Functions Test 30

Result

Functions Test 30

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

Consider the following functions are odd function in their default domains
(i) $$\cfrac { { 2 }^{ x }-1 }{ { 2 }^{ x }+1 } $$
(ii) $$\cfrac { { x }^{ 2 }+1 }{ x\sin { x } } $$
(iii) $$\ln { \left( \cfrac { 1+x }{ 1-x } \right) } $$
(iv) $$x{ e }^{ \left| x \right| +\cos { x } }\quad $$
Which of these is/are odd

A
(i) and (iii)

B
(i) and (iv)

C
all four

D
(i), (iii) and (iv)

Solution

$$(i)$$
$$f(x)=\dfrac{2^x-1}{2^x+1}$$

   $$f(-x)=\dfrac{2^{-x}-1}{2^{-x}+1}=\dfrac{1-2^x}{1+2^x}$$

  $$\implies f(-x)=-\dfrac{2^x-1}{2^x+1}=-f(x)$$

Hence is an odd function.

$$(ii)$$
$$f(x)=\dfrac{x^2+1}{x\sin x}$$

$$f(-x)=\dfrac{(-x)^2+1}{(-x)\sin {(-}x)}$$

  $$\implies f(-x)=\dfrac{x^2+1}{x\sin x}=f(x)$$

Hence is an even function.

$$(iii)$$
$$f(x)=\ln{\left(\dfrac{1+x}{1-x}\right)}$$

  $$f(-x)=\ln{\left(\dfrac{1-x}{1+x}\right)}$$

  $$\implies f(-x)=-\ln{\left(\dfrac{1+x}{1-x}\right)}=-f(x)$$

Hence is an odd function.

$$(iv)$$
$$f(x)=xe^{|x|+\cos{x}}$$

$$f(-x)=(-x)e^{|-x|+\cos{(-x)}}$$

$$f(-x)=-xe^{|x|+\cos{x}}=-f(x)$$

Hence is an odd function.

Hence (i), (iii) and (iv) are odd functions.

Answer-(D)
Question 2
1 / -0

Let $$A = \{ 1,2,3,4,5,6\} .$$ The number of onto functions from $$A$$ to$$A$$ such that.$$f\left( x \right) \ne x$$ for all $$x \in A$$ is

A
$$720$$

B
$$240$$

C
$$245$$

D
$$265$$

Solution

$$A=\{1,2,3,4,5,6\}$$

$$f:A\rightarrow A$$

$$\therefore \ $$Number of onto function$$=n!$$

$$ =6!$$

$$ =6\times 5\times 4\times 3\times 2\times 1$$

$$\therefore \ $$Number of onto function$$ =720$$
Question 3
1 / -0

Let $$f:A \to b$$ be a function defined by f(x) =$$\sqrt {1 - {x^2}} $$

A
f(x) is one-one if A =[0,1]

B
f(x) is onto if B = [0,1]

C
f(x) is one-one if A =[-1 , 0]

D
f(x) is onto if B = [-1,1]

Solution

$$f:A\rightarrow B$$
$$ f(x)=\sqrt { 1-{ x }^{ 2 } } $$
When $$ x=0$$
$$ f(0)=\sqrt { 1-0 } $$
$$ =1$$
$$ f(1)=\sqrt { 1-{ 1 }^{ 2 } } $$
$$ =0$$
$$ f(x)$$ is one - one function when $$A=[0,1]$$
Question 4
1 / -0

Let $$f$$, $$g:R\rightarrow {R}$$ be two functions defined as $$ f\left( x \right) =\left| x \right| +x$$, $$ g\left( x \right) =\left| x \right| -x, \forall x\in R$$. Then, find $$fog(x)$$

A
$$||x|-x|-|x|-x$$

B
$$||x|-x|+|x|-x$$

C
$$||x|-x|-|x|+x$$

D
None of thesse

Solution

$$f\left( x \right) = \left| x \right| + x$$
$$g\left( x \right) = \left| x \right| - x$$
$$fog\left( x \right) = f\left( {g\left( x \right)} \right)$$
$$ = f\left( {\left| x \right| - x} \right)$$
$$ = \left| {\left| x \right| - x} \right| + \left| x \right| - x$$
Question 5
1 / -0

Consider set $$A={1,2,3,4}$$ and set $$B={0,2,4,6,8}$$, then the number of one-one function from set $$A$$ to set $$B$$ is ?

