CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 6 - MCQExams.com

Consider the functions $$\displaystyle f\left( x \right) =\sqrt { x } $$ and $$\displaystyle g\left( x \right) =7x+b$$. Find the value of $$b$$, if the composite function, $$\displaystyle y=f\left( g\left( x \right)  \right) $$ passes through $$(4, 6)$$. 
  • $$8$$
  • $$-8$$
  • $$-25$$
  • $$-26$$
  • $$\displaystyle 4-7\sqrt { 6 } $$
Find $$g(x)$$, if $$f(x) = 7x + 12$$ and $$f(g(x) = 21x^{2} + 40$$
  • $$21x^{2} + 28$$
  • $$21x^{2}$$
  • $$7x^{2} + 4$$
  • $$3x^{2} + 28$$
  • $$3x^{2} + 4$$
Given a function $$f(x) = \dfrac {1}{2}x - 4$$ and the composite function $$f(g(x)) = g(f(x))$$, determine which among the following can be $$g(x)$$:
I. $$2x - \dfrac {1}{4}$$
II. $$2x + 8$$
III. $$\dfrac {1}{2}x - 4$$
  • I only
  • II only
  • III only
  • II and III only
  • I, II, and III
Extraction of a cube root of a given number is
  • binary operation
  • relation
  • unary operation
  • relation in some set
If $$f(x) = 4x - 3$$ and $$g(x) = x - 4$$, determine which of the following composite function has a value of $$-11$$.
  • $$f(g(2))$$
  • $$g(f(2))$$
  • $$g(f(3))$$
  • $$f(g(3))$$
  • $$f(g(4))$$
$$*$$ is a binary operation on $$Z$$ such that:
$$a * b = a + b + ab$$.
The solution of $$(3* 4) *x = -1$$ is
  • $$1$$
  • $$-1$$
  • $$4$$
  • $$3$$
If $$f(x) = 3x - 5$$ and $$g(x) = x^2 + 1, f [g(x)] =$$
  • $$3x^2-5$$
  • $$3x^2+6$$
  • $$x^2-5$$
  • $$3x^2-2$$
  • $$3x^2+5x-2$$
If $$a*b = |a-b|$$ then $$6*8$$ will be
  • $$-2$$
  • $$14$$
  • $$2$$
  • Cannot be determined
The Set $$A$$ has $$4$$ elements and the Set $$B$$ has $$5$$ elements then the number of injective mappings that can be defined from $$A$$ to $$B$$ is
  • $$144$$
  • $$72$$
  • $$60$$
  • $$120$$
If $$f(x)=\dfrac{x+1}{x-1}$$ and $$g(x)=2x-1, f[g(x)]=$$
  • $$\dfrac{x-1}{x}$$
  • $$\dfrac{x}{x+1}$$
  • $$\dfrac{x+1}{x}$$
  • $$\dfrac{x}{x-1}$$
  • $$\dfrac{2x-1}{2x+1}$$
If $$f(x) = 2x - 6$$, then $$f^{-1}(x)$$ is
  • $$6 - 2x$$
  • $$\dfrac {1}{2}x - 6$$
  • $$\dfrac {1}{2}x - 3$$
  • $$\dfrac {1}{2}x + 3$$
  • $$\dfrac {1}{2} x + 6$$
If $$f(x) = 4/\sqrt [3]{x + 1}$$, what is $$f^{-1}(7)$$?
  • $$-7$$
  • $$8$$
  • -$$\dfrac{279}{343}$$
  • $$1.75$$
  • $$7.2$$
Let f : $$N \rightarrow N$$ defined by $$f(n)=\left\{\begin{matrix}
\dfrac{n+1}{2} & \text{if }\, n \, \text{is odd} \\
\dfrac{n}{2} & \text{if}\, n \, \text{is even}
\end{matrix}\right.$$
then $$f$$ is.
