ) \}$$
  • $$f = \{ ( 1,1 ) , ( 2,2 ) , ( 3,1 ) \}$$
If $$f ( x ) = \sin ^ { - 1 } ( \sin x ) + \cos ^ { - 1 } ( \sin x ) \text { and } \phi ( x ) = f ( f ( f ( x ) ) )$$ then $$\phi ^ { \prime } ( x )$$
  • 1
  • $$\sin x$$
  • 0
  • none of these
if $$f\left( x \right) = 3x + 2$$ , $$g\left( x \right) = {x^2} + 1$$,then the values of $$\left( {f_og} \right)\left( {{x^2} - 1} \right)$$
  • $$3{x^4} - 6{x^2} + 8$$
  • $$3{x^4} + 3x + 4$$
  • $$6{x^4} + 3{x^2} - 2$$
  • $$6{x^4} + 3{x^2} + 2$$
Let A = {1,2,3,4,5} and B={1,2,3,4,5}. If $$f:A\rightarrow B$$ is an one-one function and $$f(x)=x$$ holds only for one value of  $$x\epsilon \{ 1,2,3,4,5\} ,$$ then the number of such possible function is  
  • $$120$$
  • $$36$$
  • $$45$$
  • $$44$$
Difference between the greatest and the least values of the function
$$f(x) = x(ln x - 2)$$ on $$[1, e^{2}]$$ is
  • $$2$$
  • $$e$$
  • $$e^{2}$$
  • $$1$$
The function $$f :\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]\rightarrow \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$$ defined by $$f(x)=\sin^{-1}(3x-4x^{3})$$ is 
  • both one-one and onto
  • onto but not one-one
  • one-one but not onto
  • neither one-one nor onto
If $$f\left( x \right) = \frac{{x - 1}}{{x + 1}}$$, then $$f^{-1}\left( x \right)$$ is
  • $$\frac{{f\left( x \right) + 1}}{{f\left( x \right) + 3}}$$
  • $$\frac{{3f\left( x \right) + 1}}{{f\left( x \right) + 3}}$$
  • $$\frac{{f\left( x \right) + 3}}{{f\left( x \right) + 1}}$$
  • $$\frac{{f\left( x \right) + 3}}{{3f\left( x \right) + 1}}$$
Let g be the inverse function of differentiable function f and $$G\left( x \right) =\frac { 1 }{ g\left( x \right)  } if\quad f\left( 4=2 \right) $$ and $$f'\left( 4 \right) =\frac { 1 }{ 16 } $$, then the value of $${ \left( G'\left( 2 \right)  \right)  }^{ 2 }$$ equals to:
  • 1
  • 4
  • 16
  • 64
If $$f:( - 1,1) \to B$$ , is a function defined by $$f(x) = {\tan ^{ - 1}}\dfrac{{2x}}{{1 - {x^2}}}$$, then find $$B$$ when $$f(x)$$ is both one-one and onto function. 
  • $$\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]$$
  • $$\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$$
  • $$\left( {0,\frac{\pi }{2}} \right)$$
  • $$\left[ {0,\frac{\pi }{2}} \right)$$
If $$f(x)=x^{3}+x^{2}f'(1)+xf''(2)+f'''(3)\ \forall x\ \epsilon \ R$$, then $$f(x)$$ is
  • one-one and onto
  • one-one and into
  • many-one and onto
  • non-invertible
The multiplicative inverse of the product of the additive inverse of x+1 is ________________.
  • $$x-1$$
  • $$\dfrac { 1 }{ 1-x } $$
  • $${ x }^{ 2 }-1$$
  • $$\dfrac { 1 }{ 1-{ x }^{ 2 } } $$
Let S be a non-empty set and P(S) be the power set of set S. Find the identity element for the union $$(\cup)$$ as a binary operation on $$P(S)$$.
  • $$\phi$$
  • $$1$$
  • $$0$$
  • None of these
If $$\begin{bmatrix} \sin { \left( \dfrac { \pi  }{ 2 }  \right)  }  & \cos { \left( \dfrac { \pi  }{ 3 }  \right)  }  \\ 2\tan { \left( \dfrac { \pi  }{ 4 }  \right)  }  & 2k \end{bmatrix}$$ is not invertible, then $$k=$$
  • $$2$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$3$$
Let $$N$$ be the set of natural numbers and two functions $$f$$ and $$g$$ be defined as $$f,g : N\to N$$ such that :
$$f (n)= \begin{cases}\dfrac{n+1}{2}& \text{if n is odd}\\ \dfrac{n}{2} & \text{in n is even} \end{cases}$$
and $$g(n) = n - (-1)^n$$. The fog is:
  • Both one-one and onto
  • One-one but not onto
  • Neither one-one nor onto
  • onto but not one-one
The numbers system which uses alphabets as well as numbers is-
  • Binary numbers system
  • Octal numbers system
  • Decimal numbers system
  • Hexadecimal numbers system
 $$f : R \rightarrow R , f ( x ) = e ^ { | x | } - e ^ { - x }$$  is many-one into function.
