The value of $\lambda$ such that $\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$ represents a pair of straight lines is:

$A$ straight line passing through the origin $O$ meets the parallel lines $4x + 2y = 9$ and $2x + y + 6 = 0$ at points $P$ and $Q$ respectively. In what ratio does the point $O$ divide the line segment $PQ$?

The spring balance $A$ reads $2 \ kg$ with a block $m$ suspended from it. $A$ balance $B$ reads $5 \ kg$ when a beaker with liquid is placed on the pan of the balance. The two balances are now arranged such that the hanging mass is inside the liquid in the beaker as shown in the figure. In this situation:

If $\alpha, \beta, \gamma$ are the roots of $x^3+4x+1=0$,then the equation whose roots are $\frac{\alpha^2}{\beta+\gamma}, \frac{\beta^2}{\gamma+\alpha}, \frac{\gamma^2}{\alpha+\beta}$ is

Water rises to a height $h$ in a capillary tube. If the length of the capillary tube above the surface of water is made less than $h$,then

$A$ cannon shell fired breaks into two equal parts at its highest point. One part retraces its path to the cannon with kinetic energy $E_1$,and the kinetic energy of the second part is $E_2$. The relation between $E_1$ and $E_2$ is: