Relations and Functions Test

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Relations and Functions Test - 47

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Weekly Quiz Competition

Question 1
1 / -0

If $$R$$ is a relation on the set $$A=\left\{ 1,2,3,4,5,6,7,8,9 \right\} $$ given by $$xRy\Leftrightarrow y=3x$$, then $$R=$$

A
$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 8,2 \right) ,\left( 9,3 \right) \right\} $$

B
$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 9,3 \right) \right\} $$

C
$$\left\{ \left( 3,1 \right) ,\left( 2,6 \right) ,\left( 3,9 \right) \right\} $$

D
none of these

Solution

Given, $$R$$ is a relation on the set $$A=\left\{ 1,2,3,4,5,6,7,8,9 \right\} $$ given by $$xRy\Leftrightarrow y=3x$$, then $$R=\{(1,3),(6,2),(9,3)\}$$.
Question 2
1 / -0

A is a set having $$6$$ distinct elements. The number of distinct function from A to A which are not bijections is?

A
$$6!-6$$

B
$$6^6-6$$

C
$$6^6-6!$$

D
$$6!$$

Solution

Total$$-$$ bijections

$$6^6-6!$$.
Question 3
1 / -0

If $$f(x) = log_e \left(\dfrac{1 - x}{1 + x} \right) $$ then $$f \left(\dfrac{2x}{1 + x^2} \right)$$ is equal to

A
$$f(x)$$

B
$$2f(x)$$

C
$$-2 f(x)$$

D
$$(f(x))^2$$

Solution

$$f(x) = \ell n \left(\dfrac{1 - x}{1 + x} \right)$$
$$f \left(\dfrac{2x}{1 + x^2} \right) = \ell n \left(\dfrac{1 - \dfrac{2x}{1 + x^2}}{1 + \dfrac{2x}{1 + x^2}} \right) = \ell n \left(\dfrac{1 + x^2 - 2x}{1 + x^2 + 2x} \right) = \ell n \left(\left(\dfrac{1 - x}{1 + x} \right)^2 \right) = 2 \ell n \left(\dfrac{1 - x}{1 + x} \right) = 2f(x)$$
Question 4
1 / -0

The number of ordered pairs (a, b) of positive integers such that $$\dfrac{2a - 1}{b}$$ and $$\dfrac{2b - 1}{a}$$ are both integers is

A
$$1$$

B
$$2$$

C
$$3$$

D
more than $$3$$

Solution

The number of ordered pairs $$\left(a, b\right)$$ of positive integers where $$a,b\in{I}^{+}$$
$$\Rightarrow\left(\dfrac{2a-1}{b},\dfrac{2b-1}{a}\right)\in{I}^{+}$$
Let $$\dfrac{2a-1}{b}=1$$
$$\Rightarrow\,2a-1=b$$

and
$$\Rightarrow\,\dfrac{2b-1}{a}=\dfrac{2\left(2a-1\right)-1}{a}$$
$$=\dfrac{4a-2-1}{a}$$
$$=\dfrac{4a-3}{a}$$

$$\therefore \dfrac{2b-1}{a}=4-\dfrac{3}{a}$$
For integer,$$a=1,3$$ we get
$$b=2a-1=2-1=1$$ for $$a=1$$
$$b=2a-1=2\times 3-1=6-1=5$$
$$\therefore\,b=1,5$$
So total sets are $$\left(1,1\right),\left(3,5\right),\left(5,3\right)$$
Hence there are totally $$3$$ sets.
Question 5
1 / -0

Let $$f : \rightarrow $$ satisfy $$f(x) f(y) = f(xy)$$ for all real numbers x and y. If $$f(2) = 4$$ then $$f\left(\dfrac{1}{2} \right) = $$

A
$$0$$

B
$$\dfrac{1}{4}$$

C
$$\dfrac{1}{2}$$

D
$$1$$

E
$$2$$

Solution

Given
$$f(x) f(y) = f(xy) $$ __(i)
On taking $$x = 1, y = 1$$
$$f(1) f(1) = f(1\times1) = f(1^2)= f(1) = 1$$
Now, $$x = 2, y = \dfrac{1}{2}$$, then from equation (i)
$$f(2) f\left(\dfrac{1}{2} \right) = f\left(2. \dfrac{1}{2} \right)$$
$$\Rightarrow 4 f\left(\dfrac{1}{2} \right) = f(1) $$ $$[\because f(2) = 4]$$
$$\Rightarrow f \left(\dfrac{1}{2} \right) = \dfrac{1}{4} f(1)$$
On putting the value of $$f(1)$$,
$$\Rightarrow f \left(\dfrac{1}{2} \right) = \dfrac{1}{4} \times 1 = \dfrac{1}{4}$$
Question 6
1 / -0

For all rest numbers $$x$$ and $$y$$, it is known as the real valued function $$f$$ satisfies $$f(x) +f(y) = f(x+y)$$. If $$f(1) = 7$$. then $$\displaystyle \sum_{r=1}^{100} f(r)$$ is equal to

A
$$7\times 51 \times 102$$

B
$$6\times 50 \times 102$$

C
$$7\times 50 \times 102$$

D
$$7\times 50 \times 101$$

Solution

We have,
$$f(x+y) = f(x) + f(y)$$ ...(i)

