The least integer $k$ which makes the roots of the equation ${x^2} + 5x + k = 0$ imaginary is

For what value of $k$ will the equation $x^2 - (3k - 1)x + 2k^2 + 2k - 11 = 0$ have equal roots?

$Quantity \, 1$: Value of $y$. $A$ vessel contains $2.5 \, \text{liters}$ of water and $10 \, \text{liters}$ of milk. $20 \%$ of the contents of the vessel are removed. To the remaining contents,$x$ liters of water are added to reverse the ratio of water and milk. Then $y$ liters of milk are added again to reverse the ratio of water and milk. $Quantity \, 2$: $120 \, \text{liters}$.

If the ratio of the roots of the equation $ax^2 + bx + c = 0$ is $p:q$,then:

If the product of the roots of the equation $(a + 1)x^2 + (2a + 3)x + (3a + 4) = 0$ is $2$,then the sum of the roots is:

For what value of $\lambda$ is the sum of the squares of the roots of ${x^2} + (2 + \lambda )x - \frac{1}{2}(1 + \lambda ) = 0$ minimum?