Relations and Functions Test

Result

Relations and Functions Test - 57

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Weekly Quiz Competition

Question 1
1 / -0

If $$f(x)$$ is invertible and twice differentiable function satisfying
$$\int _{ 0 }^{ f\left( x \right) }{ f^{ -1 }\left( 1 \right) dt,\forall x\in R }$$ and $$f^{ \prime }\left( 0 \right)=1$$ then $$f^{ \prime }\left( 1 \right)$$ can be-

A
$$e$$

B
$${e}^{2}$$

C
$$\dfrac{1}{e}$$

D
None of these

Solution
Question 2
1 / -0

If the expansion in powers of x of the function $$\frac{1}{(1-ax)(1-bx)}$$ is $$a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...., $$ then $$a_{n}$$ is

A
$$\frac{a^{n}-b^{n}}{b-a}$$

B
$$\frac{a^{n+1}-b^{n+1}}{b-a}$$

C
$$\frac{b^{n+1}-a^{n+1}}{b-a}$$

D
$$\frac{b^{n}-a^{n}}{b-a}$$

Solution
Question 3
1 / -0

If $$f(x)=\dfrac{7^{1}+\ln x}{x^{\ln 7}}$$ then $$f(2015)=$$

A
$$20$$

B
$$7$$

C
$$2015$$

D
$$100$$

Solution
Question 4
1 / -0

Let $$f\left( x \right) =1+\sqrt { x } $$ and $$g\left( x \right) =\dfrac { 2x }{ { x }^{ 2 }+1 } $$, then

A
$$dom(f+g)=(0,\infty)$$

B
range of $$f\bigcap$$ range of $$g=\left\{1\right\}$$

C
range of $$f\bigcup$$ range of $$g=\left[ -1,\infty \right] $$

D
$$All\ of\ the\ above$$

Solution
Question 5
1 / -0

If $$n(A)=3,n(B)=4$$ and $$f:A\rightarrow B,$$ Then

A
Total number of function is $$128$$

B
Number of one-one functions is $$64$$

C
Total number of function is $$64$$

D
Number of one-one functions is $$24$$

Solution
Question 6
1 / -0

The number of real solutions of the equation $$f(x)=0$$, where $$f(x)=(x-1)^{3}+(x-2)^{3}+(x-3)^{3}$$, is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$zero$$

Solution
Question 7
1 / -0

The set of values of $$x$$ for which $$f(x)=x^{12}-x^{9}+x^{4}-x+1>0$$ is

A
$$(-4,10]$$

B
$$(0,1)$$

C
$$[-100,100]$$

D
$$R$$

Solution
Question 8
1 / -0

If f(x + ay, x - ay) = axy then f(x,y) is equal to

A
$$\dfrac{x^2-y^2}{4}$$

B
$$\dfrac{x^2 + y^2}{4}$$

C
4xy

D
axy

Solution

$$f(x+a y, x-a y)=a x y$$

Let $$x+a y=m-i$$ and $$x-a y=n-i 1$$

Now, an adding (i) and(ii),

\begin{array}{l}2 x=m+n \\ x=\dfrac{m+n}{2}\end{array}

and on subtraction (ii) from(i)

\begin{array}{l}2 a y=m-n \\ y=\dfrac{m-n}{2 a}\end{array}

Now, $$\begin{aligned} f(m, n) &=a \times \dfrac{m+n}{2} \times \dfrac{m-n}{2 a} \\ &=\dfrac{m^{2}-n^{2}}{4} \end{aligned}$$

Replace $$m$$ by $$x$$ and $$n$$ by $$y$$

$$f(x, y)=\dfrac{x^{2}-y^{2}}{4}$$
Question 9
1 / -0

Let F(x) = $$= \int e^{sin^{-1}x}(1-\frac{x}{\sqrt{1-x^{2}}})dx$$ and $$ F(0) = 1, $$ If $$ F(1/2) = \frac{k\sqrt{3}e^{\pi /6}}{\pi }, $$ then the value of k is

A
$$\pi $$

B
$$\frac{\pi }{3}$$

C
$$\frac{\pi }{4}$$

D
$$\frac{\pi }{2}$$

Solution
Question 10
1 / -0

If $$f(x)={ \left( \dfrac { { x }^{ l } }{ { x }^{ m } } \right) }^{ l+m }{ \left( \dfrac { { x }^{ m } }{ { x }^{ nd} } \right) }^{ m+n }{ \left( \dfrac { { x }^{ n } }{ { x }^{ l } } \right) }^{ n+l }$$, then $$f'(x)$$ is equal to

A
$$1$$

B
$$0$$

C
$$x^{l+m+n}$$

D
$$None\ of\ these$$

Solution