Relations and Functions Test

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Relations and Functions Test - 18

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Weekly Quiz Competition

Question 1
1 / -0

Let $$R$$ be a relation such that $$R = \{(1,4), (3, 7), (4, 5), (4, 6), (7, 6) \}$$ then $$(R^{-1} oR)^{-1} =$$

A
$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4), (4, 3)\}$$

B
$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4) \}$$

C
$$\{(1, 1), (3, 3), (4, 4) \}$$

D
$$\phi$$

Solution
Question 2
1 / -0

The population of a town increases by $$5\%$$ every year. If the present population is $$5,40,000$$ find the population after 2 years.

A
5,95,350

B
5,94,000

C
5,95,000

D
5,94,350

Solution

Population after 1 year = 5,40,000 + 5% of 5,40,000 = 5,67,000
Population after 2 years = 5,67,000 + 5% of 5,67,000 = 5,95,350
Question 3
1 / -0

If $$f(x+y)=2f(x)\times f(y)$$, $$f$$ is differentiable and $$f(2)=8$$ then $$f'(3)$$ equals

A
$$64 (ln 2)$$

B
$$128 (ln 2)$$

C
$$256 (ln 2)$$

D
$$1024 (ln 2)$$
Question 4
1 / -0

Find x and y, if $$(x+3,5)=(6,2x+y)$$.

A
x=3, y=-1

B
x=6, y=2

C
x=2,y=3

D
none of these

Solution

By the definition of equality of ordered pairs
$$(x+3,5)=(6,2x+y)$$
$$x+3=6$$ and $$5=2x+y$$
$$x=3$$ and $$5=2x+y$$
$$x=3$$ and $$5=6+y$$
$$x=3$$ and $$y=-1$$
Question 5
1 / -0

If $$f\left( {\sqrt {\left( {\dfrac{{x - 1}}{{x + 1}}} \right)} } \right) = 3x,$$ then the value of $$f\left( 3 \right)$$ is :

A
$$ - \dfrac{{15}}{4}$$

B
$$\dfrac{{\sqrt 2 }}{2}$$

C
$$9$$

D
$$0$$

Solution

$$1(\sqrt{\frac{x-1}{x+1}})=3x$$
$$\Rightarrow \sqrt{\frac{x-1}{x+1}}=3\Rightarrow \frac{x-1}{x+1}=9$$
$$x-1=9x+9$$
$$-8x=10 \Rightarrow x=\frac{-5}{4}$$
$$f(3)=3(\frac{-5}{4})=\frac{-15}{4}$$
Question 6
1 / -0

The function $$f\left( x \right) = \log x - \dfrac{{2x}}{{2 + x}}$$ is increasing in the interval

A
$$\left( { - \infty ,0} \right)$$

B
$$\left( {0,\infty } \right)$$

C
$$\left( {1,\infty } \right)$$

D
$$\left( { - \infty ,1} \right)$$

Solution

$$f(x)=log x-\frac{2x}{x+2}$$
$$f'(x)=\frac{1}{x}-(\frac{1\times 2x-2(x+2)}{(x+2)^{2}})$$
$$\Rightarrow \frac{1}{x}-(\frac{2x-2x-4}{x^{2}+4+4x})\geq 0$$
$$\Rightarrow \frac{x^{2}+4x+4+4x}{x(x+2)^{2}}\geq 0$$
$$\Rightarrow \frac{x^{2}+8x+4}{x(x+2)^{2}}\geq 0$$
so, $$x\epsilon (0,\infty )$$
Question 7
1 / -0

A relation $$R$$ is defined from $$\left\{ 2,3,4,5 \right\} $$ to $$\left\{ 3,6,7,10 \right\} $$ by:
$$xRy\Leftrightarrow x$$ is relatively prime to $$y$$. Then, domain of $$R$$ is

