Functions Test 28

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Functions Test 28

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Weekly Quiz Competition

Question 1
1 / -0

If $$A = \left \{ 1 , 3 , 5 , 7 \right \} $$ and $$ B = \left \{ 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 \right \} $$ then the number of one-to-one functions from $$A$$ into $$B$$ is

A
1340

B
1860

C
1430

D
1880

E
1680

Solution

Given, $$A = \{1, 3, 5, 7\}$$
and $$B =\{1, 2, 3, 4, 5, 6, 7, 8\}$$
Here, $$n(A) = 4$$ and $$n(B) = 8$$
Thus number of one-one function from $$A$$ into $$B$$
$$= \,^8 P_4 = 8.7.6.5 = 1680$$
Question 2
1 / -0

If $$f(x)=3x+5$$ and $$g(x)={ x }^{ 2 }-1$$, then $$\left( f\circ g \right) $$ $$({ x }^{ 2 }-1)$$ is equal to

A
$$3{ x }^{ 4 }-3x+5$$

B
$$3{ x }^{ 4 }-6{ x }^{ 2 }+5$$

C
$$6{ x }^{ 4 }+3{ x }^{ 2 }+5$$

D
$$6{ x }^{ 4 }-6x+5$$

E
$$3{ x }^{ 2 }+6x+4$$

Solution

$$f(x)=3x+5$$
and $$g(x)={x}^{2}-1$$
$$\therefore f\circ g\left( { x }^{ 2 }-1 \right) =f[g\left( { x }^{ 2 }-1 \right) ]=f\left[ { \left( { x }^{ 2 }-1 \right) }^{ 2 }-1 \right] $$
$$=3\left( { x }^{ 4 }-2{ x }^{ 2 } \right) +5=3{ x }^{ 4 }-6{ x }^{ 2 }+5$$
Question 3
1 / -0

If $$g(x)={ x }^{ 2 }+x-2$$ and $$\cfrac { 1 }{ 2 } (g\circ f)(x)=2{ x }^{ 2 }-5x+2$$, then $$f(x)$$ is

A
$$2x-3$$

B
$$2x+3$$

C
$$2{ x }^{ 2 }+3x+1$$

D
$$2{ x }^{ 2 }+3x-1$$

Solution

$$\cfrac { 1 }{ 2 } \left( g\circ f \right) (x)=2{ x }^{ 2 }-5x+2$$

$$\Rightarrow \cfrac { 1 }{ 2 } g\left[ f(x) \right] =2{ x }^{ 2 }-5x+2\quad $$

$$\left[ { \left\{ f(x) \right\} }^{ 2 }+\left\{ f(x) \right\} -2 \right] =2\left[ 2{ x }^{ 2 }-5x+2 \right] \quad \left[ \because g(x)={ x }^{ 2 }+x-2 \right] $$

$$\Rightarrow f{ \left( x \right) }^{ 2 }+f(x)-\left( 4{ x }^{ 2 }-10x+6 \right) =0$$ ....represents quadratic equation

We have a formula for solving quadratic equation $$ax^2+bx+c=0$$ is

$$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$

$$\therefore f(x)=\cfrac { -1\pm \sqrt { 1+4\left( 4{ x }^{ 2 }-10x+6 \right) } }{ 2 } $$

$$=\cfrac { -1\pm(4x-5) }{ 2 } =2x-3$$ or $$2-2x$$
Question 4
1 / -0

If $$(ax^2 + bx + c)y +a'x^2+b'x+c=0$$, then the condition that x may be a rational function of y is

A
$$(ac'-a'c)^2=(ab'-a'b)(bc'-b'c)$$

B
$$(ab'-a'b)^2=(ab'-a' c)(bc'-b'c)$$

C
$$(bc'-b'c)^2=(ab'-a'b)(ac'-a'c)$$

D
None of these

Solution
Question 5
1 / -0

Let $$f(x) = |x - 2|$$, where $$x$$ is a real number. Which one of the following is true?

A
$$f$$ is periodic

B
$$f(x + y) = f(x) + f(y)$$

C
$$f$$ is an odd function

D
$$f$$ is not one-one function

E
$$f$$ is an even function

Solution

From the graph of the function it can be observed clearly that $$f$$ is not one-one function.
So option $$D$$ is correct.
Question 6
1 / -0

If $$f\left( x \right) $$ and $$g\left( x \right) $$ are two functions with $$g\left( x \right) =x-\dfrac { 1 }{ x } $$ and $$f\circ g\left( x \right) ={ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } $$, then $$f^{ ' }\left( x \right) $$ is equal to

