Vertex

Q. If length of tangent at any point on the curve $y=f\left(x\right)$ intercepted between the point and the $x-$ axis is of length $1$ units, then the equation of the curve is:

1. $\ln \left|\frac{1-\sqrt{1-y^2}}{y}\right|+\sqrt{1-y^2}=\pm x+c$
2. $\ln \left|\frac{1-\sqrt{1-x^2}}{x}\right|+\sqrt{1-x^2}=\pm y+c$
3. $\ln \left|\frac{1-\sqrt{1-y^2}}{y}\right|+\sqrt{1-x^2}=\pm x+c$
4. $\ln \left|\frac{1-\sqrt{1-x^2}}{x}\right|+\sqrt{1-y^2}=\pm y+c$

Q. For the parabola
$x^2+8x+12y+4=0$, which of the following options is/are correct?

1. length of latus rectum is $12$ units
2. coordinates of vertex are $(-4,1)$
3. length of latus rectum is $16$ units
4. coordinates of vertex are $(4,2)$

Q. If the curve $y=a{x}^2+bx+c=0$ has $y$-intercept $6$ and vertex as $(\frac{5}{2},\frac{49}{4}),$ then the value of $a+b+c$ is

Q. Equation of a hyperbola, conjugate hyperbola and pair of their asymptote differs by a constant only.

1. True
2. False

Q. Select the correct statements for the quadratic polynomial $y=-(x-2{)}^2.$

1. $D=0$
2. $\text{Vertex}\equiv (2,0)$
3. $\text{Number of distinct real roots}=2$
4. The graph of $y=-(x-2{)}^2$ is given as:
   

Q. $\mathrm{The}\mathrm{maximum}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{quadratic}\mathrm{function}y=-5x^2+7x+2\mathrm{is}\frac{7}{10}$ .

1. False
2. True

Q. Vertex of the parabola formed by taking reflection of $y=4x^2-4x+3$ along the line $y=x$ will be

1. $(\frac{-1}{2},2)$
2. $(\frac{1}{2},-2)$
3. $(2,\frac{1}{2})$
4. $(\frac{-1}{2},-2)$

Q. Match the following quadratic expressions with it's $y-$ intercepts.

1. $2$
2. $-1$
3. $0$
4. $9$

Q.

$\alpha ,\beta$ are the roots of the equation $x^2+bx+c=0$ and $\alpha +2,\beta +2$ are the roots of the equation $x^2+px+q=0$. If the minimum value of the expression $x^2+bx+c$ is $2$, find the minimum value of $x^2+px+q$

Q. If least value of $f\left(x\right)=x^2+bx+c$ be $-\frac{1}{4}$ and maximum value of $g\left(x\right)=-x^2+bx+2$ occurs at $\frac{3}{2}$, then $c$ is equal to

Q.

If the graph of $a{x}^2+bx+c$ is given as

Then the graph of $a{(x-h)}^2+b{(x-h)}^2+c=0$

Where $h>0$, will be

1. 

2. 

3. 

4. 

Q. Match the following quadratic polynomials with the coordinates of their vertex.

1. $(-1,-2)$
2. $(\frac{11}{14},\frac{-149}{28})$
3. $(\frac{1}{3},\frac{8}{3})$
4. $(\frac{2}{9},\frac{-41}{9})$

Q. The vertex of the quadratic expression $y=3x^2+2x+5$ is given as:

1. $(\frac{14}{3},-\frac{1}{3})$
2. $(-\frac{14}{3},\frac{1}{3})$
3. $(-\frac{1}{3},\frac{14}{3})$
4. $(\frac{1}{3},-\frac{14}{3})$

Q. $$\begin{gathered} \mathrm{The}\mathrm{maximum}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{quadratic} \\ \mathrm{function}y=-5x^2+7x+2\mathrm{is}\frac{7}{10} \end{gathered}$$ .

Q.

If the vertex of parabola y = $x^2$-8x + c lies on x - axis, then the value of c is

1. -16

2. -4

3. 4

4. 16

Q. The graph of the quadratic polynomial \(y=ax^2+bx+c\) has its vertex at \((4, -5)\) and two \(x\) intercepts, one positive and one negative. Which of the following hold(s) good?

