Relations and Functions Test

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Relations and Functions Test - 36

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Weekly Quiz Competition

Question 1
1 / -0

If $$f(x+1)=3x-9$$, then what will be the value of $$f\left( { x }^{ 2 }-1 \right) $$?

A
$$3{ x }^{ 2 }-9$$

B
$$3{ x }^{ 2 }-15$$

C
$${ x }^{ 2 }-10$$

D
$$3{ x }^{ 2 }-10$$

Solution

Let

$$x+1=t$$

$$x=t-1$$

$$f(x+1)=3x-9$$

$$f(t)=3(t-1)-9$$

$$f(t)=3t-12$$

Putting $$t=x^2-1$$

$$f(x^2-1)=3(x^2-1)-12$$

$$=3x^2-15$$

Hence, option $$B$$ is the correct answer.
Question 2
1 / -0

On differentiating an identity function, we get?

A
Signum function

B
Sinc function

C
Constant function

D
None

Solution

Derivative of an Identity function gives a Constant function

$$\dfrac{d(cx)}{dx} = c$$
Where $$c$$ is a constant.
Question 3
1 / -0

If $$f(x + 1) = 3x - 9$$, then what will be the value of $$f(x^{2} - 1)$$?

A
$$3x^{2} - 9$$

B
$$3x^{2} - 15$$

C
$$x^{2} - 10$$

D
$$3x^{2} - 10$$

Solution

Replace $$x\rightarrow x - 1\Rightarrow f(x) = 3x - 12$$

$$\therefore f(x^{2} - 1) = 3(x^{2} - 1) - 12$$

$$= 3x^{2} - 15$$.
Question 4
1 / -0

If $$\left |x^2-2x-8 \right | + \left |x^2+x-2 \right | = 3 \left |x +2 \right |$$, then the set of all real values of x is

A
$$[1,4] \cup {-2}$$

B
$$ [1, 4 ]$$

C
$$ [-2, 1] \cup [4, \infty]$$

D
$$[ - \infty , -2] \cup [1, 4] $$

Solution

We have $$|x^2-2x-8|+|x^2+x-2|=3|x+2|$$
Above is possible when

$$x^2-2x-8\le0$$ and $$x^2+x-2\ge0$$
or
$$x^2-2x-8\ge0$$ and $$x^2+x-2\le0$$

For Case 1 when $$x^2-2x-8<0$$ and $$x^2+x-2>0$$

$$(x-4)(x+2)\le0$$ and $$(x+2)(x-1)\ge0$$

$$x\in [-2,4]$$ and $$x\in R-(-2,1)$$

$$\therefore x\in [1,4]\cup \{-2\}$$

For Case 2 when $$x^2-2x-8\ge0$$ and $$x^2+x-2\le0$$

$$(x-4)(x+2)\ge0$$ and $$(x+2)(x-1)\le0$$

$$x\in R- (-2,4)$$ and $$x\in [-2,1]$$

So, for the case 2, $$x\in \phi$$

$$\therefore x\in [1,4]\cup \{-2\}$$
Question 5
1 / -0

If log $$x - 5$$ log $$3$$ = $$-2$$, then x equals

A
$$1.25$$

B
$$0.81$$

C
$$2.43$$

D
$$0.8$$

E
either $$0.8$$ or $$1.25$$

Solution

Given : $$\log x- 5 \log 3=-2$$
Find : $$x=?$$
solution :
$$\log x- 5 \log m=-2$$
$$5 \log 3 - \log x=2$$
$$\log 3 - \log x=2$$
Now use $$\log x- \log y= \log \dfrac{x}{y}$$
$$\log_{10}\left(\dfrac{35}{x}\right)=2$$
$$\dfrac{35}{x}=10^{2}$$
$$\dfrac{243}{x}=100$$
$$x=\dfrac{243}{100}$$
.$$x=2.43$$
Question 6
1 / -0

If $$f(x) = \dfrac {x(x - 1)}{2}$$ then $$f(x + 2)$$ equals.

A
$$f(x) + f(2)$$

B
$$(x + 2)f(x)$$

C
$$x(x + 2)f(x)$$

D
$$\dfrac {xf(x)}{x + 2}$$

E
$$\dfrac {(x + 2)f(x + 1)}{x}$$

Solution

$$f(x)=\dfrac{x(x-1)}{2}$$

$$f(x+2)=\dfrac{(x+2)(x+2-1)}{2}$$

$$f(x+2)=\dfrac{(x+2)(x+1)}{2}$$ ---- ( 1 )
Now,
$$f(x)=\dfrac{x(x-1)}{2}$$

$$f(x+1)=\dfrac{(x+1)(x+1-1)}{2}$$

$$f(x+1)=\dfrac{(x+1)x}{2}$$

$$\dfrac{f(x+1)}{x}=\dfrac{(x+1)}{2}$$ --- ( 2 )
Substituting value of ( 2 ) in ( 1 ) we get,

$$f(x+2)=\dfrac{(x+2)f(x+1)}{x}$$
Question 7
1 / -0

If $${C}_{r}$$ stands for $${ _{ }^{ n }{ C } }_{ r }$$ then $$\left( { C }_{ 0 }+{ C }_{ 1 } \right) +\left( { C }_{ 1 }+{ C }_{ 2 } \right) +....\left( { C }_{ n-1 }+{ C }_{ n } \right) \quad $$ is equal to