  • Many-one and onto
  • One-one and not onto
  • Onto but not one-one
  • Neither one-one nor onto
If the operation $$\oplus$$ is defined by $$a\oplus b = a^{2} + b^{2}$$ for all real numbers $$'a'$$ and $$'b'$$, the $$(2\oplus 3)\oplus 4 = $$ 
  • $$120$$
  • $$175$$
  • $$129$$
  • $$185$$
  • $$312$$
If $$f: R\rightarrow R$$ is defined by $$f(x) = \dfrac {x}{x^{2} + 1}$$, find $$f(f(2))$$
  • $$\dfrac {1}{29}$$
  • $$\dfrac {10}{29}$$
  • $$\dfrac {29}{10}$$
  • $$29$$
If $$f : IR \rightarrow IR$$ is defined by $$f(x) = 2x + 3$$, then $$f^{-1}(x)$$
  • Is given by $$\dfrac{x-3}{2}$$
  • Is given by $$\dfrac{1}{2x + 3}$$
  • Does not exist because 'f' is not injective
  • Does not exist because 'f' is not surjective
Let $$f(x)=2x-\sin x$$ and $$g(x)=\sqrt[3] x$$, then
  • Range of $$gof$$ is $$R$$
  • $$gof$$ is one-one
  • both $$f$$ and $$g$$ are one-one
  • both $$f$$ and $$g$$ are onto
If the function $$f : R \rightarrow R$$ is defined by $$f(x) = (x^2+1)^{35} \forall \in R$$, then $$f$$ is
  • One-one but not onto
  • Onto but not one-one
  • Neither one-one nor onto
  • Both one-one and onto
Let $$f(x) = 2^{100}x+1$$
$$g(x) = 3^{100}x+1$$
Then the set of real numbers x such that $$f(g(x)) = x$$ is
  • Empty
  • A singleton
  • A finite se with more than one element
  • Infinite
The number of real linear functions $$f(x)$$ satisfying $$f(f(x))=x+f(x)$$ is
  • $$0$$
  • $$4$$
  • $$5$$
  • $$2$$
Let $$f : R\rightarrow R$$ be defined by $$f(x) = \dfrac {1}{x} \ \   \forall  \ x \ \in \ R$$, then $$f$$ is _____
  • One-one
  • Onto
  • Bijective
  • $$f$$ is not defined
In the set of integers under the operation $$\ast $$ defined by $$a\ast b=a+b-1$$, the identity element is:
  • $$0$$
  • $$1$$
  • $$a$$
  • $$b$$
Which of the following is not a binary operation on $$R$$?
  • $$a \times b = ab$$
  • $$a \times b = a - b$$
  • $$a \times b = \sqrt{ab}$$
  • $$a \times b = \sqrt{a^2 + b^2}$$
If $$N$$ is a set of natural numbers, then under binary operation $$a\cdot b = a + b, (N, .)$$ is
  • Quasi-group
  • Semi-group
  • Monoid
  • Group
If $$f(x) = \log_{e}\left (\dfrac {1 + x}{1 - x}\right ), g(x) = \dfrac {3x + x^{3}}{1 + 3x^{2}}$$ and $$go f(t) = g(f(t))$$, then what is $$go f\left (\dfrac {e - 1}{e + 1}\right )$$ equal to?
  • $$2$$
  • $$1$$
  • $$0$$
  • $$\dfrac {1}{2}$$
Let $$f(x)=\dfrac{x+1}{x-1}$$ for all $$x \neq 1$$. 
Let
$$f^1(x)=f(x), f^2(x)=f(f(x))$$ and generally
$$f^n(x)=f(f^{n-1}(x)) $$ for $$n > 1$$
Let $$P= f^1(2)f^2(3)f^3(4)f^4(5)$$
Which of the following is a multiple of P ?
  • $$125$$
  • $$375$$
  • $$250$$
  • $$147$$
Consider the following statements :
Statement 1 : The function $$f:R \rightarrow R$$ such that $$f(x)=x^3$$ for all $$x\in R$$ is one-one.
Statement 2 : $$f(a) = f(b) \Rightarrow a=b$$ for all $$a, b \in R$$ if the function $$f$$ is one-one.
Which one of the following is correct in respect of the above statements?
  • Both the statements are true and Statement 2 is the correct explanation of Statement 1.
  • Both the statements are true and Statement 2 is not the correct explanation of Statement 1.
  • Statement 1 is true but Statement 2 is false.
  • Statement 1 is true but Statement 2 is true.
Consider the function $$f(x)=\displaystyle\frac{x-1}{x+1}$$. What is $$f(f(x))$$ equal to?
