Higher Math 1st Paper CQ-Dhaka Board-2022

Find the derivative of $f^{\prime}(x) \log 2 x f(2 x)$.

Find the derivative of $f(\mathrm{mx})$ with respect to $x$ using first principle.

If $y=f\left(\sec ^{-1} x\right)$, then show that, $x^{2}\left(x^{2}-1\right) y_{2}+$ $x\left(2 x^{2}-1\right) y_{1}-y=0$.

Show that, $\mathrm{B}$ is an idempotent matrix?

If $|A|=0$, then find the value of $x$.

Determine $\left(A^{T}\right)^{-1}$ where $\mathrm{x}=0$.

Show that, $\cos 75^{\circ}=\frac{1}{4}(\sqrt{6}-\sqrt{2})$.

Find the value of $\frac{\cot \theta+\cos (-\theta)}{\operatorname{cosec}(-\theta)+\tan \theta}$; where $\frac{\pi}{2}<\theta<$ $\pi$

Prove that, $\tan A \cdot \tan 3 A \cdot \tan 5 A \cdot \tan 7 A \cdot \tan 11 \mathrm{~A}=1$.

Find the intercepted portion from the $y$-axis by the circle $g(x, y)=0$.

Show that, according to the scenario-1, the straight line $f(x, y)=0$ is a tangent to the circle $g(x, y)=0$.

Find the equation of circle according to the scenario-2. Also find the equation of the circle which passes through origin and the intersection of the line $f(x, y)=0$ and the derived circle.

Show that, $\frac{b+c}{b-c}=\frac{\sin B+\sin C}{\sin B-\sin C}$

Prove that, $\frac{\cos A}{\sin B \sin C}+\frac{\cos B}{\sin C \sin A}+\frac{\cos C}{\sin A \sin B}=2$

If $\alpha=45^{\circ}, \beta=60^{\circ}$ and $a=(\sqrt{3}+1) \mathrm{cm}$ then, determine the area of $\triangle \mathrm{ABC}$.