Put, $$x = y= 1$$ in equation (i),

We get, $$f(2) = f(1) + f(1) = 2f(1) = 2\times 7$$

Again, put $$x = 1, y = 2$$ in Equation (i), we get

$$f(3) = f(1) + f(2) = 7 + 2 \times 7 = 3\times 7$$

$$\therefore f(n) = n \times 7$$

Now, $$\displaystyle \sum_{r=1}^{100} f(r) = f(1) + f(2) + f(3)+...+f(100)$$

$$= 7 + 2 \times 7 + 3 \times 7 + ...+ 100\times 7$$

$$= 7[1+2+3...+100]$$

$$= 7\times \dfrac{100(100+1)}{2}\left[ \because \sum n = \dfrac{n(n+1)}{2}\right]$$

$$= 7\times 50\times 101$$
Question 7
1 / -0

If $$f(x) = \dfrac{1}{x^2 + 4x + 4} - \dfrac{4}{x^4 + 4x^2 + 4x^2} + \dfrac{4}{x^3 + 2x^{2'}}$$ then $$f\left(\dfrac{1}{2}\right)$$ is equal to

A
$$1$$

B
$$2$$

C
$$-1$$

D
$$4$$

Solution

We have,
$$f(x) = \dfrac{1}{x^2 + 4x + 4} - \dfrac{4}{x^4 + 4x^2 + 4x^2} + \dfrac{4}{x^3 + 2x^{2}}$$

$$= \dfrac{1}{(x+2)^2} - \dfrac{4}{x^2(x+2)^2} + \dfrac{4}{x^2(x+2)}$$

$$= \dfrac{x^2 - 4+4(x+2)}{(x+2)^2 .x^2}$$

$$= \dfrac{x^2 -4+4x+8}{(x+2)^2.x^2}$$

$$= \dfrac{x^2+4x+4}{(x+2)^2.x^2}$$

$$=\dfrac{(x+2)^2}{(x+2)^2.x^2}$$

$$= \dfrac{1}{x^2}$$

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

$$\Rightarrow f\left(\dfrac{1}{2}\right) = \dfrac{1}{\left(\dfrac{1}{2}\right)^2} = 4$$
Question 8
1 / -0

If $$A=\{x\in R:x^2-5|x|+6=0\}$$, then $$n(A)=$$ ________.

A
$$2$$

B
$$0$$

C
$$1$$

D
$$4$$

Solution

The question indirectly wants to know the number of real roots for the given quadratic equation,

Let's solve for roots,

$$x^2-5|x|+6=0\rightarrow (|x|-2)(|x|-3)=0$$ which gives $$x=\{2,-2,3,-3\}$$

Thus $$n(A)=4$$
Question 9
1 / -0

If $$f(x)=3x-2$$ and $$g(x)=x^2$$, then $$fog(x)=$$_______.

A
$$3x^2-2$$

B
$$3x^2+2$$

C
$$3x-2$$

D
$$2-3x^2$$

Solution

$$f(x) = 3x - 2 $$ and $$g(x) = x^2$$
Now, $$fog(x) = f (g(x))$$
$$ = f(x^2)$$
$$= 3(x^2) - 2$$

$$\therefore\ fog(x) = 3x^2 - 2$$
Question 10
1 / -0

Let $$f: (-1, 1) \rightarrow (-1, 1)$$ be continuous , $$f(x) = f(x^2)$$ for all $$x \in (-1, 1)$$ and $$f(0) = \dfrac{1}{2}$$ , then the value of $$4 f \left(\dfrac{1}{4} \right)$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

E
$$5$$

Solution

Given,
$$f(x)=f(x^2)$$

Now, $$f(1)=f(-1)=f(1)$$

similarly, $$f(\dfrac12)=f(\dfrac{1}4)=f(\dfrac{-1}{2})$$

which is only possible when the function is a constant function.

So, $$f(0)=\dfrac12=f(\dfrac14)$$

Therefore, $$4f\left(\dfrac{1}{4}\right)$$$$=4.\dfrac{1}{2}=2$$

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Relations and Functions Test - 47

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Relations and Functions Test - 47

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Question 1

$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 8,2 \right) ,\left( 9,3 \right) \right\} $$

$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 9,3 \right) \right\} $$

$$\left\{ \left( 3,1 \right) ,\left( 2,6 \right) ,\left( 3,9 \right) \right\} $$

none of these

Solution

Question 2

$$6!-6$$

$$6^6-6$$

$$6^6-6!$$

$$6!$$

Solution

Question 3

$$f(x)$$

$$2f(x)$$

$$-2 f(x)$$

$$(f(x))^2$$

Solution

Question 4

$$1$$

$$2$$

$$3$$

more than $$3$$

Solution

Question 5

$$0$$

$$\dfrac{1}{4}$$

$$\dfrac{1}{2}$$

$$1$$

$$2$$

Solution

Question 6

$$7\times 51 \times 102$$

$$6\times 50 \times 102$$

$$7\times 50 \times 102$$

$$7\times 50 \times 101$$

Solution

Question 7

$$1$$

$$2$$

$$-1$$

$$4$$

Solution

Question 8

$$2$$

$$0$$

$$1$$

$$4$$

Solution

Question 9

$$3x^2-2$$

$$3x^2+2$$

$$3x-2$$

$$2-3x^2$$

Solution

Question 10

$$1$$

$$2$$

$$3$$

$$4$$

$$5$$

Solution

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