If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).

For what value of $k$ can the quadratic polynomial $3z^{2} + 5z + k$ be factored into a product of real linear factors?

$PBA$ and $PDC$ are two secants. $AD$ is the diameter of the circle with center at $O$. $\angle A = 40^{\circ}$,$\angle P = 20^{\circ}$. Find the relationship between Quantity $I$ $(\angle DBC)$ and Quantity $II$ $(\angle ADB)$.

If the equations $ax^2 + bx + c = 0$ $(a, b, c \in R, a \ne 0)$ and $2x^2 + 3x + 4 = 0$ have a common root,then $a : b : c$ equals

Given that $x^{3}+y^{3}=72$ and $xy=6$ with $x>y$,then the value of $x-y$ is:

$Quantity \, 1$: Time taken by Manoj and Shubham to complete the work together. When Manoj works alone,he takes $4.8 \, \text{hrs}$ more than the time taken by Manoj and Shubham to complete the work together. When Shubham works alone,he takes $10.8 \, \text{hrs}$ more than the time taken by both of them to complete the work together.
$Quantity \, 2$: $7.4 \, \text{hrs}$.