Consider all the structural isomers with molecular formula $\mathrm{C}_5 \mathrm{H}_{11} \mathrm{Br}$ are separately treated with $\mathrm{KOH}(\mathrm{aq})$ to give respective substitution products, without any rearrangement. The number of products which can exhibit optical isomerism from these is $\_\_\_\_$。

Let the domain of the function $f(x)=\log _3 \log _5 \log _7\left(9 x-x^2-13\right)$ be the interval $(\mathrm{m}, \mathrm{n})$. Let the hyperbola $\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ have eccentricity $\frac{\mathrm{n}}{3}$ and the length of the latus rectum $\frac{8 \mathrm{~m}}{3}$. Then $\mathrm{b}^2-\mathrm{a}^2$ is equal to :

A rectangle is formed by the lines $x=0, y=0, x=3$ and $y=4$. Let the line L be perpendicular to $3 x+y+6=0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\frac{1}{2},-5\right)$ from the line $L$ is equal to :

Let the direction cosines of two lines satisfy the equations : $4 l+m-n=0$ and $2 m n+10 n l+3 l m=0$.

Then the cosine of the acute angle between these lines is :