  • True
  • False
Number of one-one functions from A to B where $$n(A)=4, n(B)=5$$.
  • $$4$$
  • $$5$$
  • $$120$$
  • $$90$$
Let $$f(x)=x^ {135}+x^ {125}-x^ {115}+x^ {5}+1$$. If $$f(x)$$ divided by $$x^ {3}-x$$, then the remainder is some function of $$x$$ say $$g(x)$$. Then $$g(x)$$ is an:-
  • one-one function
  • many one function
  • into function
  • onto function
$$f : R \rightarrow R , f ( x ) = 2 x + | \sin x |$$  is one-one onto.
  • True
  • False
If $$  f : R \rightarrow R  $$ be given by $$  f(x)=\left(3-x^{3}\right)^{\dfrac{1}{3}},  $$ then $$fof(x)$$ is
  • $$
    x^{\dfrac{1}{3}}
    $$
  • $$
    1^{3}
    $$
  • x
  • $$
    \left(3-x^{3}\right)
    $$
Let : $$R\rightarrow R$$ defined as $$f\left( x \right) =\dfrac { x\left( x+1 \right) \left( { x }^{ 4 }+1 \right) +{ 2x }^{ 4 }+{ x }^{ 2 }+2 }{ { x }^{ 2 }+x+1 } $$
  • odd and one-one
  • even and one-one
  • many to one and even
  • many to one and neither even nor odd
If a binary operation is defined $$a\star b=a^b$$ then 2$$\star 2$$ is equal to:
  • $$4$$
  • $$2$$
  • $$9$$
  • $$8$$
If is a binary operation such that $$a * b = a^2 + b^2$$ then $$3 * 5$$ is 
  • 34
  • 9
  • 8
  • 25
Consider $$f(x) = \dfrac{x^2}{1 + x^3}$$ ; $$g(t) = \displaystyle \int f(t) dt$$ . If $$g(1) = 0$$ then $$g(x)$$ equals 
  • $$\dfrac{1}{3} ln(1 + x^3)$$
  • $$\dfrac{1}{3} ln\left ( \dfrac{1 + x^3}{2} \right )$$
  • $$\dfrac{1}{2} ln\left ( \dfrac{1 + x^3}{3} \right )$$
  • $$\dfrac{1}{3}l n\left ( \dfrac{1 + x^3}{3} \right )$$
Let f : $$R\rightarrow R$$ be a function defined by f(x) = $${ x }^{ 3 }+{ x }^{ 2 }+3x+sin\times .$$ Then f is.
  • one-one & onto
  • one-one & into
  • many one & onto
  • many one & into
A function $$f$$ from the set of natural numbers to integers defined by $$f(n)=\begin{cases} \cfrac { n-1 }{ 2 } ,\quad \text{when n is odd} \\ -\cfrac { n }{ 2 } ,\quad \text{when n is even} \end{cases}$$  is
  • neither one-one nor onto
  • one-one but not onto
  • onto but not one-one
  • one-one and onto both
Let $$f:[2,\infty )\rightarrow X$$ be defined by $$f(x)=4x-{x}^{2}$$. Then, $$f$$ is invertible, if $$X=$$
  • $$[2,\infty)$$
  • $$(-\infty,2]$$
  • $$(-\infty,4]$$
  • $$[4,\infty)$$
If $$g(x)={ x }^{ 2 }+x-2$$ and $$\cfrac { 1 }{ 2 } (g\circ f(x))=2{ x }^{ 2 }-5x+2$$, then $$f(x)$$ is equal to
  • $$2x-3$$
  • $$2x+3$$
  • $$2{x}^{2}+3x+1$$
  • $$2{x}^{2}-3x-1$$
If $$g(x)=x^2+x-1$$ and 
$$(gof)(x)=4x^2-10x+5$$, then
$$f\left(\dfrac{5}{4}\right)$$ is equal to:
  • $$\dfrac{3}{2}$$
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{3}{2}$$
  • $$-\dfrac{1}{2}$$
The inverse function of
$$f(x)  = \dfrac{8^{2x}-{8^{-2x}}} {8^{2x}+8^{-2x}}\in (-1, 1)$$, is ________.
