Tag: functions and their graphs

Questions Related to functions and their graphs

If $f(x)=\left | \sin x \right |$, then domain of $f$ for the existence of inverse is  

business maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. $[0,\pi ]$

  2. $\left [ 0,\dfrac{\pi }{2} \right ]$

  3. $\left [ -\dfrac{\pi }{4},\dfrac{\pi }{4} \right ]$

  4. $\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]$

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Correct Option: B
Explanation:

We know that $-1 \leq \sin x \leq 1$ for $x \in \left [ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right] $. 

For $| \sin x |$ to be invertible, the function has to be one-to-one. 
Thus, we need unique values of $x$ that give unique values of $f$ and vice versa.
For $x \in \left [0, \dfrac{\pi}{2} \right]$, $0 \leq \sin x \leq 1 \Rightarrow  0 \leq | \sin x \leq 1$
For $x \in \left [-\dfrac{\pi}{2},0 \right]$, $-1 \leq \sin x \leq 0 \Rightarrow  0 \leq | \sin x \leq 1$.
So, we have
$ \left [0, \dfrac{\pi}{2} \right] \rightarrow\left [0, 1 \right]$
$ \left [- \dfrac{\pi}{2},0 \right] \rightarrow\left [0, 1 \right]$
Since both the domains of $|\sin x|$ map to$\left [0, 1 \right]$, we consider only one of them for $x$ to be unique. 
Here, according to the options, the domain of $f$ must be$\left [0, \dfrac{\pi}{2} \right]$.

If $|z-1|+ |z+3| \le 8$, then the range of values of $|z-4|$ is

business maths limits and continuity of a function graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. $(0, 7)$

  2. $(1,8)$

  3. $[1,9]$

  4. $[2,5]$

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Correct Option: C

If all the roots of $z^3 +az^2 +bz+c=0$ are of unit modulus, then

business maths limits and continuity of a function graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. $|a| \le 3$

  2. $|b| > 3$

  3. $|c| < 3$

  4. $None\ of\ these$

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Correct Option: A

The solution set of the equation $(x+1)^2+[x-1]^2=(x-1)^2+[x+1]^2$ where $[x]$ and $(x)$ are the greatest integer and nearest integer to $x$, is 

business maths limits and continuity of a function graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. $ x\in R$

  2. $x\in N$

  3. $x\in I$

  4. $x\in Q$

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Correct Option: C

Two lines $L _{1} :2x+3y-5=0$ and $L _{2} :3x-4y+1=0$ intersect a point $P$ and make an angle $\theta$ with each other. Equation of a line which passes through $P$ and makes an angle $(\pi/2-\theta)$ with the line $L _{1}$ is

maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. $16x+64y+79=0$ and $4x+3y+7=0$

  2. $16x+63y-79=0$ and $4x+3y+7=0$

  3. $16x-63y+79=0$ and $4x-3y+7=0$

  4. $16x-63y-79=0$ and $4x-3y+7=0$

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Correct Option: A

If the line $3x+4y=\sqrt{7}$ touches the ellipse $3x^{2}+4y^{2}=1$, then the point of contact is 

maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. $(\dfrac{1}{\sqrt{7}},\dfrac{1}{\sqrt{7}})$

  2. $(\dfrac{1}{\sqrt{3}},-\dfrac{1}{\sqrt{3}})$

  3. $(\dfrac{1}{\sqrt{7}},-\dfrac{1}{\sqrt{7}})$

  4. None of these

Show answer
Correct Option: A
Explanation:

$\begin{array}{l} 3{ x^{ 2 } }+4{ y^{ 2 } }=1 \ Equation\, of\, \tan  gent\, to\, ellipse\, at\, \left( { { x _{ 1 } },{ y _{ 1 } } } \right)  \ 3x{ x _{ 1 } }+4y{ y _{ 1 } }=1 \ \frac { { 3x } }{ { \sqrt { 7 }  } } +\frac { { 4y } }{ { \sqrt { 7 }  } } =1 \ { x _{ 1 } }=\frac { 1 }{ { \sqrt { 7 }  } } \, \, \, \, \, { y _{ 1 } }=\frac { 1 }{ { \sqrt { 7 }  } }  \ \left( { \frac { 1 }{ { \sqrt { 7 }  } } ,\frac { 1 }{ { \sqrt { 7 }  } }  } \right)  \ Hence, \ option\, \, A\, is\, correct\, answer. \end{array}$

A graph with all vertices having equal degree is known as a .....

maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. Multi Graph

  2. Regular Graph

  3. Simple Graph

  4. Complete Graph

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Correct Option: B

The column sum in an incidence matrix for a simple graph is ________________.

maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. depends on number of edges

  2. always greater than 2

  3. equal to 2

  4. equal to the number of edges

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Correct Option: C

Which of the following ways can be used to represent a graph?

maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

  1. Adjacency List and Adjacency Matrix

  2. Incidence Matrix

  3. Adjacency List, Adjacency Matrix as well as Incidence Matrix

  4. None of the mentioned

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Correct Option: C

The equation of a line parallel to $x+2y=1$ and passing through the point of intersection of the lines $x-y=4$ and $3x+y=7$ is ?

graphs to solve linear and non linear equations functions and their graphs curved graphs sets, relations and functions maths

  1. $x+2y=5$

  2. $4x+8y-1=0$

  3. $4x+8y+1=0$

  4. none of these

Show answer
Correct Option: B
Explanation:

Let the equation of line parallel to $x+2y=1$ be $y=mx+c$

intersection point of line $x-y=4$ and $3x+y=7$ is given by solving two equation we get $x=\dfrac { 11 }{ 4 } $ and $y=\dfrac { -5 }{ 4 } $
from equation $x+2y=1\ \Rightarrow y=-\dfrac { 1 }{ 2 } x+\dfrac { 1 }{ 2 } $
we get ${ m } _{ 1 }=-\dfrac { 1 }{ 2 } $
since $y=mx+c$ is parallel to $x+2y=1$
$\therefore { m } _{ 2 }=-\dfrac { 1 }{ 2 } $
also, $y=mx+c\ \Rightarrow \dfrac { -5 }{ 4 } =\dfrac { -1 }{ 2 } .\dfrac { 11 }{ 4 } +c\ \Rightarrow c=\dfrac { 1 }{ 8 } $
therefore equation of line parallel to $x+2y=1$ is given as 
$y=mx+c\ \Rightarrow y=\dfrac { -1 }{ 2 } x+\dfrac { 1 }{ 8 } \ \Rightarrow 4x+8y-1=0$