Higher Math 2nd Paper CQ-Dinajpur Board-2021

Find the equation of a hyperbola whose parametric coordinates is $(\sqrt{3} \sec \theta, 2 \tan \theta)$.

From scenario-1, show that the equation indicates and ellipse, find the equation of its latus rectum.

From scenatio-2, find the equation of the hyperbola.

If $\alpha, \beta$ are two roots of the equation $x^{2}+5 x+3=$ 0 , then find the value of $\frac{1}{\beta}-\frac{1}{\alpha}$.

From scenario-2, if $3+i$ is a root of the equation $g(x)=0$, find the other roots.

From scenario-1, if 1 is a root of the equation $f(x)$ $=0$ and other roots are $\alpha, \beta, \gamma$, then find $\alpha^{3}+\beta^{3}+$ $r^{3}$.

If $a=1, b=-2, c=1$, then find the nature of the roots of the equation $f(x)=0$.

From the scenario, if $\alpha, \beta$ are the roots of the equation $f(x)=0$, then express the roots of the equation $c x^{2}-$ $\left(\frac{b^{2}}{a}-2 c\right) x+c=0$ in terms of $\alpha, \beta$.

From scenario if $a=1, b=-2 n, c=n^{2}-m^{2}$ then form an equation whose roots are the sum and absolute value of the difference of the roots of the equation $f(x)=0$.

If two equal forces of magnitude $F$ acting at a point at an angle $120^{\circ}$, determine the magnitude and direction of their resultant.

How much equal forces of same magnitudes are to be added to each of the forces in scenario-1, so that the point of action of their resultant moves aside $5 \mathrm{~cm}$ ?

From scenario-2 if $Q=13 \mathrm{~N}$ and $R=12 \mathrm{~N}$ is the resultant of $P$ and $Q$, find the value of $P$.

If $x=\frac{1}{2} \cos ^{-1} \frac{3}{4}$, then what is the value of $\tan x$.

Show that, $N=\frac{1}{2} \tan ^{-1} x$.

Solve: $f(\theta)+f(2 \theta)+f(3 \theta)=0$, when $-2 \pi \leq \theta \leq$ $2 \pi$.