If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 2x + 3 = 0$,then the equation whose roots are $\frac{1}{\alpha^2}$ and $\frac{1}{\beta^2}$ is:

If $\alpha$ and $\beta$ are the roots of $x^2 - ax + b = 0$ and if $\alpha^n + \beta^n = V_n$,then

$Quantity$ $1$: Height of the tank if the volume of a cylindrical tank is $12320 \, cm^3$. Its radius and height are in the ratio of $7:10$ respectively. $Quantity$ $2$: Height of the kerosene level in a jar if a cone of radius $2 \, cm$ and height $3 \, cm$ is melted and recast into a cylinder of radius $2 \, cm$.

Number of rational roots of the equation $x^{2016} - x^{2015} + x^{1008} + x^{1003} + 1 = 0$ is equal to

Solve the given system of three linear equations and select the correct relationship between $x, y,$ and $z$ from the given options.
$I. 7x + 6y + 4z = 122$
$II. 4x + 5y + 3z = 88$
$III. 9x + 2y + z = 78$

If one root of $ax^2 + bx + c = 0$ is the square of the other,then the value of $b^3 + ac^2 + a^2c$ is