  • $$x$$
  • $$-x$$
  • $$-\displaystyle\frac{1}{x}$$
  • None of the above
On the set $$Z$$, of all integers $$\ast$$ is defined by $$a\ast b = a + b - 5$$. If $$2\ast (x\ast 3) = 5$$ then $$x =$$
  • $$0$$
  • $$3$$
  • $$5$$
  • $$10$$
If $$f(x) = 8x^3, g(x) = x^{1/3}$$, then fog (x) is
  • $$8^3x$$
  • $$(8x)^{1/3}$$
  • $$8x^3$$
  • $$8x$$
If $$g(x)=\dfrac{1}{f(x)}$$ and $$f(x)=x, x\ne 0,$$ then which one of the following is correct?
  • $$f(f(f(g(g(f(x))))))=g(g(f(g(f(x)))))$$
  • $$f(g(f(g(g(f(g(x)))))))=g(g(f(g(f(x)))))$$
  • $$f(g(f(g(g(f(g(x)))))))=f(g(f(g(f(x)))))$$
  • $$f(f(f(g(g(f(x))))))=f(f(f(g(f(x)))))$$
Let $$f (x) = \sqrt {2 - x - x^2}$$ and g(x) = cos x. Which of the following statements are true?
(I) Domain of $$f((g(x))^2) = $$ Domain of f(g(x))
(II) Domain of f(g(x)) + g(f(x)) = Domain of g(f(x))
(III) Domain of f(g(x)) = Domain of g(f(x))
(IV) Domain of $$g((f(x))^3) = $$ Domain of f(g(x))
  • Only (I)
  • Only (I) and (II)
  • Only (III) and (IV)
  • Only (I) and (IV)
If $$f:R\rightarrow R, g:R \rightarrow R$$ be two functions given by $$f(x)=2x-3$$ and $$g(x)=x^3+5$$, then $$(fog)^{-1}(x)$$ is equal to
  • $$\begin{pmatrix}\dfrac{x+7}{2}\end{pmatrix} ^{\dfrac{1}{3}}$$
  • $$\begin{pmatrix}\dfrac{x-7}{2}\end{pmatrix} ^{\dfrac{1}{3}}$$
  • $$\begin{pmatrix}x-\dfrac{7}{2}\end{pmatrix} ^{\dfrac{1}{3}}$$
  • $$\begin{pmatrix}x+\dfrac{7}{2}\end{pmatrix} ^{\dfrac{1}{3}}$$
If $$f : R \rightarrow R$$ is defined by $$f(x) = x^{3}$$ then $$f^{-1}(8) =$$
  • $$\left \{2\right \}$$
  • $$\left \{2, 2\omega, 2\omega^{2}\right \}$$
  • $$\left \{2, -2\right \}$$
  • $$\left \{2, 2\omega\right \}$$
Let $$A=\left\{ x\in R|x\ge 0 \right\} $$. A function $$f:A\rightarrow A$$ is defined by $$f(x)={ x }^{ 2 }$$. Which one of the following is correct?
  • The function does not have inverse
  • $$f$$ is its own inverse
  • The function has an inverse but $$f$$ is not its own inverse
  • None of the above
If $$f:[0, \infty)\to [0,\infty)$$ and $$f(x) = \dfrac{x}{1+x}$$, then $$f$$ is 
  • One-one and onto
  • One-one but not onto
  • Onto but not one-one
  • Neither one-one nor onto
If $$\ast $$ is the operation defined by $$a\ast b={ a }^{ b }$$ for $$a,b\in N$$, then $$\left( 2\ast 3 \right) \ast 2$$ is equal to
  • $$81$$
  • $$512$$
  • $$216$$
  • $$64$$
  • $$243$$
The function $$f:A\rightarrow B$$ given by $$f(x) = x ,x\in A$$, is one to one but not onto. Then;
  • $$B\subset A$$
  • $$A=B$$
  • $$A'\subset B'\quad $$
  • $$A\subset B$$
  • $$A'\cap B'=\phi $$
If $$fog = |\sin x|$$ and $$gof = \sin^{2}\sqrt {x}$$, then $$f(x)$$ and $$g(x)$$ are
  • $$f(x) = \sqrt {\sin x}, g(x) = x^{2}$$
  • $$f(x) = |x|, g(x) = \sin x$$
  • $$f(x) = \sqrt {x}, g(x) = \sin^{2}x$$
  • $$f(x) = \sin \sqrt {x}, g(x) = x^{2}$$
Let $$f(x) = |x - 2|$$, where $$x$$ is a real number. Which one of the following is true?
