Relations and Functions Test

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Relations and Functions Test - 17

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Weekly Quiz Competition

Question 1
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Let $$A = \left \{x, y, z\right \}$$ and $$B = \left \{p, q, r, s\right \}$$. What is the number of distinct relations from $$B$$ to $$A$$?

A
$$4096$$

B
$$4094$$

C
$$128$$

D
$$126$$

Solution

Since, No. of elements in set A$$=3$$
No. of elements in set B$$=4$$
$$\therefore$$ No. of elements in B$$\times$$A$$=3\times 4=12$$
$$\therefore$$ No. of distinct relations from B to A $$= 2^{12}=4096$$
Option A is correct.
Question 2
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Let $$ f(x)=1+2x^2+ 2^2x^4 +..... + 2^{10}x^{20}$$. Then f(x) has

A
more than one minimum

B
exactly one minimum

C
at least one maximum

D
none of these

Solution

$$f'(x)=2 . 2 x+2^2 . 4 x^3+. . . . . +2^{10} . 20 x^{19}$$

$$\Rightarrow f'(x)=x[2 . 2+2^2 . 4x^2+. . . . .+2^{10} . 20x^{18}]$$

For maxima or minima, $$f'(x)=0$$

$$\Rightarrow x=0$$

The function increases strictly for both $$x>0$$ and $$x<0$$ and has minimum value at $$x=0$$

$$\therefore$$ Correct option is B
Question 3
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Let $$A=\left\{ u,v,w,z \right\} ;B=\left\{ 3,5 \right\} $$, then the number of relations from $$A$$ to $$B$$ is

A
$$256$$

B
$$1024$$

C
$$512$$

D
$$64$$

Solution

Given
$$A=\left\{ u,v,w,z \right\} ;B=\left\{ 3,5 \right\} $$
Here, number of element in set $$A$$ is $$n(A)=4$$
And number of element in set $$B$$ is $$ n(B)=2$$
$$\therefore$$ Number of relations from $$A$$ to $$B$$
$$={ 2 }^{ n(A).n(B) }={ 2 }^{ 4\times 2 }={ 2 }^{ 8 }=256$$
Question 4
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Consider two sets $$A=\{a, b, c\}, B=\{e, f\}$$. If maximum numbers of total relations from A to B; symmetric relation from A to A and from B to B are $$l, m, n$$ respectively, then the value of $$2l+m-n$$ is

A
$$212$$

B
$$184$$

C
$$240$$

D
$$64$$

Solution

We have $$n(A)=3$$ and $$n(B)=2$$.
Now cartesian product of $$A$$ and $$B$$ can contain maximum $$3\times 2=6$$ elements.
Since relation is the possible subsets of cartesian product.
So maximum number of relations from $$A$$ to $$B$$ be $$(l)=2^{6}=64$$.
Now maximum possible symmetric relation from $$A$$ to $$A$$ be $$(m)=2^{\cfrac{3(3+1)}{2}}=2^6=64$$ and that of from $$B$$ to $$B$$ be $$(m)=2^{\cfrac{2(2+1)}{2}}=2^3=8$$. [Using formula for number of symmetric relation]
Now $$2l+m-n=128+64-8=184$$.
Question 5
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Let R be a relation from a set A to a set B then

A
$$R = A \cup B$$

B
$$ R = A \cap B$$

C
$$ R \subseteq A\times B$$

D
$$ R\subseteq B\times A$$

Solution

Let R be a relation from a set A to a set B then

$$R\subseteq A\times B$$
Question 6
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If $$f:R-\left\{ 2 \right\} \rightarrow R$$ is a function defined by $$f(x)=\cfrac { { x }^{ 2 }-4 }{ x-2 } $$, then its range is

A
$$R$$

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

C
$$R-\left\{ 4 \right\} $$

D
$$R-\left\{ -2,2 \right\} $$

Solution

Given function is $$f\left( x \right) =\dfrac { { x }^{ 2 }-4 }{ x-2 } $$,
As the domain does not include $$2$$ we can simplyfy,
$$f\left( x \right) =x+2$$,
$$\therefore$$ $$x$$ can take any real value in the domain,
So,the range is $$R-\left\{ 4 \right\} $$,
As is $$x+2=4\Longrightarrow x=2$$ which is not in the domain.
Question 7
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If $$n(A) = 2, n(B) = m$$ and the number of relation from $$A$$ to $$B$$ is $$64$$, then the value of $$m$$ is

A
$$6$$

B
$$3$$

C
$$16$$

D
$$8$$

Solution

The number of relations between sets can be calculated using $$ 2^{mn}$$ where m and n represent
the number of members in each set.
Given that $$ n(A) = 2 $$ and $$ n(B) = m $$
So,
$$ 2^{2m} = 64 $$
$$ 2^{2m} = 2^{6} $$
On comparing powers of 2 we get,
$$ 2m = 6 $$
$$ m = 3 $$
Question 8
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If $$f(x)=\dfrac{1-x}{1+x}$$ then the value of $$(fof)(x)$$.

