Higher Math 2nd Paper CQ-Barishal Board-2023

Find the equation of directrix of the conic $y^{2}=8 x+5$

From scenario-1, the vertex of the conic $\mathrm{x}=f(\mathrm{y})$ is at the point $(3,-2)$ and it passes through the point $(5,0)$, find the value of $a, b, c$.

From scenario-2, find the equation of the parabola.

Solve: $\tan 2 \theta \tan \theta=1$.

Show that, $f\left(\sqrt{2} g\left(\frac{\pi}{2}-\theta\right)\right)+f(\sqrt{g(2 \theta)})=\frac{\pi}{2}$.

Solve: $\mathrm{g}(\mathrm{x})+\sqrt{3} \mathrm{~g}^{\prime}(\mathrm{x})=\sqrt{2}$, when $-\pi<\mathrm{x}<\pi$.

Find the square root of $z_{1}$.

Show that, $\operatorname{Arg}\left(\frac{z_{1}}{z_{2}}\right)=\operatorname{Arg} z_{1}-\operatorname{Arg} z_{2}$.

Prove that, $\left(\frac{1}{2} \bar{z}_{1}\right)^{n}+\left(\frac{1}{2} z_{1}\right)^{n}=2$, when $n$ is divisible by 3 or, -1 , when $n$ is any other integer.

Find the value of $\sqrt[4]{-81}$.

If $f(1)=0$, then prove that, $\{f(\omega)\}^{3}+\left\{f\left(\omega^{2}\right)\right\}^{3}=27 \mathrm{abc}$, when $\omega$ is one of the complex cube roots of unity.

If the roots of the equation $\mathrm{g}(\mathrm{x})=0$ are $\gamma$ and $\delta$, then express the roots of $\mathrm{rp}\left(\mathrm{x}^{2}+1\right)-\left(\mathrm{q}^{2}-2 \mathrm{rp}\right) \mathrm{x}=0$ in terms of $\gamma, \delta$

If the resultant of the forces $\mathrm{a}$ and $\mathrm{b}(\mathrm{a}>\mathrm{b})$ trisects the angle between them, find the angle between two forces.

If the forces $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ act at the incentre $\mathrm{I}$ are in equilibrium show that, $\mathrm{P}: \mathrm{Q}: \mathrm{R}=\sin \left(\frac{\pi}{2}-\frac{\mathrm{A}}{2}\right): \sin \left(\frac{\pi}{2}-\frac{\mathrm{B}}{2}\right): \sin$ $\left(\frac{\pi}{2}-\frac{C}{2}\right)$.

If the forces $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ of the stem are not acted, only three like parallel forces $U, V, W$ act at the vertrices $A, B, C$ and their resultant force passes through the circumcentre $O$, prove that, $\mathrm{U}: \mathrm{V}: \mathrm{W}=\operatorname{acos} \mathrm{A}: \mathrm{b} \cos \mathrm{B}: \cos \mathrm{C}$.