Relations Test 28

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Relations Test 28

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Weekly Quiz Competition

Question 1
1 / -0

Find the domain of the function defined as $$f(x)=\dfrac{x+1}{2x+3}$$.

A
$$x\epsilon$$ R$$-$${$$\dfrac{3}{2}$$}

B
$$x\epsilon$$ R$$-$${$$\dfrac{-3}{2}$$}

C
$$x\epsilon$$ R
D

$$x\epsilon$$ R$$-$${$$\dfrac{2}{3}$$}
Solution

Given:

$$ f\left ( x \right ) = \frac{x + 1}{2x + 3} $$

$$ 2x + 3 = 0 $$

$$ x = -\frac{3}{2} $$

The domain is $$ R - \left \{ -\frac{3}{2} \right \} $$.

Hence, the domain of the function is $$ R - \left \{ -\frac{3}{2} \right \} $$.
Question 2
1 / -0

Let R be a relation from$$ A=\left\{ 1,2,3,4 \right\} to B=\left\{ 1,3,5 \right\}$$ such that $$R=[(a,b):a<b,where\ a\epsilon A\ and\ b\epsilon B]$$. What is $$R$$ equal to?

A
$$(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)$$

B
$$(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)$$

C
$$(3,3),(3,5),(5,3),(5,5)$$

D
$$(3,3),(3,4),(4,5)$$

Solution

$$A=\{ 1,2,3,4\} \quad ,\quad B=\{ 1,3,5\} \\ R=[(a,b):a<b,where\quad a\epsilon A\quad \& \quad b\epsilon B]\\ R=[(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)]$$
Question 3
1 / -0

$$x*y=\sqrt {\dfrac {(x+y)(y^{2}-12x)}{(x-2)(y-7)}}$$ then what will be the value of $$5*9$$?

A
$$7$$

B
$$8$$

C
$$10$$

D
$$9$$

Solution

$$x\ast y=\sqrt{\dfrac{(x+y)(y^{2}-12x)}{(x-2)(y-7)}}$$

$$5\ast 9=\sqrt{\dfrac{(5+9)(81-60)}{(3)(2)}}=\sqrt{\dfrac{(14)(21)}{(3)(2)}}=\sqrt{(7)(7)}$$

$$=\sqrt{49}$$

$$=7$$
Question 4
1 / -0

Let $$R$$ be a reflexive relation in a finite set having $$n$$ elements and let there be $$m$$ ordered pairs in $$R$$. Then,

A
$$m \ge n$$

B
$$m \le n$$

C
$$m = n$$

D
None of these

Solution

$$\underset{(n\quad values\quad possible)}{\underset{\downarrow}{a R a}}$$ $$\underset{n\quad elements}{\underset{\downarrow}{a\in A}}$$
Given m ordered pairs. [We know n values possible that is at least n values cut of m]
$$\therefore m\geq n$$.
Question 5
1 / -0

Let S be a non-empty set. Let R be a relation on P(S), defined as ARB$$\Leftrightarrow A\cap B\neq \phi$$. The relation R is?

A
Reflexive

B
Symmetric

C
Transitive

D
None of these

Solution
Question 6
1 / -0

If $$f:\,R\, \to R\,$$ is an even function which is differentiable on R and $${f^N}\left( \pi \right) = 1\,the\,{f^N}\left( { - \pi } \right)\,{\rm{is}}$$

A
-1

B
0

C
1

D
2

Solution
Question 7
1 / -0

Let R be a relation from N to N defined by
$$R = \left\{ {\left( {a,\,b} \right):a,\,b\, \in \,N\,\,and\,\,a = {b^2}} \right\}$$

A
$$\left( {a,a} \right) \in R\,$$, for all $$\,a \in N$$

B
$$\left( {a,b} \right) \in R$$, implies $$\left( {b,a} \right) \in R$$

C
$$\left( {a,b} \right) \in R,\,\left( {b,c} \right) \in \,R$$ implies $$\left( {a,c} \right) \in R$$

D
None of these

Solution
Question 8
1 / -0

Let $$A = \left\{ {a,\,b,\,c} \right\}$$ and $$B = \left\{ {4,\,5} \right\}$$. Consider a relation defined from set A to set B, then R is equal to

A
A

B
B

C
$$A \times B$$

D
$$B \times A$$

Solution

A relation from set $$A$$ to set $$B$$ is called
$$A\times B$$ and have following elements
$$A\times B=\left\{(a, 4),\ (a, 5),\ (b, 4),\ (b, 5),\ (c, 4),\ (c, 5)\right\}$$
Total $$6$$ elements
$$C$$ is correct
Question 9
1 / -0

If $$A = \{2, 3, 5\}$$ and $$B = \{5, 7\}$$, find the set with highest number of elements:

A
$$A \times B$$

B
$$ B \times A$$

C
$$A \times A$$

D
$$B \times B$$

Solution

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

$$B=\left \{ 5,7 \right \}$$

$$A\times B=\left \{ (2,5),(2,7),(3,5),(3,7),(5,5),(5,7) \right \}$$

$$B\times A=\left \{ (5,2),(5,3),(5,5),(7,2),(7,3),(7,5) \right \}$$

$$A\times A=\left \{ (2,2),(2,3),(2,5),(3,2),(3,3),(3,5),(5,2),(5,3),(5,5) \right \}$$

$$B\times B=\left \{ (5,5),(5,7),(7,5),(7,7) \right \}$$

$$\therefore A\times A$$ has the highest number of elements
Question 10
1 / -0

If $$f:R \to R$$ is defined by $$f\left( x \right) = {x^2} - 3x + 2$$ and $$f\left( {{x^2} - 3x - 2} \right) = a{x^4} + b{x^3} + c{x^2} + dx + e$$ then $$a + b + c + d + e = $$

A
$$1$$

B
$$2$$

C
$$30$$

D
$$20$$

Solution

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Re-Attempt Weekly Quiz Competition

Relations Test 28

Result

Relations Test 28

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SHARING IS CARING

Question 1

$$x\epsilon$$ R$$-$${$$\dfrac{3}{2}$$}

$$x\epsilon$$ R$$-$${$$\dfrac{-3}{2}$$}

$$x\epsilon$$ R

$$x\epsilon$$ R$$-$${$$\dfrac{2}{3}$$}

Solution

Question 2

$$(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)$$

$$(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)$$

$$(3,3),(3,5),(5,3),(5,5)$$

$$(3,3),(3,4),(4,5)$$

Solution

Question 3

$$7$$

$$8$$

$$10$$

$$9$$

Solution

Question 4

$$m \ge n$$

$$m \le n$$

$$m = n$$

None of these

Solution

Question 5

Reflexive

Symmetric

Transitive

None of these

Solution

Question 6

-1

0

1

2

Solution

Question 7

$$\left( {a,a} \right) \in R\,$$, for all $$\,a \in N$$

$$\left( {a,b} \right) \in R$$, implies $$\left( {b,a} \right) \in R$$

$$\left( {a,b} \right) \in R,\,\left( {b,c} \right) \in \,R$$ implies $$\left( {a,c} \right) \in R$$

None of these

Solution

Question 8

A

B

$$A \times B$$

$$B \times A$$

Solution

Question 9

$$A \times B$$

$$ B \times A$$

$$A \times A$$

$$B \times B$$

Solution

Question 10

$$1$$

$$2$$

$$30$$

$$20$$

Solution

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