Higher Math 1st Paper CQ-Barishal Board-2021

Find the equation of the straight line passing through the point $(3,5)$ and parallel to the line joining the points $(0,-3)$ and $(5,0)$.

Find the perpendicular distance of the line $A B$ from the point $Q$.

Two straighte lines pass through the point $Q$ and are at a distance of 2 units from $P$. Find the angle between them.

Prove that, $\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1} \frac{x}{a}+c$, where $c$ is a constant of integration.

Find the derivative of $\frac{f(2 x)}{f\left(\frac{\pi}{2}-2 x\right)}$ by using first principle.

Show that the maximum value of $\frac{2 \ln \{g(x)\}}{\{g(x)\}^{2}}$ is $\frac{1}{\mathrm{e}}$.

Find the derivative of $\frac{1}{x}$.

Show that the tangents of the curve $f(x, y)=0$ are perpendicular on $x$-axis at the points $(0,0)$ and $(-p$, $0)$.

If $\tan ^{-1} y=n g(x)$, then prove that, $\left(1+x^{2}\right) y_{2}-2(n y-x) y_{1}$ $=0$.

If two vertices of a triangle are $(-3,4)$ and $(5,2)$ and its centroid is $(1,4)$, then find the third vertex.

Find the foot point of the perpendicular drawn from $\mathrm{C}$ to $\mathrm{AB}$.

Show that, the straight lines are perpendicular which pass through the point $(1,-1)$ and make an angle $45^{\circ}$ with the line $\mathrm{AB}$.

If $P=\left[\begin{array}{rr}3 & 3 \\ -3 & -3\end{array}\right]$, then show that, $P$ is a nilpotent matrix.

Find $f(A)$.

Prove that, $\mathrm{A}^{-1} \mathrm{~A}=\mathrm{I}_{3}$.