Relations and Functions Test

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Relations and Functions Test - 35

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Weekly Quiz Competition

Question 1
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If $$ n (A) = 5 $$ and $$ n (B) = 7 , $$ then the number of relations on $$ A \times B $$ is :

A
$$ 2^{35} $$

B
$$ 2^{49} $$

C
$$ 2^{25} $$

D
$$ 2^{70} $$

E
$$ 2^{35 \times35 } $$

Solution

Given, $$ n (A) = 5 $$ and $$ n (B) = 7 $$
Therefore, number of relations on $$ A \times B = 2^{[ n(A) \times n(B) ]} $$
$$ = 2^{(5 \times 7)} = 2^{(35)} $$
Question 2
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The functions $$f, g$$ and $$h$$ satisfy the relations $$f^{ ' }\left( x \right) =g\left( x+1 \right) $$ and $$g^{ ' }\left( x \right) =h\left( x-1 \right) $$. Then, $$f^{ '' }\left( 2x \right) $$ is equal to

A
$$h\left( 2x \right) $$

B
$$4h\left( 2x \right) $$

C
$$h\left( 2x-1 \right) $$

D
$$h\left( 2x+1 \right) $$

E
$$2h^{ ' }\left( 2x \right) $$

Solution

We have $$f^{ ' }\left( x \right) =g\left( x+1 \right) $$
$$\Rightarrow f^{ '' }\left( x \right) =g^{ ' }\left( x+1 \right) $$
But $$g^{ ' }\left( x \right) =h\left( x-1 \right) $$
$$\Rightarrow g^{ ' }\left( x+1 \right) =h\left( x+1-1 \right) $$
$$=h\left( x \right) $$
Therefore, $$ f^{ '' }\left( x \right) =h\left( x \right) $$
$$\Rightarrow f^{ '' }\left( 2x \right) =h\left( 2x \right) $$
Question 3
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Let $$ \phi (x) = \dfrac {b(x-a)}{b-a} + \dfrac {a(x-b)}{a-b} , $$ where $$ x \epsilon R $$ and $$a$$ and $$b$$ are fixed real numbers with $$ a \ne b. $$ Then $$ \phi ( a + b) $$ is equal to :

A
$$ \phi (ab) $$

B
$$ \phi (-ab) $$

C
$$ \phi (a) | \phi (b) $$

D
$$ \phi (a-b) $$

E
$$ \phi (0) $$

Solution

Given, $$ \phi (x) = \dfrac {b(x-a)}{b-a} + \dfrac {a(x-b)}{a-b} $$
Therefore, $$ \phi (a+b) = \dfrac {b(a+b-a)}{b-a} + \dfrac {a(a+b-b)}{a-b} $$
$$ = \dfrac {b^2}{b-a} + \dfrac {a^2}{(a-b)} $$
$$ = - \dfrac {b^2}{(a-b)} + \dfrac {a^2}{(a-b)} $$
$$ = (a-b) \dfrac {(a+b)}{a-b} $$
$$ = a+b $$
$$= \phi (a) + \phi (b) $$
Question 4
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If two sets $$A$$ and $$B$$ are having $$39$$ elements in common, then the number of elements common to each of the sets $$A\times B$$ and $$B\times A$$ are

A
$${ 2 }^{ 39 }$$

B
$${ 39 }^{ 2 }$$

C
$$78$$

D
$$351$$

Solution

If set $$A$$ and set $$B$$ have $$39$$ common elements, then the number of common elements in set $$A\times B$$ and set $$B\times A\,=39^2$$
Question 5
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Let $$n$$ be a fixed positive integer. Define a relation $$R$$ in the set $$Z$$ of integers by $$aRb$$ if and only if $$\dfrac {n}{a - b}$$. The relation $$R$$ is

A
Reflexive

B
Symmetric

C
Transitive

D
An equivalence relation

Solution

Given $$aRb$$ such that $$R:\dfrac{n}{a-b}\epsilon Z$$
$$(A) aRa:$$ $$\dfrac{n}{a-a}=\infty$$ and $$\infty$$ does not belongs to $$Z$$
So not a reflexive relation and hence (A) and (D) options are ruled out.
$$(B)aRb\epsilon Z $$ we need to check whether $$bRa\epsilon Z$$ or not.
$$\dfrac{n}{a-b}\epsilon Z$$ Now, $$\dfrac{n}{b-a}\rightarrow -\dfrac{n}{a-b}$$
As $$\dfrac{n}{a-b}$$ is integer so negative of it also integer
$$\therefore bRa\epsilon Z$$
Hence relation $$R$$ is Symmetric.
$$(C)$$ Now $$aRb\epsilon Z$$ and $$bRc\epsilon Z $$ but we can't say anything about $$aRc\epsilon Z.$$ Hence not a transitive relation.
Question 6
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A function $$f$$ satisfies the relation $$f\left( { n }^{ 2 } \right) =f\left( n \right) +6$$ for $$n\ge 2$$ and $$f\left( 2 \right) =8$$. Then, the value of $$f\left( 256 \right) $$ is

