Relations and Functions Test

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Relations and Functions Test - 46

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Weekly Quiz Competition

Question 1
1 / -0

Directions For Questions

Column-I Column-II Column-II
$$f(x)=[x+[x]]]$$ $$-1$$ $$-3$$
$$f(x)= sgn(x-[x])$$ $$0$$ $$-1$$
$$f(x)=[x]+[-x]$$ $$1$$ $$0$$
$$f(x)=\left[\dfrac{[x]}{1+x^2}\right]$$ $$3$$ $$1$$

...view full instructions

Which of the following options is the only correct combination?

A
$$A\ II\ Q$$

B
$$B\ III\ S$$

C
$$AIR$$

D
$$D\ II\ R$$
Question 2
1 / -0

If $$f(x)=(a-x^{n})^{1/n}$$, when $$a > 0$$ and $$n \in N$$, then $$fo\ f(x)$$ is equal to

A
$$a$$

B
$$x$$

C
$$x^{n}$$

D
$$a^{n}$$

Solution
Question 3
1 / -0

Directions For Questions

Column-I Column-II Column-II
$$f(x)=[x+[x]]]$$ $$-1$$ $$-3$$
$$f(x)= sgn(x-[x])$$ $$0$$ $$-1$$
$$f(x)=[x]+[-x]$$ $$1$$ $$0$$
$$f(x)=\left[\dfrac{[x]}{1+x^2}\right]$$ $$3$$ $$1$$

...view full instructions

Which of the following options is the correct combination?

A
$$A\ IV\ P$$

B
$$C\ I\ Q$$

C
$$A\ IV\ Q$$

D
$$D\ I\ P$$
Question 4
1 / -0

Directions For Questions

Column-I Column-II Column-II
$$f(x)=[x+[x]]]$$ $$-1$$ $$-3$$
$$f(x)= sgn(x-[x])$$ $$0$$ $$-1$$
$$f(x)=[x]+[-x]$$ $$1$$ $$0$$
$$f(x)=\left[\dfrac{[x]}{1+x^2}\right]$$ $$3$$ $$1$$

...view full instructions

Which of the following options is the correct combination?

A
$$A\ II\ P$$

B
$$D\ II\ Q$$

C
$$C\ I\ Q$$

D
$$B\ II\ Q$$
Question 5
1 / -0

The set of values of 'a' for which the function $$f(x)=(4a-3)(x+ln5)+(a-7)sinx$$ does not posses critical points is _____________________.

A
$$\left( -\infty ,-\dfrac { 4 }{ 3 } \right] \cup \left[ 2,\infty \right) $$

B
$$\left( -\infty ,2 \right) $$

C
$$[1,\infty )$$

D
$$(1,\infty )$$

Solution

$$f\left(x\right)=\left(4a-3\right)\left(x+\ln{5}\right)+\left(a-7\right)\sin{x}$$
$${f}^{\prime}\left(x\right)=\left(4a-3\right)+\left(a-7\right)\cos{x}=0$$
$$\Rightarrow\,\left(4a-3\right)+\left(a-7\right)\cos{x}=0$$
$$\Rightarrow\,\left(4a-3\right)=-\left(a-7\right)\cos{x}$$
$$\Rightarrow\,\cos{x}=\dfrac{3-4a}{a-7}$$ where $$a\neq\,7$$
We know that range of $$\cos{x}$$ is $$\left[-1,1\right]$$
$$\Rightarrow\,\left|\dfrac{3-4a}{a-7}\right|\le 1$$ is a solution
$$\Rightarrow\,-1\le\dfrac{3-4a}{a-7}\le 1$$
$$\Rightarrow\,7-a \le\,3-4a\le a-7$$
$$\Rightarrow\,a-7 \ge\,4a-3\ge 7-a$$
$$\Rightarrow\,a-7+3 \ge\,4a\ge 7-a+3$$
$$\Rightarrow\,a-4 \ge\,4a\ge -a+10$$
$$\Rightarrow\,a-4 \ge\,4a\ge -a+10$$
$$\Rightarrow\,a\le-\dfrac{4}{3},\,a\ge 2$$
$$\therefore\,a\in\left(-\infty,-\dfrac{4}{3}\right]\cup\left[2,\infty\right)$$
Question 6
1 / -0

Let $$A$$ be set of first ten natural numbers and $$R$$ be a relation on A, defined by $$(x,y)\in R\Rightarrow x+2y=10$$, then domain of $$R$$ is

A
{1, 2, 3, ......,10}

B
{2, 4, 6, 8}

C
{1, 2, 3, 4}

D
{2, 4, 6, 8, 10}
Question 7
1 / -0

Let $$f ( x )$$ is a function such that $$f ^ { \prime \prime } ( x )$$ exists and $$f ^ { \prime } ( x )$$ is not a constant and $$f ( 1 ) , f ( 2 ), f ( 3 ), f ( 4 ) $$ are in $$A . P .$$ If domain of $$f ( x )$$ is $$[ 1,4 ] ,$$ then -

