Relations and Functions Test

Result

Relations and Functions Test - 78

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Weekly Quiz Competition

Question 1
1 / -0

If $$ \phi (x) = 3
f(\frac{x^2}{3} ) + f(3-x^2) \forall x \in (3,4)$$ where $$f(x) >0 \forall x (-3,4)$$ then $$\phi (x)$$ is ____________.

A
(a) increasing in $$( \frac{3}{2} ,4)$$

B
(b) decreasing in $$( 3, \frac{3}{2} )$$

C
(c) increasing $$( -\frac{3}{2} , 0)$$

D
decreasing in $$( 0, \frac{3}{2})$$

Solution
Question 2
1 / -0

Let $$f: R\rightarrow R; f(x)=2x^3-3(p+2)x^2+12px-7$$, $$-5\leq p \leq 5$$, $$p\in I$$. Then the number of values of p for $$f(x)$$ to be invertible is?

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$
Question 3
1 / -0

Directions For Questions

Let f: R$$\rightarrow$$R is a function satisfying $$f(2-x)=f(2+x)$$ and $$f(20-x)=f(x)$$, $$\forall x\in R$$. On the basis of information, answer the following questions.

...view full instructions

If $$f(0)=5$$, then minimum possible number of values of x satisfying $$f(x)=5$$, for $$x\in [0, 170]$$ is?

A
$$21$$

B
$$12$$

C
$$11$$

D
$$22$$

Solution
Question 4
1 / -0

A real valued function $$f(x)$$ satisfies the function equation $$f(x-y)=f(x)f(y)-f(a-x)f(a+y)$$ where a is a given constant and $$f(0)=1, f(2a-x)$$ is equal to?

A
$$f(A)+f(a-x)$$

B
$$f(-x)$$

C
$$-f(x)$$

D
$$f(x)$$

Solution
Question 5
1 / -0

Let $$f\left( x \right)={2^{10}}x + 1$$ and $$g\left( x \right) = {3^{10}}x - 1$$ , If $$\left( {fog} \right)\left( x \right) = x$$ , then $$x$$ is equal to

A
$$\dfrac{{{3^{10}} - 1}}{{{3^{10}} - {2^{ - 10}}}}$$

B
$$\dfrac{{{2^{10}} - 1}}{{{2^{10}} - {3^{ - 10}}}}$$

C
$$\dfrac{{1 - {3^{10}}}}{{{2^{10}} - {3^{ - 10}}}}$$

D
$$\dfrac{{1 - {2^{-10}}}}{{{3^{10}} - {2^{ - 10}}}}$$

Solution

Given:$$f\left(x\right)={2}^{10}x+1$$ and $$g\left(x\right)={3}^{10}x-1$$

$$f\left(g\left(x\right)\right)=x$$

$$\Rightarrow\,f\left({3}^{10}x-1\right)=x$$

$$\Rightarrow\,{2}^{10}\left({3}^{10}x-1\right)+1=x$$

$$\Rightarrow\,{2}^{10}{3}^{10}x-{2}^{10}+1=x$$

$$\Rightarrow\,-{2}^{10}+1=x-{2}^{10}{3}^{10}x$$

$$\Rightarrow\,-{2}^{10}+1=x\left(1-{2}^{10}{3}^{10}\right)$$

$$\Rightarrow\,x=\dfrac{-{2}^{10}+1}{1-{2}^{10}{3}^{10}}$$

$$\Rightarrow\,x=\dfrac{{2}^{10}-1}{{2}^{10}{3}^{10}-1}$$

$$\Rightarrow\,x=\dfrac{{2}^{10}-1}{{2}^{10}{3}^{10}-1}$$

$$\Rightarrow\,x=\dfrac{{2}^{10}-1}{{2}^{10}\left({3}^{10}-{2}^{-10}\right)}$$

$$\Rightarrow\,x=\dfrac{1-{2}^{-10}}{\left({3}^{10}-{2}^{-10}\right)}$$
Question 6
1 / -0

$$f : R^+ \rightarrow R$$ defined by $$f(x) = 2^x , \, x \in (0, 1), \, f(x) = 3^x , \, x \in [1, \, \infty)$$ is

A
onto

B
one-one

C
neither one-one nor onto

D
one one onto

Solution
Question 7
1 / -0

Let f(x) and g(x) be the differentiable functions for $$1\le x\le 3$$ such that f(1)=2=g(1) and f(3)=10. Let there exist exactly one real number $$cE (1,3)$$ such that 3f'(c)=g'(c), then the value of g(3) must be

A
12

B
13

C
16

D
26

Solution
Question 8
1 / -0

If $$f(x)=|x|$$ and $$g(x)=[x]$$, then value of $$fog \left(-\dfrac {1}{4}\right)+gof \left(-\dfrac {1}{4}\right)$$ is ?

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$1/4$$

Solution
Question 9
1 / -0

The function $$f : R\rightarrow R$$ given by, then $$f(x)=3-2\sin x$$ is

A
one-one

B
onto

C
bijective

D
None of these

Solution
Question 10
1 / -0

If $$f\left( x \right) = \sin ^{ 2 }{ x } + \sin ^{ 2 }({ { x }+\frac { \pi }{ 3 }) + \cos { x\cos { \left( { x } + \frac { \pi }{ 3 } \right), ~g(\frac { 5 }{ 4 }) = 1, \text{then} \left( gof \right)\left( x \right) } }\ \text{ is}\ \text{equal}\ \text{to} } $$

A
$$1$$

B
$$0$$

C
$$\frac{1}{4}$$

D
$$\frac{1}{2}$$

Solution