A
$$24$$

B
$$26$$

C
$$22$$

D
$$28$$

E
$$32$$

Solution

Given that, $$f\left( { n }^{ 2 } \right) =f\left( n \right) +6$$

On putting $$n=2$$, we get
$$f\left( 4 \right) =f\left( { 2 }^{ 2 } \right) =f\left( 2 \right) +6=8+6=14$$ $$\left[ \because f\left( 2 \right) =8 \right] $$

On putting $$n=4$$, we get
$$f\left( 16 \right) =f\left( { 4 }^{ 2 } \right) =f\left( 4 \right) +6=14+6=20$$

On putting $$n=16$$, we get
$$f\left( 256 \right) =f\left( { 16 }^{ 2 } \right) =f\left( 16 \right) +6=20+6=26$$
Question 7
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Directions For Questions

Let a real valued polynomial function $$f(x)$$ with leading coefficient $$'x'$$ satisfied the relation $$(x^{2} - 9x + 18)f(x) - (x^{2} + 3x)f(x - 3) = 0$$.

...view full instructions

Number of distinct real roots of the equation $$f(x) = 0$$, is

A
$$3$$

B
$$4$$

C
$$8$$

D
$$12$$
Question 8
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Let $$ A = \{ 1,2,3,4 \} $$ and $$R$$ be a relation in $$A$$ given by $$ R = \{ (1,1) , (2,2) (3,3) , (4,4) , (1,2) , (3,1) , (1,3) \} $$ then $$R$$ is :

A
Reflexive and transitive only

B
Transitive and symmetric only

C
An equivalence relation

D
None of the above

Solution

$$ A = \{ 1, 2, 3, 4\} $$
$$ R = \{ (1,1) , (2,2),(3,3),(4,4),(1,2),(3,1)(1,3) \} $$
clearly $$ (1,1) \epsilon R, (1,2) \epsilon (2,2) \epsilon R , (3,3) \epsilon R, (4,4) \epsilon R $$
$$ \Rightarrow R $$ is Reflective.
also $$ (1,1) \epsilon R, (1,2) \epsilon R \Rightarrow (1,2) R $$
and $$(1,2) \epsilon R, (1,3) \epsilon R \Rightarrow (1,3) \epsilon R \Rightarrow R $$ is transitive
But for $$ (1,2) \epsilon R (2,1) \notin R, \Rightarrow R $$ is not symmetric
$$ \Rightarrow R $$ is reflective and transitive only.
Question 9
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The graph of a constant function $$f(x)=k$$ is?

A
A straight line parallel to $$X$$-axis

B
A straight line parallel to $$Y$$-axis

C
A straight line passing through orgin

D
None

Solution

A graph of a constant function is always a horizontal line.
Horizontal line is a line which is parallel to $$x$$-axis.
Question 10
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If $$f,g,h$$ are three functions from a set of positive real numbers into itself satisfying the condition,
$$f(x) \cdot g(x)=h \sqrt{x^2 + y^2}$$ such that $$x,y \epsilon (0,\infty)$$.then, $$\dfrac{f(x)}{g(x)}$$ is a?

A
Constant function

B
Identity function

C
Zero function

D
Signum function

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Relations and Functions Test - 35

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Relations and Functions Test - 35

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Question 1

$$ 2^{35} $$

$$ 2^{49} $$

$$ 2^{25} $$

$$ 2^{70} $$

$$ 2^{35 \times35 } $$

Solution

Question 2

$$h\left( 2x \right) $$

$$4h\left( 2x \right) $$

$$h\left( 2x-1 \right) $$

$$h\left( 2x+1 \right) $$

$$2h^{ ' }\left( 2x \right) $$

Solution

Question 3

$$ \phi (ab) $$

$$ \phi (-ab) $$

$$ \phi (a) | \phi (b) $$

$$ \phi (a-b) $$

$$ \phi (0) $$

Solution

Question 4

$${ 2 }^{ 39 }$$

$${ 39 }^{ 2 }$$

$$78$$

$$351$$

Solution

Question 5

Reflexive

Symmetric

Transitive

An equivalence relation

Solution

Question 6

$$24$$

$$26$$

$$22$$

$$28$$

$$32$$

Solution

Question 7

$$3$$

$$4$$

$$8$$

$$12$$

Question 8

Reflexive and transitive only

Transitive and symmetric only

An equivalence relation

None of the above

Solution

Question 9

A straight line parallel to $$X$$-axis

A straight line parallel to $$Y$$-axis

A straight line passing through orgin

None

Solution

Question 10

Constant function

Identity function

Zero function

Signum function

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