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

B
$${ x }^{ 2 }-\dfrac { 1 }{ { x }^{ 2 } } $$

C
$$1+\dfrac { 1 }{ { x }^{ 2 } } $$

D
$$3{ x }^{ 2 }+\dfrac { 3 }{ { x }^{ 4 } } $$

Solution

$$f\circ g\left( x \right) ={ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } $$
Writing $${ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } $$ using $${ \left( a-b \right) }^{ 3 }={ a }^{ 3 }-{ b }^{ 3 }-3ab\left( a-b \right) $$, we have
$${ \left( x-\dfrac { 1 }{ x } \right) }^{ 3 }={ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } -3x\cdot \dfrac { 1 }{ x } \left( x-\dfrac { 1 }{ x } \right) $$
$$\Rightarrow { x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } ={ \left( x-\dfrac { 1 }{ x } \right) }^{ 3 }+3\left( x-\dfrac { 1 }{ x } \right) $$
We have,
$$f\left( g\left( x \right) \right) ={ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } ={ \left( x-\dfrac { 1 }{ x } \right) }^{ 3 }+3\left( x-\dfrac { 1 }{ x } \right) $$
As $$g\left( x \right) =x-\dfrac { 1 }{ x } $$, this yields
$$f\left( x-\dfrac { 1 }{ x } \right) ={ \left( x-\dfrac { 1 }{ x } \right) }^{ 3 }+3\left( x-\dfrac { 1 }{ x } \right) $$
On putting $$x-\dfrac { 1 }{ x } =t$$, we get
$$f\left( t \right) ={ t }^{ 3 }+3t$$
Thus, $$f\left( x \right) ={ x }^{ 3 }+3x$$
and $$f^{ ' }\left( x \right) =3{ x }^{ 2 }+3$$
Question 7
1 / -0

If $$f(x)=\left| x \right| ,x\in R$$, then

A
$$f(x)=\left( f\times f \right) \left( x \right) $$

B
$$f(x)=x$$

C
$$f(x)=\left( f\times f \right) \left( x^2 \right) $$

D
$$f(x)=\left( f\circ f \right) \left( x \right) $$

Solution
Question 8
1 / -0

$$f,g:R\rightarrow R$$ are functions such that $$f(x)=3x-\sin { \left( \cfrac { \pi x }{ 2 } \right) } ,g(x)={ x }^{ 3 }+2x-\sin { \left( \cfrac { \pi x }{ 2 } \right) } $$
The value of $$\cfrac { d }{ dx } { f }^{ -1 }{ \left( { g }^{ -1 }(x) \right) }_{ x=12 }$$ is equal to

A
$$\cfrac { 2 }{ 30+x } $$

B
$$\cfrac { 2 }{ 30-x } $$

C
$$\cfrac { 2 }{ 3\left( 28-\pi \right) } $$

D
$$\cfrac { 2 }{ 3\left( 28+\pi \right) } $$

Solution

Given that,
$$g:R \to R$$

$$f\left( x \right) = 3x - \sin \left( {\dfrac{{\pi x}}{2}} \right)$$

$$g\left( x \right) = {x^3} + 2x - \sin \left( {\dfrac{{\pi x}}{2}} \right)$$

$$\therefore \dfrac{{d{f^{ - 1}}}}{{dx}}{\left( {g'\left( x \right)} \right)_{x = 12}} = \dfrac{2}{{30 + x}}$$

Hence, the option $$(A)$$ is correct.
Question 9
1 / -0

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
1 / -0

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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Re-Attempt Weekly Quiz Competition

Functions Test 28

Result

Functions Test 28

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Rank

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SHARING IS CARING

Question 1

1340

1860

1430

1880

1680

Solution

Question 2

$$3{ x }^{ 4 }-3x+5$$

$$3{ x }^{ 4 }-6{ x }^{ 2 }+5$$

$$6{ x }^{ 4 }+3{ x }^{ 2 }+5$$

$$6{ x }^{ 4 }-6x+5$$

$$3{ x }^{ 2 }+6x+4$$

Solution

Question 3

$$2x-3$$

$$2x+3$$

$$2{ x }^{ 2 }+3x+1$$

$$2{ x }^{ 2 }+3x-1$$

Solution

Question 4

$$(ac'-a'c)^2=(ab'-a'b)(bc'-b'c)$$

$$(ab'-a'b)^2=(ab'-a' c)(bc'-b'c)$$

$$(bc'-b'c)^2=(ab'-a'b)(ac'-a'c)$$

None of these

Solution

Question 5

$$f$$ is periodic

$$f(x + y) = f(x) + f(y)$$

$$f$$ is an odd function

$$f$$ is not one-one function

$$f$$ is an even function

Solution

Question 6

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

$${ x }^{ 2 }-\dfrac { 1 }{ { x }^{ 2 } } $$

$$1+\dfrac { 1 }{ { x }^{ 2 } } $$

$$3{ x }^{ 2 }+\dfrac { 3 }{ { x }^{ 4 } } $$

Solution

Question 7

$$f(x)=\left( f\times f \right) \left( x \right) $$

$$f(x)=x$$

$$f(x)=\left( f\times f \right) \left( x^2 \right) $$

$$f(x)=\left( f\circ f \right) \left( x \right) $$

Solution

Question 8

$$\cfrac { 2 }{ 30+x } $$

$$\cfrac { 2 }{ 30-x } $$

$$\cfrac { 2 }{ 3\left( 28-\pi \right) } $$

$$\cfrac { 2 }{ 3\left( 28+\pi \right) } $$

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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