Functions Test 34

Result

Functions Test 34

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Weekly Quiz Competition

Question 1
1 / -0

If $f ( x ) = \left( a - x ^ { n } \right) ^ { 1 / n }$ where $a > 0$ and $n$ is a positive integer then $( f o f ) ( x )$ is

A
$f ( x )$

B
$x$

C
$0$

D
$1$

Solution

$f(n)= (a-x^{n})^{1/n}$
$(fof)^{(n)}=(a-((a-x^{n})^{1/n})^{n})^{1/n}$
$=(a-a+x^{n})^{1/n}$
$fof(n)=x$
Question 2
1 / -0

$f:R \rightarrow R$ such that $f(x)=\ell n(x+\sqrt {x^{2}+1})$ . Another function $g(x)$ is defined such that $gof(x)=x\ \forall\ x \in\ R$ . Then $g(2)$ is -

A
$\dfrac {e^{2}+e^{-2}}{2}$

B
$e^{2}$

C
$\dfrac {e^{2}-e^{-2}}{2}$

D
$e^{-2}$

Solution

$f(x)= ln (x+ \sqrt{x^{2}+1})$

$g(x)= g(f(x))$ $g(2)=?$

ln $(x+ \sqrt{x^{2}+1})=y$

$\sqrt{x^2+1}= e^y-{x}$

$x^{2}+1= e^{2y}-2e^{y}x+x^{2}$

$e^{2y}-2e^{y}x-1=o \Rightarrow$

$x =\dfrac{e^{2y}-1}{2e^{y}}$

$=\dfrac{(e^{y}- e^{-y})}{2}$

$\therefore g(x) = \dfrac{e{^y}- e^{-y}}{2} \Rightarrow g (2)= \dfrac{e^{2}-e^{-2}}{2}$
Question 3
1 / -0

Let $f:R\rightarrow R$ is a function satisfying $f(2-x)=f(2+x)$ and $f(20-x)=f(x)\forall x\in R$
If $f(0)=5$ then the minimum possible no. of values of $x$ satisfying $f(x)=5$ for $x=[0.,70]$ , is

A
$21$

B
$12$

C
$11$

D
$22$

Solution

Given, $f(2 - x) = f(2 + x) .. (i)$
$\therefore f(x)$ is symmetric about $x = 2$
$f(20 - x) = f(x) ..(ii)$
$x \rightarrow x + 10$
$f(10 - x) = f(10 + x)$
$f(x)$ is symmetric about $x = 10$
$x \rightarrow x + 2$
$f(18 - x) = f(x + 2) .. (ii)$
$f(2 - x) = f(18 - x)$
$x \rightarrow -x$
$f(2 + x) = f(18 + x)$
$\therefore x + 2 \rightarrow x$
$\therefore f(x) = f(x + 16)$
$f(x)$ has period = $16$
$f(x) = 5 , \, x \in [0, 170]$
put $x = 2$ in (i) $f(0) = f(4)$
put $x = 4$ in (ii) $f(4) = f(16)$
when $x \in [0, 16] \rightarrow f(x)$ has two solution
when $x \in [0, 160]$ has $20$ solution.
When $x in [0, 170]$ has $'21'$ solution
Question 4
1 / -0

Let $f\left( x \right) = {x^2}$ and $g\left( x \right) = {2^x}$ . Then the solution of the equation $fog\left( x \right) = gof\left( x \right)$ is

A
$R$

B
$\left\{ {0} \right\}$

C
$\left\{ {0,\,2} \right\}$

D
none

Solution

$f\left(x\right)={x}^{2}$

$f{g\left(x\right)}=f{\left({2}^{x}\right)}={\left({2}^{x}\right)}^{2}={2}^{2x}$

$g\left(x\right)={2}^{x}$

$g{f\left(x\right)}=g{\left({x}^{2}\right)}={2}^{{x}^{2}}$

Given: $f{g\left(x\right)}=g{f\left(x\right)}$

$\Rightarrow\,{2}^{2x}={2}^{{x}^{2}}$

Since bases are same we can equate the powers
$\Rightarrow\,2x={x}^{2}$
$\Rightarrow\,{x}^{2}-2x=0$
$\Rightarrow\,x\left(x-2\right)=0$
$\Rightarrow\,x=0,2$

When $x=0,\,f{g\left(x\right)}=g{f\left(x\right)}\Rightarrow\,{2}^{0}={2}^{0}=1$
When $x=2,\,f{g\left(x\right)}=g{f\left(x\right)}\Rightarrow\,{2}^{2\times 2}={2}^{{2}^{2}}\Rightarrow\,16=16$
Hence $x=\left\{0,2\right\}$
Question 5
1 / -0

All values of a for which f : R $\to R$ defined by f(x)= ${x^3} + a{x^2} + 3x + 100$ is a one one functions, are

A
$( - \infty , - 2)$

B
$( - \infty ,4)$

C
$(4, - 4)$

D
$( - 3,3)$

Solution

$f ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 a x + 3$
$f ^ { \prime } ( x ) = 0$ and f ( x ) > 0 $f ^ { \prime } ( x ) < 0$
$3 x ^ { 2 } + 2 a x + 3 = 0$
$x = \dfrac { - 2 a \pm \sqrt { 4 a ^ { 2 } - 36 } } { 6 }$
$4 a ^ { 2 } < 36$
$a \in ( - 3,3 )$
Question 6
1 / -0

