Relations and Functions Test

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Relations and Functions Test - 34

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Weekly Quiz Competition

Question 1
1 / -0

If $$ f: R->R$$ is defined by $$f(x) = |x|$$, then

A
$$f^{-1}_{}(x) = -x$$

B
$$f^{-1}_{}(x) = \dfrac{1}{|x|}$$

C
The function $$f^{-1}_{}(x)$$ does not exist

D
$$f^{-1}_{}(x) = \dfrac{1}{x}$$

Solution

For a function $$f(x)$$ to be invertible, the function must be one-one and onto.

The range of $$f(x) = |x|$$ is $$[0, \infty)$$, while the co-domain of $$f(x)$$ is given as $$\mathbb{R}$$. Hence $$f(x)$$ is not onto.

Also, since $$f(x) = f(-x)$$, $$f(x)$$ is also not one-one in its domain.

Hence, f(x) is not invertible, ie, the function $$f^{-1}(x)$$ does not exist.
Option C is the right answer.
Question 2
1 / -0

If $$a \times b =2 a - 3b + ab$$, then $$3 \times 5+5\times 3$$ is equal to

A
$$22$$

B
$$24$$

C
$$26$$

D
$$28$$

Solution

$$ a\times b = 2a-3b+ab $$

find $$ 3\times 5+5\times 3$$

$$ 3\times 5[a = 3, b = 5]$$

$$ \therefore 3\times 5 = 2(3)-3(5)+3(5)$$

$$ 3\times 5= 6 $$

$$ 5\times 3 [a = 5, b = 3]$$

$$ \therefore 5\times 3 = 2(5)-3(3)+(5)(3)$$

$$ = 10-9+15$$

$$ 5\times 3= 16$$

$$ \therefore 3\times 5+5\times 3 = 6+16$$

$$ 3\times 5+5\times 3= 22 $$
Question 3
1 / -0

If $$x \times y = x^{2}+y^{2}-xy$$ then the value of $$9 \times 11$$ is :

A
$$93$$

B
$$103$$

C
$$113$$

D
$$121$$

Solution

$$ x \times y = x^{2}+y^{2}-xy $$
$$ 9\times 11 = 9^{2}+11^{2}-9(11) $$
$$ = 81+121-99 $$
$$ 9\times 11 = 103 $$
Question 4
1 / -0

Let $$R$$ be the relation on $$Z$$ defined by $$R = \{(a, b): a, b \in z, a - b$$ is an integer$$\}$$. Find the domain and Range of $$R$$.

A
$$z, z$$

B
$$z^+, z$$

C
$$z, z^-$$

D
None of these

Solution

Given:

$$R=\{ (a, b) : a, b, \in z, a-b \text{ is an integer}\}$$

As difference of integers are also integers so,

Domain of $$R = z$$

Range of $$R = z$$, as

$$a-b$$ spans the whole integer values.
Question 5
1 / -0

Let $$f(x)={x}^{3}-6{x}^{2}+15x+3$$. Then,

A
$$f(x)> 0$$ for all $$x\in R$$

B
$$f(x)> f(x+1)$$ for all $$x\in R$$

C
$$f(x)$$ is invertible

D
$$f(x)< 0$$ for all $$x\in R$$

Solution

$$f(x)=x^3-6x^2+15x+3$$

$$f'(x)=3x^2-12x+15$$
$$=3(x^2-4x+5)$$
$$=3(x^2-4x+4+1)$$

$$f'(x)=3(x-2)^2+{3} > 0$$

Therefore $$f(x)$$ is strictly increasing function

$$\Rightarrow f^{-1}(x)$$ exists

Hence $$f(x)$$ is a invertible function.
Question 6
1 / -0

Read the following information and answer the three items that follow :
Let $$f(x) = x^2 + 2x - 5 $$ and $$g(x) = 5x + 30$$
If $$h(x) = 5f(x) - xg (x)$$, then what is the derivative of $$h(x)$$ ?

A
$$-40$$

B
$$-20$$

C
$$-10$$

D
$$0$$

Solution

Given,

$$f(x)=x^2+2x-5$$

$$g(x)=5x+30$$

$$h(x)=5f(x)-xg(x)$$

$$=5(x^2+2x-5)-x(5x+30)$$

$$=5x^2+10x-25-5x^2-30x$$

$$h(x)=-20x-25$$

$$\dfrac{d}{dx}[h(x)]=\dfrac{d}{dx}(-20x-25)$$

$$=-20$$
Question 7
1 / -0

The number of binary operation on {1, 2, 3... n} is..