A
$${ 2 }^{ n }-1$$

B
$${ 2 }^{ n+1 }+1$$

C
$${ 2 }^{ n+1 }-1$$

D
$${ 2 }^{ n+1 }-2$$

Solution

$$(1+x)^{n}=^{}C_0+^{}C_{1}x+^{}C_2x^2+C_{3}x^{3}...........+^{}C_{n}x^{n}$$

Put x =1

Using the relation of sum of coefficent
$$(C_0+C_1+$$ $$C_2+C_3....C_n)=2^n$$
According to the question ,

$$(C_0+C_1)+$$ $$(C_1+C_2)+$$ $$(C_2+C_3)+$$ $$(C_{n-1}+C_n)\rightarrow$$

on simplifying
$$(C_0+C_1+$$ $$C_2+C_3....C_n)+$$ $$(C_1+C_2+C_3+....$$ $$C_{n-1})\rightarrow$$

Adding and subtracting $$C_0and C_n$$ to above equation

$$(C_0+C_1+$$ $$C_2+C_3....C_n)+$$ $$(C_0+C_1+C_2+C_3+....$$ $$C_{n})-C_0-C_n\rightarrow$$

$$=2^n+2^n-1-1=2^{n+1}-2$$
Question 8
1 / -0

Let $$A=\left\{ a,b,c \right\} $$ and $$B=\left\{ 1,2 \right\} $$. Consider a relation $$R$$ defined from set $$A$$ to set $$B$$. Then $$R$$ is equal to set

A
$$A$$

B
$$B$$

C
$$A\times B$$

D
$$B\times A$$

Solution

The relation defined from a set $$A$$ to set $$B$$ is defined as a subset of $$A\times B$$.

$$R:A\rightarrow B=\{(x,y)|x\in A, y\in B , (x,y)\in A\times B\}$$

Here, $$A\times B=\{(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)\}$$

Relation from A to B will be a subset of this $$A\times B$$.
Question 9
1 / -0

If $$f(x)=\cos { \left( \log { x } \right) } $$ then $$f(x)f(y)-\cfrac { 1 }{ 2 } \left[ f\left( \cfrac { x }{ y } \right) +f(xy) \right] $$ has value

A
$$-1$$

B
$$2$$

C
$$-2$$

D
$$0$$

Solution

$$f\left( x \right) = \cos \left( {\log x} \right)$$
$$f\left( x \right)f\left( y \right) - \frac{1}{2}\left[ {f\left( {\frac{x}{y}} \right) + f\left( {xy} \right)} \right]$$
$$= \cos \left( {\log x} \right)\cos \left( {\log y} \right) - \frac{1}{2}\left[ {\cos \left( {\log \frac{x}{y}} \right) + \cos \left( {\log xy} \right)} \right]$$
$$ = \cos \left( {\log x} \right)\cos \left( {\log y} \right) - \frac{1}{2}\left[ {\cos \left( {\log x - \log y} \right) + \cos \left( {\log x + \log y} \right)} \right]$$
$$\cos \left( {\log x} \right)\cos \left( {\log y} \right) - \frac{1}{2}\left[ {2\cos \left( {\log x} \right)\cos \left( {\log y} \right)} \right]$$
$$[ \because \cos A + \cos B = 2\cos \frac{{A + B}}{2}\cos \frac{{A - B}}{2}]$$
$$\cos \left( {\log x} \right)\cos \left( {\log y} \right) - \cos \left( {\log x} \right)\cos \left( {\log y} \right)$$
$$ = 0$$
Question 10
1 / -0

If $$f(x)=\cfrac { x }{ { x }^{ 2 }+1 } $$ is increasing function then the value of $$x$$ lies in

A
$$R$$

B
$$\left( -\infty ,-1 \right) $$

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

D
$$(-1,1)$$

Solution

Given function is $$f\left( x \right) =\dfrac { x }{ { x }^{ 2 }+1 } $$,

Given that this is increasing in nature,

$$\Longrightarrow f^{ ' }\left( x \right) >0\\ \Longrightarrow \dfrac { 1-{ x }^{ 2 } }{ { \left( 1+{ x }^{ 2 } \right) }^{ 2 } } >0\\ \Longrightarrow { x }^{ 2 }<1$$

$$x$$ lies in the interval $$(-1,1)$$.

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Re-Attempt Weekly Quiz Competition

Relations and Functions Test - 36

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Relations and Functions Test - 36

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SHARING IS CARING

Question 1

$$3{ x }^{ 2 }-9$$

$$3{ x }^{ 2 }-15$$

$${ x }^{ 2 }-10$$

$$3{ x }^{ 2 }-10$$

Solution

Question 2

Signum function

Sinc function

Constant function

None

Solution

Question 3

$$3x^{2} - 9$$

$$3x^{2} - 15$$

$$x^{2} - 10$$

$$3x^{2} - 10$$

Solution

Question 4

$$[1,4] \cup {-2}$$

$$ [1, 4 ]$$

$$ [-2, 1] \cup [4, \infty]$$

$$[ - \infty , -2] \cup [1, 4] $$

Solution

Question 5

$$1.25$$

$$0.81$$

$$2.43$$

$$0.8$$

either $$0.8$$ or $$1.25$$

Solution

Question 6

$$f(x) + f(2)$$

$$(x + 2)f(x)$$

$$x(x + 2)f(x)$$

$$\dfrac {xf(x)}{x + 2}$$

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

Solution

Question 7

$${ 2 }^{ n }-1$$

$${ 2 }^{ n+1 }+1$$

$${ 2 }^{ n+1 }-1$$

$${ 2 }^{ n+1 }-2$$

Solution

Question 8

$$A$$

$$B$$

$$A\times B$$

$$B\times A$$

Solution

Question 9

$$-1$$

$$2$$

$$-2$$

$$0$$

Solution

Question 10

$$R$$

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

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

$$(-1,1)$$

Solution

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