If $f ( x ) = \sin ^ { - 1 } ( \sin x ) + \cos ^ { - 1 } ( \sin x ) \text { and } \phi ( x ) = f ( f ( f ( x ) ) )$ then $\phi ^ { \prime } ( x )$

A
1

B
$\sin x$

C
0

D
none of these

Solution

$f(x)=\sin ^{ -1 }{ \left( \sin { x } \right) } +\cos ^{ -1 }{ \left( \sin { x } \right) } =\cfrac { \pi }{ 2 } \left[ \because \sin ^{ -1 }{ \theta } +\cos ^{ -1 }{ \theta } =\cfrac { \pi }{ 2 } \right]$
$f(f(x))=f\left(\cfrac { \pi }{ 2 }\right )=\cfrac { \pi }{ 2 }$
$\phi (x)=f(f(f(x)))=f\left(\cfrac { \pi }{ 2 }\right )=\cfrac { \pi }{ 2 }$
$\phi (x)=\cfrac { \pi }{ 2 }$
$\phi '(x)=0$
Question 7
1 / -0

if $f\left( x \right) = 3x + 2$ , $g\left( x \right) = {x^2} + 1$ ,then the values of $\left( {f_og} \right)\left( {{x^2} - 1} \right)$

A
$3{x^4} - 6{x^2} + 8$

B
$3{x^4} + 3x + 4$

C
$6{x^4} + 3{x^2} - 2$

D
$6{x^4} + 3{x^2} + 2$

Solution

$f\left(x\right)=3x+2$

$g\left(x\right)={x}^{2}+1$
$f.g\left(x\right)=f\left({x}^{2}+1\right)$
$f.g\left(x\right)=3\left({x}^{2}+1\right)+2=3{x}^{2}+5$

$f.g\left(x\right)=3{x}^{2}+5$
$f.g\left({x}^{2}-1\right)=3{\left({x}^{2}-1\right)}^{2}+5$

$=3\left({x}^{4}-2{x}^{2}+1\right)+5$

$=3{x}^{4}-6{x}^{2}+3+5$

$=3{x}^{4}-6{x}^{2}+8$
Question 8
1 / -0

Let A = {1,2,3,4,5} and B={1,2,3,4,5}. If $f:A\rightarrow B$ is an one-one function and $f(x)=x$ holds only for one value of $x\epsilon \{ 1,2,3,4,5\} ,$ then the number of such possible function is

A
$120$

B
$36$

C
$45$

D
$44$

Solution
Question 9
1 / -0

The domain function $\sqrt{log_{10}(\frac{5x-x^{2}}{4})}$ is

A
[1,4]

B
[-1,4]

C
[0,5]

D
[-1,5]

Solution
Question 10
1 / -0

$f(x) = \mid |x|^2 - 2 |x| -3 \mid$ is non differentiable at $k$ points in its domain , the value of $k$ is

A
$3$

B
$2$

C
$4$

D
$1$

Solution

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Functions Test 34

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Functions Test 34

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Question 1

f(x)f ( x )f(x)

xxx

000

111

Solution

Question 2

e2+e−22\dfrac {e^{2}+e^{-2}}{2}2e2+e−2​

e2e^{2}e2

e2−e−22\dfrac {e^{2}-e^{-2}}{2}2e2−e−2​

e−2e^{-2}e−2

Solution

Question 3

212121

121212

111111

222222

Solution

Question 4

RRR

{0}\left\{ {0} \right\}{0}

{0, 2}\left\{ {0,\,2} \right\}{0,2}

none

Solution

Question 5

(−∞,−2)( - \infty , - 2)(−∞,−2)

(−∞,4)( - \infty ,4)(−∞,4)

(4,−4)(4, - 4)(4,−4)

(−3,3)( - 3,3)(−3,3)

Solution

Question 6

1

sin⁡x\sin xsinx

0

none of these

Solution

Question 7

3x4−6x2+83{x^4} - 6{x^2} + 83x4−6x2+8

3x4+3x+43{x^4} + 3x + 43x4+3x+4

6x4+3x2−26{x^4} + 3{x^2} - 26x4+3x2−2

6x4+3x2+26{x^4} + 3{x^2} + 26x4+3x2+2

Solution

Question 8

120120120

363636

454545

444444

Solution

Question 9

[1,4]

[-1,4]

[0,5]

[-1,5]

Solution

Question 10

333

222

444

111

Solution

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$f ( x )$

$x$

$0$

$1$

$\dfrac {e^{2}+e^{-2}}{2}$

$e^{2}$

$\dfrac {e^{2}-e^{-2}}{2}$

$e^{-2}$

$21$

$12$

$11$

$22$

$R$

$\left\{ {0} \right\}$

$\left\{ {0,\,2} \right\}$

$( - \infty , - 2)$

$( - \infty ,4)$

$(4, - 4)$

$( - 3,3)$

$\sin x$

$3{x^4} - 6{x^2} + 8$

$3{x^4} + 3x + 4$

$6{x^4} + 3{x^2} - 2$

$6{x^4} + 3{x^2} + 2$

$120$

$36$

$45$

$44$

$3$

$2$

$4$

$1$