A
$$\left\{ 2,3,5 \right\} $$

B
$$\left\{ 3,5 \right\} $$

C
$$\left\{ 2,3,4 \right\} $$

D
$$\left\{ 2,3,4,5 \right\} $$

Solution

Let's first write R as a set of ordered pairs,

$$R=\{(2,3),(2,7),(3,7),(3,10),(4,3),(4,7),(5,3),(5,6),(5,7)\}$$

So, Domain of R is $$\{2,3,4,5\}$$
Question 8
1 / -0

Let $$R$$ be a relation on the set $$N$$ given by $$R=\left\{ \left( a,b \right) :a=b-2,b>6 \right\}$$. Then

A
$$(2,4)\in R$$

B
$$(3,8)\in R$$

C
$$(6,8)\in R$$

D
$$(8,7)\in R$$

Solution

$$a=b-2,b>6$$

Above relation also imply $$a>6-2\rightarrow a>4$$

Among the options

$$C$$ is correct answer as $$6>4$$ and aslo $$6=8-2$$ .
Question 9
1 / -0

If $$A=\left\{ 1,2,3 \right\} , B=\left\{ 1,4,6,9 \right\} $$ and $$R$$ is a relation from $$A$$ to $$B$$ defined by $$x$$ is greater than $$y$$. The range of $$R$$ is

A
$$\left\{ 1,4,6,9 \right\} $$

B
$$\left\{ 4,6,9 \right\} $$

C
$$\left\{ 1 \right\} $$

D
none of these

Solution

First Let's write R as set of ordered pairs,
$$R=\{(2,1),(3,1)\}$$
We can simply see from set that range of $$R = \{1\}$$
Question 10
1 / -0

A relation $$\phi$$ from $$C$$ to $$R$$ is defined by $$x\phi y\Leftrightarrow \left| x \right| =y$$. Which one is correct?

A
$$(2+3i)\phi 13$$

B
$$3\phi (-3)$$

C
$$(1+i)\phi 2$$

D
$$i\phi 1$$

Solution

Given, a relation $$\phi$$ from $$C$$ to $$R$$ is defined by $$x\phi y\Leftrightarrow \left| x \right| =y$$.

Now, $$|(2+3i)|=\sqrt{2^2+3^2}=\sqrt{13}$$.

So $$(2+3i)$$ and $$13$$ are not related.

Again, since $$y>0$$ this gives $$3$$ not related to $$-3$$.

$$|1+i|=\sqrt{1+1}=\sqrt 2$$

So $$1+i$$ and $$2$$ are not related.

Also $$|i|=1$$ so $$i\phi 1$$ holds.

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Relations and Functions Test - 18

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Relations and Functions Test - 18

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Question 1

$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4), (4, 3)\}$$

$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4) \}$$

$$\{(1, 1), (3, 3), (4, 4) \}$$

$$\phi$$

Solution

Question 2

5,95,350

5,94,000

5,95,000

5,94,350

Solution

Question 3

$$64 (ln 2)$$

$$128 (ln 2)$$

$$256 (ln 2)$$

$$1024 (ln 2)$$

Question 4

x=3, y=-1

x=6, y=2

x=2,y=3

none of these

Solution

Question 5

$$ - \dfrac{{15}}{4}$$

$$\dfrac{{\sqrt 2 }}{2}$$

$$9$$

$$0$$

Solution

Question 6

$$\left( { - \infty ,0} \right)$$

$$\left( {0,\infty } \right)$$

$$\left( {1,\infty } \right)$$

$$\left( { - \infty ,1} \right)$$

Solution

Question 7

$$\left\{ 2,3,5 \right\} $$

$$\left\{ 3,5 \right\} $$

$$\left\{ 2,3,4 \right\} $$

$$\left\{ 2,3,4,5 \right\} $$

Solution

Question 8

$$(2,4)\in R$$

$$(3,8)\in R$$

$$(6,8)\in R$$

$$(8,7)\in R$$

Solution

Question 9

$$\left\{ 1,4,6,9 \right\} $$

$$\left\{ 4,6,9 \right\} $$

$$\left\{ 1 \right\} $$

none of these

Solution

Question 10

$$(2+3i)\phi 13$$

$$3\phi (-3)$$

$$(1+i)\phi 2$$

$$i\phi 1$$

Solution

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