Relations And Functions Test

Relations And Functions

This is Relations and functions Test-07 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
If \(f:R \to R\); \(f\left( x \right) = \sin x\;\;and\;\;;{\text{g}}\left( {\text{x}} \right) = {{\text{x}}^2}\),find gof.
  • A
    cannot be determined.
  • B
    \(Sin{\text{ }}{x^2}\)
  • C
    \(\begin{array}{*{20}{l}} {{x^2}sin{\text{ }}x\;\;\;\;\;\;\;\;\;\;\;} \end{array}\)
  • D
    \({\text{Si}}{{\text{n}}^{\text{2}}}{\text{x}}\)
    Correct
2
Let be defined as f(x) = ax + b, a 0, then \({f^{ - 1}}\;\left( x \right)\)
  • A
    does not exist as f is not onto
  • B
    is given by \(\frac{{x - b}}{a}\)
    Correct
  • C
    is given by \(\frac{1}{{ax + b}}\)
  • D
    does not exist as f is not one – one.
3
Let f :\(R \to R\) be given by \(f\left( x \right){\text{ }} = {\text{ }}{x^2}\;--{\text{ }}3,{\text{ }}then{\text{ }}{f^{ - 1}}\;\) is given by
  • A
    \(x + \sqrt 3 \)
  • B
    \(\sqrt x + 3\)
  • C
    \(\sqrt {x + 3} \)
  • D
    none of these
    Correct
4
Let f :\(A \to B\), then f is invertible iff
  • A
    none of these.
  • B
    f is both one – one and onto
    Correct
  • C
    f is one – one
  • D
    f is onto
5
If f = {(1, 4), (2, 5), (3, 6)} and g = {(4, 8), (5, 7), (6, 9)}, then gof is
  • A
    {(1, 8), (2, 7), (3, 9)}
    Correct
  • B
    {(1, 7), (2, 8), (3, 9)}
  • C
    { }
  • D
    none of these.
6
The value of \(\alpha \in R,\) for which the function \(f(x) = 1 + \alpha x\)is inverse of itself, is
  • A
    -1
    Correct
  • B
    1
  • C
    2
  • D
    – 2
7
Let f and g be two functions such that \({R_g} \subset {D_f},\) then domain of the function (fog) (x) = f (g (x)) is
  • A
    Dg
    Correct
  • B
    none of these
  • C
    Df
  • D
    \(\phi \)
8
If \(f:{\text{ }}R \to R{\text{ }}and{\text{ }}g:{\text{ }}R \to R\) defined by f(x) = 2x + 3 and \(g\left( x \right){\text{ }} = {\text{ }}{x^2}\; + {\text{ }}7\), then the value of x for which f(g(x)) = 25 is
  • A
    \( \pm {\text{ 4}}\;\;\;\;\;\;\)
  • B
    \( \pm {\text{ }}2\;\;\;\;\;\;\)
    Correct
  • C
    \( \pm {\text{ 1}}\;\;\;\;\;\;\)
  • D
    \( \pm {\text{ }}3\;\;\;\;\;\)
9
Let f and g be two functions from R to R defined as \(f(x) = \left\{ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}\begin{array}{*{20}{c}} , \\ , \end{array}\begin{array}{*{20}{c}} {(x} \\ {(x} \end{array}\begin{array}{*{20}{c}} {is} \\ {is} \end{array}\begin{array}{*{20}{c}} {rational} \\ {irrational} \end{array}} \right\},\\ g(x) = \left\{ {\begin{array}{*{20}{c}} { - 1} \\ 0 \end{array}\begin{array}{*{20}{c}} , \\\\\\ , \end{array}\begin{array}{*{20}{c}} {(x} \\\\\\ {(x} \end{array}\begin{array}{*{20}{c}} {is} \\ {is} \end{array}\begin{array}{*{20}{c}} {rational} \\ {irrational} \end{array}} \right\}\)then , \(\left( {gof} \right)\left( e \right){\text{ }} + {\text{ }}\left( {fog} \right)(\pi )\) = .
