The value of $a$ for which the equations $x^2 - 3x + a = 0$ and $x^2 + ax - 3 = 0$ have a common root is

Number of integral values of $m$ for which ${x}^2 + 5m{x} - 3m + 1 < 0$ $\forall x \in R$ is (where ${.}$ denotes the fractional part function).

Wheels of diameters $7 \,cm$ and $14 \,cm$ start rolling simultaneously from $X$ and $Y$,which are $1990.50 \,cm$ apart,towards each other in opposite directions. Both of them make the same number of revolutions per second. Both of them meet after $10 \,s$.
Quantity $1$: Speed of the smaller wheel.
Quantity $2$: $21 \pi \,cm/s$.

Determine $k$ such that the quadratic equation $x^{2}-2(1+3k)x+7(3+2k)=0$ has equal roots.

If $\alpha$ and $\beta$ are the roots of the equation $lx^2 + mx + n = 0$,then the equation whose roots are $\alpha^3\beta$ and $\alpha\beta^3$ is:

If $l, m, n$ are real and $l \ne m$,then the roots of the equation $(l - m)x^2 - 5(l + m)x - 2(l - m) = 0$ are