  • $$f$$ is periodic
  • $$f(x + y) = f(x) + f(y)$$
  • $$f$$ is an odd function
  • $$f$$ is not one-one function
  • $$f$$ is an even function
The value of $$\alpha (\neq 0)$$ for which the function $$f(x) = 1 + \alpha x$$ is the inverse of itself is
  • $$-2$$
  • $$2$$
  • $$-1$$
  • $$1$$
If $$A = \left \{ 1 , 3 , 5 , 7 \right \} $$ and $$ B = \left \{ 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 \right \} $$ then the number of one-to-one functions from $$A$$ into $$B$$ is 
  • 1340
  • 1860
  • 1430
  • 1880
  • 1680
If $$f(x)=3x+5$$ and $$g(x)={ x }^{ 2 }-1$$, then $$\left( f\circ g \right) $$ $$({ x }^{ 2 }-1)$$ is equal to
  • $$3{ x }^{ 4 }-3x+5$$
  • $$3{ x }^{ 4 }-6{ x }^{ 2 }+5$$
  • $$6{ x }^{ 4 }+3{ x }^{ 2 }+5$$
  • $$6{ x }^{ 4 }-6x+5$$
  • $$3{ x }^{ 2 }+6x+4$$
If $$\ast$$ is defined by $$a\ast b = a - b^{2}$$ and $$\oplus$$ is defined by $$a\oplus b = a^{2} + b$$, where $$a$$ and $$b$$ are integers, then $$(3\oplus 4)\ast 5$$ is equal to
  • $$164$$
  • $$38$$
  • $$-12$$
  • $$-28$$
  • $$144$$
If $$g(x)=1+\sqrt{x}$$ and $$f\{g(x)\}=3+2\sqrt{x}+x$$, then $$f(x)$$ is equal to
  • $$1+2x^2$$
  • $$2+x^2$$
  • $$1+x$$
  • $$2+x$$
Let $$f(x)=\cot ^{ -1 }{ \left( \cfrac { 1-{ x }^{ 2 } }{ 2x }  \right)  } +\cot ^{ -1 }{ \left( \cfrac { 1-3{ x }^{ 2 } }{ 3x-{ x }^{ 3 } }  \right)  } -\cot ^{ -1 }{ \left( \cfrac { 1-6{ x }^{ 2 }+{ x }^{ 4 } }{ 4x-4{ x }^{ 3 } }  \right)  } $$, the $$F'(x)$$ equals
  • $$\cfrac { -1 }{ \sqrt { 1-{ x }^{ 2 } } } $$
  • $$\cfrac { -1 }{ 1+{ x }^{ 2 } } $$
  • $$\cfrac { 1 }{ 1+{ x }^{ 2 } } $$
  • $$\cfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } $$
If $$f(x)=\left| x \right| ,x\in R$$, then
  • $$f(x)=\left( f\times f \right) \left( x \right) $$
  • $$f(x)=x$$
  • $$f(x)=\left( f\times f \right) \left( x^2 \right) $$
  • $$f(x)=\left( f\circ f \right) \left( x \right) $$
If $$g(x)={ x }^{ 2 }+x-2$$ and $$\cfrac { 1 }{ 2 } (g\circ f)(x)=2{ x }^{ 2 }-5x+2$$, then $$f(x)$$ is
  • $$2x-3$$
  • $$2x+3$$
  • $$2{ x }^{ 2 }+3x+1$$
  • $$2{ x }^{ 2 }+3x-1$$
If $$(ax^2 + bx + c)y +a'x^2+b'x+c=0$$, then the condition that x may be a rational function of y is
  • $$(ac'-a'c)^2=(ab'-a'b)(bc'-b'c)$$
  • $$(ab'-a'b)^2=(ab'-a' c)(bc'-b'c)$$
  • $$(bc'-b'c)^2=(ab'-a'b)(ac'-a'c)$$
  • None of these
If $$f(x)=\sin ^{ 2 }{ x } +\sin ^{ 2 }{ \left( x+\cfrac { \pi  }{ 3 }  \right)  } +\cos { x } \cos { \left( x+\cfrac { \pi  }{ 3 }  \right)  } $$ and $$g\left( \cfrac { 5 }{ 4 }  \right) =1$$, then $$g\circ f(x)$$ is equal to
  • $$0$$
  • $$1$$
  • $$\sin { { 1 }^{ o } } $$
  • None of these
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