A
$$2^n$$

B
$$n^2$$

C
$$n^3$$

D
$$n^{2n}$$

Solution

we have general formula,

for the $$n$$ number of series, the number of binary operation is given by,

$$2^n$$
Question 8
1 / -0

Read the following information and answer the three items that follow :
Let $$f(x) = x^2 + 2x - 5 $$ and $$g(x) = 5x + 30$$
Consider the following statements:
1. $$f[g(x)]$$ is a polynomial of degree 3.
2. $$g[g(x)]$$ is a polynomial of degree 2.
Which of the above statements is/are correct ?

A
1 only

B
2 only

C
Both 1 and 2

D
Neither 1 nor 2

Solution

Neither 1 nor 2

Given,

$$f(x)=x^2+2x-5$$

$$g(x)=5x+30$$

(i)

$$f[g(x)]$$

$$=f[5x+30]$$

$$=(5x+30)^2+2(5x+30)-5$$

upon solving the above equation, we get,

$$f[g(x)]=25x^2+310x+955$$

degree $$=2$$

(ii)

$$g[g(x)]$$

$$=g[5x+30]$$

$$=5(5x+30)+30$$

$$=25x+150+30$$

$$=25x+180$$

$$g[g(x)]=25x+180$$

degree $$=1$$
Question 9
1 / -0

Let $$f(x)=\cfrac { 1 }{ 1-x } $$. Then $$\left\{ f\circ \left( f\circ f \right) \right\} (x)$$

A
$$x$$ for all $$x\in R$$

B
$$x$$ for all $$x\in R-\left\{ 1 \right\} $$

C
$$x$$ for all $$x\in R-\left\{ 0,1 \right\} $$

D
none of these

Solution

$$f(x)=\dfrac{1}{1-x}$$ for $$x\in R-\{1\}$$

$$\{fof\}(x)=\dfrac{1}{1-\dfrac{1}{1-x}}=\dfrac{1}{\dfrac{1-x-1}{1-x}}=\dfrac{x-1}{x}=1-\dfrac{1}{x}$$ for $$x\in R - \{0,1\}$$

$$\{fofof\}(x)=\dfrac{1}{1-fof(x)}=\dfrac{1}{1-\dfrac{x-1}{x}}=\dfrac{1}{\dfrac{x-x+1}{x}}=x$$ for $$x\in R-\{0,1\}$$
Question 10
1 / -0

Read the following information and answer the three items that follow :
Let $$f(x) = x^2 + 2x - 5 $$ and $$g(x) = 5x + 30$$
What are the roots of the equation $$g[f(x)] = 0$$ ?

A
$$1, -1$$

B
$$-1, -1$$

C
$$1, 1$$

D
$$0, 1$$

Solution

Given,

$$f(x)=x^2+2x-5$$

$$g(x)=5x+30$$

$$g[f(x)]=0$$

$$g[x^2+2x-5]=0$$

$$5(x^2+2x-5)+30=0$$

$$5x^2+10x-25+30=0$$

$$5x^2+10x+5=0$$

$$x^2+2x+1=0$$

$$(x+1)^2=0$$

$$\therefore x=-1,-1$$

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Relations and Functions Test - 34

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Relations and Functions Test - 34

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Question 1

$$f^{-1}_{}(x) = -x$$

$$f^{-1}_{}(x) = \dfrac{1}{|x|}$$

The function $$f^{-1}_{}(x)$$ does not exist

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

Solution

Question 2

$$22$$

$$24$$

$$26$$

$$28$$

Solution

Question 3

$$93$$

$$103$$

$$113$$

$$121$$

Solution

Question 4

$$z, z$$

$$z^+, z$$

$$z, z^-$$

None of these

Solution

Question 5

$$f(x)> 0$$ for all $$x\in R$$

$$f(x)> f(x+1)$$ for all $$x\in R$$

$$f(x)$$ is invertible

$$f(x)< 0$$ for all $$x\in R$$

Solution

Question 6

$$-40$$

$$-20$$

$$-10$$

$$0$$

Solution

Question 7

$$2^n$$

$$n^2$$

$$n^3$$

$$n^{2n}$$

Solution

Question 8

1 only

2 only

Both 1 and 2

Neither 1 nor 2

Solution

Question 9

$$x$$ for all $$x\in R$$

$$x$$ for all $$x\in R-\left\{ 1 \right\} $$

$$x$$ for all $$x\in R-\left\{ 0,1 \right\} $$

none of these

Solution

Question 10

$$1, -1$$

$$-1, -1$$

$$1, 1$$

$$0, 1$$

Solution

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