Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\left(1+x+x^2\right)\left(1-y+y^2\right), y(0)=\frac{1}{2}$. Then $(2 y(1)-1)$ is equal to

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96 p$ is equal to $\_\_\_\_$ .

Consider the parabola $\mathrm{P}: y^2=4 k x$ and the ellipse $\mathrm{E}: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Let the line segment joining the points of intersection of P and E , be their latus rectums. If the eccentricity of E is $e$, then $e^2+2 \sqrt{2}$ is equal to $\_\_\_\_$ .

If $\mathrm{A}=\frac{\sin 3^{\circ}}{\cos 9^{\circ}}+\frac{\sin 9^{\circ}}{\cos 27^{\circ}}+\frac{\sin 27^{\circ}}{\cos 81^{\circ}}$ and $\mathrm{B}=\tan 81^{\circ}-\tan 3^{\circ}$, then $\frac{\mathrm{B}}{\mathrm{A}}$ is equal to

$\_\_\_\_$ .