  • $$\dfrac{1}{4} \log_e\left(\dfrac{1-x}{1+x}\right)$$
  • $$\dfrac{1}{4} \log_e\left(\dfrac{1+x}{1-x}\right)$$
  • $$\dfrac{1}{4} (\log_e)\log_e\left(\dfrac{1-x}{1+x}\right)$$
  • $$\dfrac{1}{4} (\log_e)\log_e\left(\dfrac{1+x}{1-x}\right)$$
If $$f(x) = \dfrac{x+1}{x-1}$$, then the valueof $$f(f(x))$$ is equal to
  • $$x$$
  • $$0$$
  • $$-x$$
  • $$1$$
Let $$f : x \rightarrow y $$ be such that $$f(1) = 2$$ and $$f(x + y) = f(x) f(y)$$ for all natural numbers x and y. If $$\displaystyle \sum_{k= 1}^n f(a + k) = 16 (2^n - 1)$$ , then a is equal to 
  • $$3$$
  • $$4$$
  • $$5$$
  • $$6$$
  • $$7$$
If $$f(x)=\dfrac{(4x+3)}{(6x-4)}, x\neq \dfrac{2}{3}$$ then $$(f o f)(x)=?$$
  • $$x$$
  • $$(2x-3)$$
  • $$\dfrac{4x-6}{3x+4}$$
  • None of these
If $$f(x)=\sqrt[3]{3-x^3}$$ then $$(f o f)(x)=?$$
  • $$x^{1/3}$$
  • $$x$$
  • $$(1-x^{1/3})$$
  • None of these
Let $$ f : R \rightarrow R : f(x) =x +1 $$ and $$ g : R \rightarrow R : g(x) = 2x -3 $$.
Find $$(f +g) (x)$$.
  • $$3x -2$$
  • $$4x -5$$
  • $$3x -4$$
  • $$2x -3$$
If $$\displaystyle f(x) = | x - 2 |$$ and $$ g(x) = fof\,(x) $$ , then for $$ x > 20 , {g}\,'(x) = $$ 

  • $$ 2 $$
  • $$ 1 $$
  • $$ 3 $$
  • None of these
If $$\displaystyle {f}'(x) = g\,(x) $$ and $$\displaystyle {g}'(x) = - f\,(x) $$ for all $$ x $$ and $$ f\,(2) = 4 = {f}'(2) $$ then $$\displaystyle f^{2}\,(19) + g^{2} \,(19) $$ is 
  • $$ 16 $$
  • $$ 32 $$
  • $$ 64 $$
  • None of these
The value of f(0), so that the function
f(x) = $$ \dfrac{2x-sin^{-1}x}{2x+tan^{-1}x} $$ is continuous at each point in its domain, is equal to
  • 2
  • 1/3
  • 2/3
  • -1/3
let $$f(x) = sin^2 x/2 + cos ^2 x/2 $$ and $$g(x) = sec^2 x - tan ^2 x.$$ The two functions are equal over the set
  • $$\phi$$
  • $$R$$
  • $$R-{ x:x (2n+1) \frac{\pi}{2}, n\in1}$$
  • None of these
Let $$f(n)$$ denote the number of different ways in which the positive integer $$n$$ can be expressed as the sum of $$1s$$ and $$2s$$. For example, $$f(4) = 5$$, since $$4 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1$$. Note that order of $$1s$$ and $$2s$$ is important.
$$f : N\rightarrow N$$ is
  • One-one and onto
  • One-one and into
  • Many-one and onto
  • Many-one and into
The function $$f(x)= \dfrac{(3^{x}-1^{})^2}{\sin x. \ln(1+x)}, x\neq 0 $$ , is continuous at $$x=0$$. Then the value of $$f(0)$$ is 
  • 2log 3
  • $$ (\log_{e}3)^{2} $$
  • $$ \log_{e} 6 $$
  • None of these
Let f(x)= max { 1+sinx, 1, 1 -cosx}, $$x \epsilon [0, 2 \pi]$$ and g(x)= max {1, |x-1|} $$x \epsilon R$$, then
  • g(f(0))=1
  • g(f(1))=1
  • f(f(1))=1
  • f(g(0))=1+sin1
If f: $$R\rightarrow R$$ be given by $$f(x) = 3 + 4x$$ and $$a_n = A + Bx$$, then which of the following is not true?
  • A + B + 1 = $$2^{2n + 1}$$
  • | A - B| = 1`
  • $$lim_{n \to \infty} \dfrac{A}{B} = -1$$
  • None of these
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