Relations and Functions Test

Result

Relations and Functions Test - 82

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Weekly Quiz Competition

Question 1
1 / -0

l qt f(x) be a function satisfying f'(x)=f(x) with f(0)=1 and g be the function satisfying f(x)+g(x)=$$X^{2}$$, the value of the integral $$\int _{ 0 }^{ 1 }{ f(x)g(x)\quad dx\quad is } $$

A
$$\frac { 1 }{ 4 } (e-7)$$

B
$$\frac { 1 }{ 4 } (e-2)$$

C
$$\frac { 1 }{ 4 } (e-3)$$

D
none of the above

Solution
Question 2
1 / -0

$$f : R \rightarrow R$$ defined by $$f ( x ) = \dfrac { x } { x ^ { 2 } + 1 } , \forall x \in R$$ is

A
one-one

B
onto

C
bijective

D
neither one one nor onto

Solution

$$\begin{aligned} f(x) &=\dfrac{x}{1+x^{2}} \\-f(4)=& \dfrac{4}{1+16}=\dfrac{4}{17} \\ f\left(\dfrac{1}{4}\right) &=\frac{\dfrac{1}{4}}{1+\dfrac{1}{16}}=\dfrac{4}{17} \end{aligned}$$
Hence two different input has Same output $$\cdot$$ so not one-one
$$f(x)=y$$
$$y=\dfrac{x}{1+x^{2}}=1 \quad y+y x^{2}-x=0$$
$$x=\dfrac{1 \pm \sqrt{1-4 y^{2}}}{2 y}$$
$$1-4 y^{2} \geq 0$$
$$(1+2 y)(1-x y) \geq 0$$
$$\dfrac{1}{-3} \leq y<\dfrac{1}{2}$$
Range of $$f(x)$$ is $$\left[\dfrac{-1}{2}, \dfrac{1}{2}\right]$$
Range of $$f(x) \neq$$ codomain $$f(x)$$ Hence f(x) is neither one-one nor onto option D is correct.
Question 3
1 / -0

If f (x) = cosx and g (x) = x$$^2$$ then (gof) (x) is ....

A
cos$$^2$$ x

B
cosx$$^2$$

C
both (a)&(b)

D
x$$^2$$ cosx

Solution
Question 4
1 / -0

Suppose that $$g(x)=1+\sqrt { x } and\quad f(g(x))=3+2\sqrt { x } +x\quad then\quad f(x)\quad is$$

A
$$\quad 1+2{ x }^{ 2 }$$

B
$$\quad 2+{ x }^{ 2 }$$

C
$$1+x$$

D
$$2+x$$

Solution
Question 5
1 / -0

If $$f ( x ) = \sqrt { x ^ { 2 } + 1 } , g ( x ) = \dfrac { x + 1 } { x ^ { 2 } + 1 }$$ and $$h ( x ) = 2 x - 3 ,$$ then $$f ^ { \prime } \left( h ^ { \prime } \left( g ^ { \prime } ( x ) \right) =\right.$$

A
$$0$$

B
$$\dfrac { 1 } { \sqrt { x ^ { 2 } + 1 } }$$

C
$$\dfrac { 2 } { \sqrt { 5 } }$$

D
$$\dfrac { x } { \sqrt { x ^ { 2 } + 1 } }$$

Solution
Question 6
1 / -0

If the operation * is defined as $$\left( a\times b \right) ={ a }^{ 2 }+{ b }^{ 2 }$$ then $$\left( 3\times 4 \right) =5$$ is

A
650

B
125

C
625

D
3125
Question 7
1 / -0

Consider the binary operation $$*$$ on $$Q$$ defined by $$x*y=1+12x+xy,x,y\rightarrow Q$$ then $$2*3$$ equals

A
$$31$$

B
$$41$$

C
$$43$$

D
$$51$$

Solution
Question 8
1 / -0

Let A=(5,6); how many binary operations can be defined on this set?

A
8

B
10

C
16

D
20

Solution
Question 9
1 / -0

If =$$f=\left\{ (-2,4),(0,6),(2,8) \right\} $$ and
$$g=\left\{ (-2,-1),(0,3),(2,5) \right\} $$, then
$$\left( \frac { 2f }{ 3g } +\frac { 3g }{ 2f } \right) (0)=\quad $$

A
1/12

B
25/12

C
5/12

D
13/12

Solution
Question 10
1 / -0

Number of solution of the equation $$f ( x ) = g ( x )$$ are same as number of point of intersection of the curves $$y = f ( x )$$ and $$y = g ( x )$$ hence answer the following question.
Number of the solution of the equation $$x ^ { 2 } = | x - 2 | + | x + 2 | - 1$$ is

A
$$0$$

B
$$3$$

C
$$2$$

D
$$4$$

Solution