A
$$\dfrac{1-x}{1+x}$$

B
$$x$$

C
$$\dfrac{1}{x}$$

D
none of these

Solution

Given,
$$f(x)=\dfrac{1-x}{1+x}$$......(1)
Now,
$$(fof)(x)$$
$$=f(f(x))$$
$$=f\left(\dfrac{1-x}{1+x}\right)$$ [ Using (1)]
$$=\dfrac{1-\dfrac{1-x}{1+x}}{1+\dfrac{1-x}{1+x}}$$
$$=\dfrac{1+x-1+x}{1+x+1-x}$$
$$=\dfrac{2x}{2}$$
$$=x$$.
Question 9
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If $$f(x)=\dfrac {2^{x}+2^{-x}}{2}$$, then $$f(x+y).f(x-y)$$ is equal to ?

A
$$\dfrac {1}{2}[f(x+y)]+f(x-y)]$$

B
$$\dfrac {1}{2}[f(2x)+f(2y)]$$

C
$$\dfrac {1}{2}[.f(x+y)]+f(x-y)]$$

D
$$None\ of\ these$$

Solution

$$f(x+y)\cdot f(x-y)$$
$$=\left(\dfrac{2^{x+y}+2^{-(x+y)}}{2}\right)\left(\dfrac{2^{x-y}+2^{-(x-y)}}{2}\right)$$
$$=\dfrac{1}{4}\left(2^{2x}+2^{2y}+2^{-2x}+2^{-2y}\right)$$
$$=\dfrac{1}{2}\left[\dfrac{1}{2}(2^{2x}+2^{-2x})+\dfrac{1}{2}(^{2y}+2^{-2y})\right]$$
$$=\dfrac{1}{2}[f(2x)+f(2y)]$$.
Question 10
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If $$f:{R}^{+}\rightarrow {R}^{+},f(x)={x}^{2}+2$$ and $$g:{R}^{+}\rightarrow {R}^{+},g(x)=\sqrt{x+1}$$
then $$(f+g)(x)$$ equals

A
$$\sqrt{{x}^{2}+3}$$

B
$$x+3$$

C
$$\sqrt{{x}^{2}+2}+(x+1)$$

D
$${x}^{2}+2+\sqrt{(x+1)}$$

Solution

$$(f+g)(x)=f(x)+g(x)=x^2+2+\sqrt{x+1}$$

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Relations and Functions Test - 17

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Relations and Functions Test - 17

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SHARING IS CARING

Question 1

$$4096$$

$$4094$$

$$128$$

$$126$$

Solution

Question 2

more than one minimum

exactly one minimum

at least one maximum

none of these

Solution

Question 3

$$256$$

$$1024$$

$$512$$

$$64$$

Solution

Question 4

$$212$$

$$184$$

$$240$$

$$64$$

Solution

Question 5

$$R = A \cup B$$

$$ R = A \cap B$$

$$ R \subseteq A\times B$$

$$ R\subseteq B\times A$$

Solution

Question 6

$$R$$

$$R-\left\{ 2 \right\} $$

$$R-\left\{ 4 \right\} $$

$$R-\left\{ -2,2 \right\} $$

Solution

Question 7

$$6$$

$$3$$

$$16$$

$$8$$

Solution

Question 8

$$\dfrac{1-x}{1+x}$$

$$x$$

$$\dfrac{1}{x}$$

none of these

Solution

Question 9

$$\dfrac {1}{2}[f(x+y)]+f(x-y)]$$

$$\dfrac {1}{2}[f(2x)+f(2y)]$$

$$\dfrac {1}{2}[.f(x+y)]+f(x-y)]$$

$$None\ of\ these$$

Solution

Question 10

$$\sqrt{{x}^{2}+3}$$

$$x+3$$

$$\sqrt{{x}^{2}+2}+(x+1)$$

$${x}^{2}+2+\sqrt{(x+1)}$$

Solution

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