$A$ wall is made up of two layers $A$ and $B$. The thickness of the two layers is the same,but materials are different. The thermal conductivity of $A$ is double than that of $B$. In thermal equilibrium,the temperature difference between the two ends is $36\,^{\circ}C$. Then the difference of temperature at the two surfaces of $A$ will be......... $^{\circ}C$

The condition for the lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ to be conjugate with respect to the circle $x^2 + y^2 = r^2$ is

Acetaldehyde forms a white crystalline precipitate on mixing with a ...... solution of

The two surfaces of a biconvex lens have the same radii of curvature. This lens is made of glass with a refractive index of $1.5$ and has a focal length of $10 ~cm$ in air. The lens is cut into two equal halves along a plane perpendicular to its principal axis to yield two plano-convex lenses. The two pieces are glued such that the convex surfaces touch each other. If this combination lens is immersed in water (refractive index $=$ $4/3$),its focal length (in $cm$) is:

The binding energy per nucleon versus mass number curve for nuclei is shown in the figure. $W, X, Y$ and $Z$ are four nuclei indicated on the curve. The process that would release energy is

If $\alpha$ is a non-real root of the equation $x^6-1=0$,then $\frac{\alpha^2+\alpha^3+\alpha^4+\alpha^5}{\alpha+1}$ is equal to