  • A
    2
  • B
    0
  • C
    1
  • D
    -1
    Correct
10
Let f :[0, 1] \( \to \)[0, 1] and g :[0, 1] \( \to \) [0, 1] be two functions defined by \(f(x) = \frac{{1 - x}}{{1 + x}}\)and g(x) =4(x) (1 – x), then (fog)(x) equals \(\frac{{1 - 4x + 4{x^2}}}{{1 + 4x - 4{x^2}}}\) \(\frac{{4(1 - x)}}{{1 + x}}\) \(\frac{{8x(1 - x)}}{{{{(1 + x)}^2}}}\) none of these Let f :[0, 1] \( \to \)[0, 1] and g :[0, 1] \( \to \) [0, 1] be two functions defined by \(f(x) = \frac{{1 - x}}{{1 + x}}\)and g(x) =4(x) (1 – x), then (f og)(x) = f(g(x))=f(4(x) (1 – x))=\(f\left( {4x{\text{ }}--{\text{ }}4{x^2}} \right)\) =\(\frac{{1 - 4x + 4{x^2}}}{{1 + 4x - 4{x^2}}}\) .
  • A
    \(\frac{{8x(1 - x)}}{{{{(1 + x)}^2}}}\)
  • B
    none of these
  • C
    \(\frac{{1 - 4x + 4{x^2}}}{{1 + 4x - 4{x^2}}}\)
    Correct
  • D
    \(\frac{{4(1 - x)}}{{1 + x}}\)
11
Let \(f(x) = \frac{{ax}}{{x + 1}},x \ne 1.\)Then for what value of a is f (f(x)) = x ?
  • A
    1
  • B
    \( - \sqrt 2 \)
  • C
    –1
    Correct
  • D
    \(\sqrt 2 \)
12
If f(x) = \(\frac{{3x + 2}}{{5x - 3}},then\)
  • A
    \({f^{ - 1}}(x) = - \frac{1}{{19}}f(x)\)
  • B
    \({f^{ - 1}}\;\left( x \right){\text{ }} = {\text{ }}f\left( x \right)\;\;\;\;\;\;\;\;\)
    Correct
  • C
    \({f^{ - 1}}\;\left( x \right){\text{ }} = {\text{ }}--{\text{ }}f\left( x \right)\;\;\;\;\;\;\;\;\)
  • D
    \((f \circ \,f)(x) = - x\)
13
A function \(f{\text{ }}:{\text{ }}X{\text{ }} \to {\text{ }}Y\) Y is said to be one – one and onto if
  • A
    if f is onto
  • B
    if f is either one – one or onto
  • C
    if f is both one – one and onto
    Correct
  • D
    if f is one – one
14
Let \(f{\text{ }}:{\text{ }}A{\text{ }} \to {\text{ }}B{\text{ }}and{\text{ }}g{\text{ }}:{\text{ }}B{\text{ }} \to {\text{ }}C\) be two functions. Then the composition of f and g, denoted by gof, is defined as
  • A
    \(g\left( {f{\text{ }}\left( {{x^2}} \right)} \right),\forall x \in A\)
  • B
    \(f\left( {g{\text{ }}\left( x \right)} \right),\forall x \in A\)
  • C
    \(g\left( {f{\text{ }}\left( x \right)} \right),\forall x \in A\)
    Correct
  • D
    \(g\left( {f{\text{ }}\left( {x{\text{ }} + {\text{ }}2} \right)} \right),\forall x \in A\)
15
A function \(f{\text{ }}:{\text{ }}X{\text{ }} \to {\text{ }}Y\) is defined to be invertible, if
  • A
    there exists a function \(g{\text{ }}:{\text{ }}Y{\text{ }} \to {\text{ }}X{\text{ }}\\such{\text{ }}that{\text{ }}gof{\text{ }} = {\text{ }}{I_X}\)
  • B
    there exists a function \(g{\text{ }}:{\text{ }}Y{\text{ }} \to {\text{ }}X{\text{ }}\\such{\text{ }}that{\text{ }}gof{\text{ }} = {\text{ }}{I_X}and{\text{ }}fog{\text{ }}={\text{ }}{I_Y}.\)
    Correct
  • C
    there exists a function \(g{\text{ }}:{\text{ }}Y{\text{ }} \to {\text{ }}X{\text{ }}\\such{\text{ }}that{\text{ }}gof{\text{ }} = {\text{ }}{I_X}and{\text{ }}fog \ne {\text{ }}{I_Y}.\)
  • D
    there exists a function \(g{\text{ }}:{\text{ }}Y{\text{ }} \to {\text{ }}X\;fog{\text{ }} = {\text{ }}{I_Y}.\)