Higher Math 2nd Paper CQ-Sylhet Board-2023

Show that, $\sec ^{2}\left(\cot ^{-1} \frac{1}{2}\right)+\operatorname{cosec}^{2}\left(\tan ^{-1} \frac{1}{3}\right)=15$.

From scenario-1, find the value of $\sum \alpha^{2} \beta$.

From scenario-2, prove that, $a^{n}+b^{n}=2$ or -1 , where the value of $n$ is respectively divisible by 3 or any other integer.

If the square of resultant of two equal forces is equal to the thrice of the product of that two forces, then find the angle between the forces.

If three forces $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ of scenario-1 are in equilibrium, then prove that, $\frac{\mathrm{P}}{\mathrm{a}^{2}\left(\mathrm{~b}^{2}+\mathrm{c}^{2}-\mathrm{a}^{2}\right)}=\frac{\mathrm{Q}}{\mathrm{b}^{2}\left(\mathrm{a}^{2}+\mathrm{c}^{2}-\mathrm{b}^{2}\right)}=\frac{\mathrm{R}}{\mathrm{c}^{2}\left(\mathrm{a}^{2}+\mathrm{b}^{2}-\mathrm{c}^{2}\right)}$.

In scenario-2 if the forces $\mathrm{P}$ and $\mathrm{Q}$ are both increased by $\mathrm{S}$ and the resultant is displaced from the point $\mathrm{C}$ to $\mathrm{D}$, then show that, $C D=\frac{S}{P-Q} A B$.

If a particle projected with the velocity $u$ at an angle $\alpha$ to the horizon, then prove that, greatest height, $\mathrm{H}=\frac{u^{2} \sin ^{2} \alpha}{2 g} .$

In scenario-1, if a swimmer requires $t$ seconds to cross $A B$ distance and $t^{\prime}$ seconds to cross AC distance, then show that, $t: t^{\prime}=\sqrt{u+v}: \sqrt{u-v}$.

From scenario-2, find the horizontal range of projectile.

Prove that, $\cos ^{-1} x=2 \cos ^{-1} \sqrt{\frac{1+x}{2}}$.

Prove that, $A-\frac{1}{2} B+C=\tan ^{-1}\left(\frac{\sqrt{35}-1}{\sqrt{7}+\sqrt{5}}\right)$.

Find the solution of the equation $f(\mathrm{x})+f(2 \mathrm{x})+f(3 \mathrm{x})=0$ from the stem where $0 \leq \mathrm{x} \leq \pi$.

If $9 x^{2}-(k+2) x+4$ is a perfect square, then find the value of $k$.

From scenario-1, find the quadratic equation whose roots are $\mathrm{a}+\frac{1}{\mathrm{~b}}$ and $\mathrm{b}+\frac{1}{\mathrm{a}}$.

By using $\alpha$ and $\beta$ of scenario-2, express the two roots of the equation $\mathrm{r}\left(\mathrm{x}^{2}+1\right)-\left(\mathrm{q}^{2}-2 \mathrm{r}\right) \mathrm{x}=0$ in terms of $\alpha$ and $\beta$.