A
$$5$$

B
$$24$$

C
$$120$$

D
None of these

Solution

Number of one-one function from
$$\underset{(m)}{A}$$ to $$\underset{(n)}{B} = \left\{\begin{matrix} \, ^nP_m, & if \, n \ge m \\ 0, & if \ n < m \end{matrix}\right.$$
$$m = 4, \ n = 5$$
One-one function $$=\, ^5P_4 = \dfrac{5!}{1!} = 120$$
Question 6
1 / -0

If $$g\left( x \right) = {x^2} + x - 2$$ and $$\frac{1}{2}gof\left( x \right) = 2{x^2} + 5x + 2$$, then $$f\left( x \right)$$ is

A
$$2x-3$$

B
$$2x+3$$

C
$$2{x^2} + 3x + 1$$

D
$$2{x^2} - 3x -1$$

Solution

$$g(x)={ x }^{ 2 }+x-2\\ \cfrac { 1 }{ 2 } g(f(x))=2{ x }^{ 2 }+5x+2\\ \Rightarrow f^{ 2 }\left( x \right) +f(x)-2=4{ x }^{ 2 }-10x+4\\ \Rightarrow f^{ 2 }\left( x \right) +f(x)-(4{ x }^{ 2 }-10x+6)=0\\ f(x)=\cfrac { -1\pm \sqrt { 1+4(4{ x }^{ 2 }-10x+6) } }{ 2 } \\ f(x)=\cfrac { -1\pm \sqrt { 1+16{ x }^{ 2 }-40x+24 } }{ 2 } \\ f(x)=\cfrac { -1\pm (4x-5) }{ 2 } =(2x-3)$$
Question 7
1 / -0

Let f : $$R \to R$$ and g : $$R \to R$$ be two one-one and onto functions such that they are the mirror images of each other about the line y = 2. If h(x) = f(x) + g(x), then h(0) equal to

A
2

B
4

C
0

D
1

Solution

Given $$g(x)$$ and $$f(x)$$ are mirror images about $$y=2$$
$$\implies g(x)-2=2-f(x)\implies f(x)+g(x)=4\implies h(x)=4$$
$$h(0)=4$$
Question 8
1 / -0

If $$f(x)=2x+5$$ and $$g(x)=x^2+1$$ be two real function , then value of $$fog$$ at x=1

A
$$9$$

B
$$6$$

C
$$5$$

D
$$4$$

Solution
Question 9
1 / -0

$$c \to c\,\,is\,defined\,as\,f\left( x \right) = \frac{{ax + b}}{{cx + d}}\,\,bd \ne 0$$.then f is a constant function when

A
a=c

B
b=d

C
ad=bc

D
ab=cd

Solution
Question 10
1 / -0

Let $$A$$ be a set of $$4$$ elements and $$B$$ has $$3$$ elements . From the set of all functions from $$A$$ to $$B$$, the probability that it is an onto function is

A
$$\dfrac{4}{9}$$

B
$$0$$

C
$$\dfrac{29}{32}$$

D
$$1$$

Solution

No .of functions from $$A$$ to $$B$$ is $$3^4=81$$ elements
No.of onto functions $$4\times 3^2=36$$
Probability $$\dfrac{36}{81}=\dfrac 49$$

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Functions Test 30

Result

Functions Test 30

Score

-

Rank

-

SHARING IS CARING

Question 1

(i) and (iii)

(i) and (iv)

all four

(i), (iii) and (iv)

Solution

Question 2

$$720$$

$$240$$

$$245$$

$$265$$

Solution

Question 3

f(x) is one-one if A =[0,1]

f(x) is onto if B = [0,1]

f(x) is one-one if A =[-1 , 0]

f(x) is onto if B = [-1,1]

Solution

Question 4

$$||x|-x|-|x|-x$$

$$||x|-x|+|x|-x$$

$$||x|-x|-|x|+x$$

None of thesse

Solution

Question 5

$$5$$

$$24$$

$$120$$

None of these

Solution

Question 6

$$2x-3$$

$$2x+3$$

$$2{x^2} + 3x + 1$$

$$2{x^2} - 3x -1$$

Solution

Question 7

2

4

0

1

Solution

Question 8

$$9$$

$$6$$

$$5$$

$$4$$

Solution

Question 9

a=c

b=d

ad=bc

ab=cd

Solution

Question 10

$$\dfrac{4}{9}$$

$$0$$

$$\dfrac{29}{32}$$

$$1$$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.