Relations and Functions Test

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Relations and Functions Test - 14

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Weekly Quiz Competition

Question 1
1 / -0

Let $$x$$ be a real number $$\left [ x \right ]$$ denotes the greatest integer function, and $$\left \{ x \right \}$$ denotes the fractional part and $$(x)$$ denotes the least integer function,then solve the following.
$$\left [ 2x \right ]-2x=\left [ x+1 \right ]$$

A
$$\left \{ -1,\displaystyle-\frac{1}{2} \right \}$$

B
$$\left \{ -1,\displaystyle \frac{1}{2} \right \}$$

C
$$\left \{ 1,\displaystyle-\frac{1}{2} \right \}$$

D
$$\left \{ 1,\displaystyle \frac{1}{2} \right \}$$

Solution

$$\left [ 2x \right ]-2x=\left [ x+1 \right ]$$

$$\Rightarrow \left \{2x\right\}=\left [ x \right ]+1$$ (i)

But range of fractional part is $$[0,1)$$

$$\Rightarrow 0\leq [x]+1 < 1$$

$$\Rightarrow -1 \leq [x] < 0$$

$$ \Rightarrow -1 \leq x < 0$$

$$\Rightarrow [x] =-1$$

Thus (i) becomes $$\{2x\} =0 $$ it means that fraction part is zero for that,

$$\therefore x = -\cfrac{1}{2}, -1$$ in $$[-1,0)$$
Question 2
1 / -0

If $$A=\left \{x:x^2-3x+2=0\right \}$$ and $$B=\left \{x:x^2+4x-5=0\right \}$$ then the value of A-B is

A
$$\left \{1, 2\right \}$$

B
$$\left \{2\right \}$$

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

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

Solution

$$A=\left \{x:x^2-3x+2=0\right \}\Rightarrow A=\left \{1, 2\right \}$$
$$B=\left \{x:x^2+4x-5=0\right \}\Rightarrow A=\left \{1, -5\right \}$$
$$\therefore A-B=\left \{2\right \}$$
Question 3
1 / -0

If R={$$(x,y)/3x+2y=15$$ and x,y $$\displaystyle \epsilon $$ N}, the range of the relation R is________

A
$$\{1,2\}$$

B
$$\{1,2,..,5\}$$

C
$$\{1,2,.,7\}$$

D
$$\{3,6\}$$

Solution

Given, $$ 3x + 2y = 15 $$

$$ => y = \dfrac {15-3x}{2} $$
As both $$ x , y $$ are natural numbers, we should find $$ x $$ such that $$ 15-3x $$ is divisible by $$ 2 $$

We can see that when $$ x =1$$ or $$3 $$, we get $$ y = 6$$ or $$3 $$

Hence, the range is $$ {3,6} $$
Question 4
1 / -0

If $$x$$ co-ordinate of a point is $$2$$ and $$y$$ co-ordinate is $$0$$, then ordered pair for its coordinate on $$XY$$ plane is

A
$$(0, 0)$$

B
$$(2, 2)$$

C
$$(0, 2)$$

D
$$(2, 0)$$

Solution

Ordered pair for co-ordinate of a point in $$XY$$ plane is written as $$(x, y).$$
So, option D is correct.
Question 5
1 / -0

If $$(x+3, 4-y)=(1,7)$$, then $$(x-3, 4+y)$$ is equal to

A
$$(-2, -3)$$

B
$$(-5, 1)$$

C
$$(3, 4)$$

D
None of these

Solution

$$(x+3, 4-y)=(1, 7)\Rightarrow x+3=1, 4-y=7$$
$$\Rightarrow x=1-3=-2, y=4-7=-3$$
$$\therefore (x-3, 4+y)=(-2-3, 4-3)=(-5, 1)$$.
Question 6
1 / -0

An operation $$\displaystyle \theta $$ is defined by the equation
$$\displaystyle a\theta b=\frac { a-b }{ a+b } $$, for all numbers a and b such that $$\displaystyle a\neq -b$$. If $$\displaystyle a\neq -c$$ and $$\displaystyle a\ \theta\ c = 0$$, then find the value of c.

A
$$-a$$

B
$$\displaystyle -\frac { 1 }{ a } $$

C
$$0$$

D
$$\displaystyle \frac { 1 }{ a } $$

E
$$a$$

Solution

$$\cfrac{(a - c)}{(a+c)}$$ = 0 ----------Given
So, numerator $$(a - c)$$ must be equal to 0
Thus, a = c OR c = a. (option E)
Question 7
1 / -0

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

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

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

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

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

Solution

$$R=\left\{(a, b):a=b-2, b>6\right\}$$
Now, since $$b>6, (2, 4)\notin R$$
Also, as $$3\neq 8-2, (3, 8)\notin R$$
And, as $$8\neq 7-2$$
$$\therefore (8, 7)\notin R$$
Now, consider $$(6, 8)$$.
We have $$8 > 6$$ and also, $$6=8-2$$.
$$\therefore (6, 8)\in R$$
Hence the correct answer is C.
Question 8
1 / -0

If ordered pair $$(a, b)$$ is given as $$(-2, 0)$$, then $$a =$$

A
$$-2$$

B
$$0$$

C
$$2$$

D
None of the above

Solution

$$a$$ will be first entry in ordered pair $$(a, b).$$
So, option A is correct.
Question 9
1 / -0

What are the ways of representing a relation?

A
Set builder form

B
Roster form

C
Arrow diagram

D
All of these

Solution

Relation can be represented in set builder form, roster form and Arrow diagram.
Question 10
1 / -0

If $$x$$ and $$y$$ coordinate of a point is $$(3, 10)$$, then the $$x$$ co-ordinate is

A
$$0$$

B
$$3$$

C
$$10$$

D
None of the above

Solution

Here, $$x$$ co-ordinate will be the first entry in ordered pair.
So, option B is correct.

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Relations and Functions Test - 14

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Relations and Functions Test - 14

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Question 1

$$\left \{ -1,\displaystyle-\frac{1}{2} \right \}$$

$$\left \{ -1,\displaystyle \frac{1}{2} \right \}$$

$$\left \{ 1,\displaystyle-\frac{1}{2} \right \}$$

$$\left \{ 1,\displaystyle \frac{1}{2} \right \}$$

Solution

Question 2

$$\left \{1, 2\right \}$$

$$\left \{2\right \}$$

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

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

Solution

Question 3

$$\{1,2\}$$

$$\{1,2,..,5\}$$

$$\{1,2,.,7\}$$

$$\{3,6\}$$

Solution

Question 4

$$(0, 0)$$

$$(2, 2)$$

$$(0, 2)$$

$$(2, 0)$$

Solution

Question 5

$$(-2, -3)$$

$$(-5, 1)$$

$$(3, 4)$$

None of these

Solution

Question 6

$$-a$$

$$\displaystyle -\frac { 1 }{ a } $$

$$0$$

$$\displaystyle \frac { 1 }{ a } $$

$$a$$

Solution

Question 7

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

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

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

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

Solution

Question 8

$$-2$$

$$0$$

$$2$$

None of the above

Solution

Question 9

Set builder form

Roster form

Arrow diagram

All of these

Solution

Question 10

$$0$$

$$3$$

$$10$$

None of the above

Solution

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