A
$$f ^ { \prime \prime } ( x ) = 0$$ for at least two values of
$$x$$

B
$$f ^ { \prime } ( x )$$ increases in its domain

C
$$f ^ { \prime } ( x )$$ decreases in its domain

D
$$\mathrm { f } ^ { \prime } ( \mathrm { x } )$$ is non-monotonic
Question 8
1 / -0

The function f is not defined for x=0, but for all non zero real number x, f(x) + $$2f(\dfrac{1}{x})$$=3x. The equation 44,
f(x)=f(-x) is satisfied by -

A
exactly one real number

B
exactly two real numbers

C
no real numbers

D
all non zero real numbers

Solution
Question 9
1 / -0

The relation $$R$$ defined on the set $$A=\left\{ 1,2,3,4,5 \right\} $$ by $$R=\left\{ \left( a,b \right) :\left| { a }^{ 2 }-{ b }^{ 2 } \right| <16 \right\} $$, is not given by

A
$$\left\{ \left( 1,1 \right) ,\left( 2,1 \right) ,\left( 3,1 \right) ,\left( 4,1 \right) ,\left( 2,3 \right) \right\} $$

B
$$\left\{ \left( 2,2 \right) ,\left( 3,2 \right) ,\left( 4,2 \right) ,\left( 2,4 \right) \right\} $$

C
$$\left\{ \left( 3,3 \right) ,\left( 4,3 \right) ,\left( 5,4 \right) ,\left( 3,4 \right) \right\} $$

D
none of these

Solution

Given, the relation $$R$$ defined on the set $$A=\left\{ 1,2,3,4,5 \right\} $$ by $$R=\left\{ \left( a,b \right) :\left| { a }^{ 2 }-{ b }^{ 2 } \right| <16 \right\} $$.

Then $$R$$ can be written as,
$$R=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),$$ $$(4,3),(4,4),(4,5),(5,4),(5,5)\}$$.
Question 10
1 / -0

Time complexity to check if an edge exists between two vertices would be __________.

A
O(V*V)

B
O(V+E)

C
O(1)

D
O(E)

Solution

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Re-Attempt Weekly Quiz Competition

Column-I	Column-II	Column-II
$$f(x)=[x+[x]]]$$	$$-1$$	$$-3$$
$$f(x)= sgn(x-[x])$$	$$0$$	$$-1$$
$$f(x)=[x]+[-x]$$	$$1$$	$$0$$
$$f(x)=\left[\dfrac{[x]}{1+x^2}\right]$$	$$3$$	$$1$$

Relations and Functions Test - 46

Result

Relations and Functions Test - 46

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Rank

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SHARING IS CARING

Question 1

$$A\ II\ Q$$

$$B\ III\ S$$

$$AIR$$

$$D\ II\ R$$

Question 2

$$a$$

$$x$$

$$x^{n}$$

$$a^{n}$$

Solution

Question 3

$$A\ IV\ P$$

$$C\ I\ Q$$

$$A\ IV\ Q$$

$$D\ I\ P$$

Question 4

$$A\ II\ P$$

$$D\ II\ Q$$

$$C\ I\ Q$$

$$B\ II\ Q$$

Question 5

$$\left( -\infty ,-\dfrac { 4 }{ 3 } \right] \cup \left[ 2,\infty \right) $$

$$\left( -\infty ,2 \right) $$

$$[1,\infty )$$

$$(1,\infty )$$

Solution

Question 6

{1, 2, 3, ......,10}

{2, 4, 6, 8}

{1, 2, 3, 4}

{2, 4, 6, 8, 10}

Question 7

$$f ^ { \prime \prime } ( x ) = 0$$ for at least two values of $$x$$

$$f ^ { \prime } ( x )$$ increases in its domain

$$f ^ { \prime } ( x )$$ decreases in its domain

$$\mathrm { f } ^ { \prime } ( \mathrm { x } )$$ is non-monotonic

Question 8

exactly one real number

exactly two real numbers

no real numbers

all non zero real numbers

Solution

Question 9

$$\left\{ \left( 1,1 \right) ,\left( 2,1 \right) ,\left( 3,1 \right) ,\left( 4,1 \right) ,\left( 2,3 \right) \right\} $$

$$\left\{ \left( 2,2 \right) ,\left( 3,2 \right) ,\left( 4,2 \right) ,\left( 2,4 \right) \right\} $$

$$\left\{ \left( 3,3 \right) ,\left( 4,3 \right) ,\left( 5,4 \right) ,\left( 3,4 \right) \right\} $$

none of these

Solution

Question 10

O(V*V)

O(V+E)

O(1)

O(E)

Solution

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$$f ^ { \prime \prime } ( x ) = 0$$ for at least two values of
$$x$$