Higher Math 2nd Paper CQ-Chittagong Board-2023

Determine the nature of the roots of the equation $f(\mathrm{x})=0$.

If $\{f(x)\}^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots \ldots . .+a_{2 n} x^{2 n}$, then prove that, $a_{0}+a_{3}+a_{6}+\ldots . .=3^{n-1}$.

If the roots of the equation $f(\mathrm{x})=0$ are $\alpha, \beta$ then determine the quadratic equation whose roots are $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha} .$

If $x=\sqrt{3}$, then express the complex number $Z_{1}$ in polar form.

Prove that, a real value of $x$ satisfies the equation $\frac{Z_{1}}{\overline{Z_{1}}}=\overline{Z_{2}}$ where $a^{2}+b^{2}=1$.

If $\sqrt[3]{Z_{2}}=p+i q$, then prove that, $-2\left(p^{2}+q^{2}\right)=\frac{a}{p}-\frac{b}{q}$.

Determine the eccentricity of conic $x^{2}-4 y^{2}=2$.

From scenario-2, find the equation of parabola.

In scenario-1, if the vertex of the conic is at a point $(1,-2)$ and it passes through the point $(3,0)$, then find the values of $\mathrm{a}, \mathrm{b}, \mathrm{c}$.

If the greatest and least resultant of two velocities act on a particle are $14 \mathrm{~m} / \mathrm{sec}$ and $2 \mathrm{~m} / \mathrm{sec}$ respectively, then find the velocities.

In scenario-1, if $u$ be the speed of the man in still water and $v$ that of the stream, then prove that, $u: v=\left(t_{1}{ }^{2}+t_{2}{ }^{2}\right):\left(t_{1}{ }^{2}-t_{2}{ }^{2}\right) .$

In scenario-2 if $h$ be the greatest height of a particle, then find OA. $\left[\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}\right]$

If $a+b+c=0$ and $a, b, c$ are real, then show that the roots of the equation (ii) will be real and unequal.

If the equations (i) and (ii) have a common root, then show that, $a+2 b+2 c=0$.

If the difference of the roots of the equations (i) and (ii) are equal, then show that, $b=c$ and $b+c+2 a=0$.