Tag: graphs of the form y=ax^2+bx+c

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Questions Related to graphs of the form y=ax^2+bx+c

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

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

  1. $[0,\pi ]$

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

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

  4. $\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]$
Reveal answer Fill a bubble to check yourself
B Correct answer
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$\left [0, \dfrac{\pi}{2} \right]$.

Multiple choice 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

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

  1. $(0, 7)$

  2. $(1,8)$

  3. $[1,9]$

  4. $[2,5]$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation |z-1| + |z+3| = 8 represents an ellipse with foci at 1 and -3. The distance between foci is 4. The major axis 2a = 8, so a = 4. The center is at -1. The vertices are at -1 +/- 4, i.e., 3 and -5. The range of |z-4| is the distance from 4 to the ellipse. The minimum distance is |3-4| = 1 and the maximum is |-5-4| = 9.

Multiple choice 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

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

  1. $|a| \le 3$

  2. $|b| > 3$

  3. $|c| < 3$

  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If roots z1, z2, z3 have unit modulus, then |z1|=|z2|=|z3|=1. By Vieta's formulas, a = -(z1+z2+z3). By triangle inequality, |a| = |z1+z2+z3| <= |z1|+|z2|+|z3| = 1+1+1 = 3.

Multiple choice 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

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 

  1. $ x\in R$

  2. $x\in N$

  3. $x\in I$

  4. $x\in Q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is (x+1)^2 + [x-1]^2 = (x-1)^2 + [x+1]^2. Rearranging gives (x+1)^2 - [x+1]^2 = (x-1)^2 - [x-1]^2. Let f(t) = t^2 - [t]^2. We need f(x+1) = f(x-1). Since f(t) = {t}^2 + 2{t}[t], this holds when x is an integer.

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

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

  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 $4x-3y+7=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

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

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

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

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

  1. depends on number of edges

  2. always greater than 2

  3. equal to 2

  4. equal to the number of edges
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In an incidence matrix for a simple undirected graph, each column represents an edge. Since an edge connects exactly two vertices, the sum of the entries in each column is 2.

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

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

  1. Adjacency List and Adjacency Matrix

  2. Incidence Matrix

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

  4. None of the mentioned
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Graphs can be represented using adjacency lists, adjacency matrices, and incidence matrices. All these methods are standard ways to store graph data structures.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

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

  1. O(V*V)

  2. O(V+E)

  3. O(1)

  4. O(E)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In an adjacency list representation, checking if an edge exists between two vertices u and v requires iterating through the list of neighbors of u, which takes O(degree(u)) time. In the worst case, this is O(E) or O(V). O(E) is the most appropriate choice among the options.