The graph below represents the path of a ball thrown by Ankush. The maximum height, $h,$ the ball reaches with respect to time, $t,$ is represented by the polynomial $h\left(t\right)=-t^2+\frac{19}{4}t+\frac{5}{4}.$

How long will it take for the ball to reach the ground$?$ Show your steps.

Shown below is the graph of a quadratic polynomial, $y\left(x\right)=p{x}^2+qx+r.$

Find the nature of the roots of the quadratic equation, $p{x}^2+qx+r=0.$ Give a reason for your answer.

${\left(2x+1\right)}^3=8x\left(x^2+1\right)+3$

Write the zeros of the above equation in the form $\left(x,y\right).$ Show your steps. 

$3{\left(3\right)}^{2m}+11{\left(3\right)}^m=4$

Use the substitution $3^m=x$ to solve for $m.$ Show your steps.

Three students were asked how they would verify their solution of a quadratic equation, $\left(x-2\right)\left(x-5\right)=0.$ Shown below are their responses.

Student $1$ said, "In the first bracket, $x$ must equal $2,$ and in the second bracket, $x$ must equal $5.$ So $\left(2-2\right)\left(5-5\right)=0.$"

Student $2$ said, "In the first bracket,$x$ must equal $2$, but in the second bracket, $x$ can have any real number value. For example $\left(2-2\right)\left(3-5\right)=0$ or $\left(2-2\right)\left(10-5\right)=0.$"

Student $3$ said, "Both brackets should always have the same $x$ value. So, $x$ is either $2$ or $5$ in both brackets. For example, $\left(2-2\right)\left(2-5\right)=0$ and $\left(5-2\right)\left(5-5\right)=0.$"

Whose response is correct$?$

Aman solved a quadratic equation and found its roots to be real.

Which of these could represent the graph of the equation Aman solved$?$

Which of these is a QUADRATIC equation having one of its roots as zero$?$

$i) x^3+x^2=0$

$ii) x^2-2x=0$

$iii) x^2-9=0$