Numbers are selected at random, one at a time from the two-digit numbers $00,01,02,-------, 99$ with replacement. An event E occurs only if the product of the two digits of a selected number is 24. If four numbers are selected, then probability, that the event E occurs at least 3 times, is

A particular solution of $3 \mathrm{e}^x \tan y \mathrm{~d} x+\left(1-\mathrm{e}^x\right) \sec ^2 y \mathrm{~d} y=0$ with $y(1)=\frac{\pi}{4}$ is

$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are nonzero vectors such that $\overline{\mathrm{a}}$ is perpendicular to $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}},|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2,|\overline{\mathrm{c}}|=1$ and $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=1$. There is nonzero vector $\overline{\mathrm{d}}$ coplanar with $\overline{\mathrm{a}}+\overline{\mathrm{b}}$ and $2 \overline{\mathrm{~b}}-\overline{\mathrm{c}}$. If $\overline{\mathrm{d}} \cdot \overline{\mathrm{a}}=1$, then $|\overline{\mathrm{d}}|^2=$

If $\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{3}$ and $\cot ^{-1}\left(\frac{1}{x}\right)-\cot ^{-1}\left(\frac{1}{y}\right)=0$ then $2 x^2+y^2-x y=$ $\qquad$