Q. If the length of $x-$ intercept made by the graph of $f\left(x\right)=4x^2-10x+4$ is $k$, then the value of $\left[k\right]$ is
(where [.] denotes the greatest integer function)

Q. The vertex of the parabola $y+2=(x-3{)}^2$ is

1. (3, 2)
2. (3, -2)
3. (-3, -2)
4. (-3, 2)

Q. For any three rational numbers $a,b,c$ and $a+b+c=0;a,c>0.$ Now, consider the parabola represented by the quadratic expression $y=a{x}^2+bx+c.$ Then select the correct options.

1. $x=1$ is one of the roots of the equation
2. Vertex lies in $I{V}^{th}$ quadrant
3. If $a=c\Rightarrow$ vertex lies on $x$-axis
4. Vertex lies in $I^{st}$ quadrant

Q. The coordinates of the vertex of a parabola represented by $y=a{x}^2+bx+c$ is, . Take D as discriminant;

1. $(\frac{b}{2a},\frac{D}{4a})$
2. $(\frac{-b}{2a},\frac{-D}{4a})$
3. $(\frac{b}{2a},\frac{-D}{4a})$
4. $(\frac{-b}{2a},\frac{D}{4a})$

Q. For the given quadratic equation $y=x^2-3x+2,$ select the correct option(s).

1. Graph of $y=x^2-3x+2$ is
   
   
2. Graph of $y=x^2-3x+2$ is
   
   
3. $\text{vertex}\equiv (0,2);$
   $y-\text{intercept}\equiv (\frac{3}{2},-\frac{1}{4})$
4. $\text{vertex}\equiv (\frac{3}{2},-\frac{1}{4});$
   $y-\text{intercept}\equiv (0,2)$

Q. Tap on the bubbles having expression for $f\left(x\right)$ with vertex as it's point of maxima.

1. $-269x^2-5x+8$
2. $-7x^2-3x+3$
3. $-25x^2+9x+7$
4. $-54x^2+2x-53$
5. $-999x^2+2x-3$
6. $-x^2+2x+5$
7. $28x^2-992x-93$
8. $-5x^2+2x+3$
9. $55x^2-2x+3$
10. $5x^2+5x+3$

Q. The graph of $f\left(x\right)=a{x}^2+bx+c$ is shown below, such that $b^2-4ac=-4.$ If the length of segment $AB$ and $AC$ are $1$ and $4$ respectively, then the value of $(a+b+c)$ is equal to

Q. If parabola y=(x−2)2 is shifted by 3 units toward right, then will be the equation of new parabola obtained.

1. $y=(x+3{)}^2$
2. $y=(x-3{)}^2$
3. $y=(x+1{)}^2$
4. $y=(x-5{)}^2$

Q. If parabola y=(x−2)2 is shifted by 3 units toward right, then will be the equation of new parabola obtained.

1. $y=(x+1{)}^2$
2. $y=(x+3{)}^2$
3. $y=(x-3{)}^2$
4. $y=(x-5{)}^2$

Q.

If the vertex of parabola y = $x^2$-8x + c lies on x - axis, then the value of c is

1. -16

2. -4

3. 4

4. 16

Q. Select the correct statements for the quadratic polynomial $y=(x-1{)}^2$ and draw the graph.

1. Graph of $y=(x-1{)}^2$ is given as:
   
2. $\text{Vertex}\equiv (1,0)$
3. $\text{Roots}=1$
4. $D=0$

Q. For any three rational numbers $a,b,c$ and $a+b+c=0;a,c>0.$ Now, consider the parabola represented by the quadratic expression $y=a{x}^2+bx+c.$ Then select the correct options.

1. Vertex lies in $I^{st}$ quadrant
2. Vertex lies in $I{V}^{th}$ quadrant
3. If $a=c\Rightarrow$ vertex lies on $x$-axis
4. $x=1$ is one of the roots of the equation

Q. If the length of $x-$ intercept made by the graph of $f\left(x\right)=4x^2-10x+4$ is $k$, then the value of $\left[k\right]$ is
(where [.] denotes the greatest integer function)

Q. The vertex of the parabola represented by the quadratic equation $y=x^2+2x+6$ is

1. (1, -6)
2. (-1, 5)
3. (1, -5)
4. (-1, 6)