If $A(2,-1)$ and $B(6,5)$ are two points,the ratio in which the foot of the perpendicular from $P(4,1)$ to $AB$ divides $AB$ is:

$\int_0^{2 \pi} \sin^6 x \cos^5 x \, dx$ is equal to

Silk, honey, and lac are:

If $x=a\left\{\cos \theta+\log \tan \left(\frac{\theta}{2}\right)\right\}$ and $y=a \sin \theta$,then $\frac{d y}{d x}$ is equal to

$A$ 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......... $^{\circ}C$

If two events $A$ and $B$ are such that $P(A^c) = 0.3$,$P(B) = 0.4$,and $P(A \cap B^c) = 0.5$,then $P\left[ \frac{B}{A \cup B^c} \right]$ is equal to