$$\begin{gathered} What are the values of a and b for which the following identity is true? \\ \frac{3{\mathcal{x}}^2+7}{({\mathcal{x}}^2+9)({\mathcal{x}}^2+4)}=\frac{a}{{\mathcal{x}}^2+9}+\frac{b}{{\mathcal{x}}^2+4} \end{gathered}$$

$Which expression is equal to \int \frac{d\mathcal{x}}{\sqrt{4\mathcal{x}-{\mathcal{x}}^2}}?$

$$\begin{gathered} The polynomial 4{\mathcal{x}}^3+{\mathcal{x}}^2-3\mathcal{x}+5=0 has roots \alpha \beta , and \gamma . \\ Which polynomial equation has roots \frac{\alpha }{2},\frac{\beta }{2}, and \frac{\gamma }{2}. \end{gathered}$$

$$\begin{gathered} Given that z=\sqrt{3}+\frac{1+\mathcal{i}}{1-\mathcal{i}}, find \\ lm\left(z\right) \\ \overline{z} \\ z in modulus argument form \end{gathered}$$

$$\begin{gathered} Find \int \frac{t^2-1}{t^3}dt \\ \int \frac{d\mathcal{x}}{\sqrt{6-\mathcal{x}-{\mathcal{x}}^2}} \end{gathered}$$

$$\begin{gathered} Evaluate {\int }_0^1\frac{\mathcal{x}}{(\mathcal{x}+1)(2\mathcal{x}+1)}d\mathcal{x} \\ {\int }_0^{\frac{\mathrm{\pi }}{4}}\mathcal{x}{\tan }^2\mathcal{x}d\mathcal{x} \end{gathered}$$

$\mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{inequalities} \mathrm{is} \mathrm{represented} \mathrm{by} \mathrm{the} \mathrm{Argand} \mathrm{diagram}?$

$\begin{array}{r}\left(\mathrm{A}\right) |\mathbb{z}-2|\le 2\\ \left(\mathrm{B}\right) |\mathbb{z}+2|\le 2\\ \left(\mathrm{C}\right) |\mathbb{z}-2\mathrm{i}|\le 2\\ \left(\mathrm{D}\right) |\mathbb{z}+2\mathrm{i}|\le 2\end{array}$

$Find {\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{d\mathcal{x}}{1+\tan \mathcal{x}}$

$For \mathcal{x}>0, \mathcal{y}>0, z>0 show that \mathcal{x}+\mathcal{y}+z+\frac{1}{\mathcal{x}}+\frac{1}{\mathcal{y}}+\frac{1}{z}\ge 6$

$Solve z^2=-8-6\mathcal{i}$

$$\begin{gathered} Which of the following is the locus of the point P representing the complex number z on the Argand plane such that \\ |z-2i|=2+lmz? \\ A circle \\ B hyperbola \\ C parabola \\ D straight line \end{gathered}$$

$If z=5i , find the value of z\overline{z} .$

$$\begin{gathered} The locus of all complex numbers for \left|z-2\right|=Re\left(z\right) is represented by which of the following \\ diagrams. \end{gathered}$$

$Express Z=\sqrt{3}+\mathcal{i} and W=1+\mathcal{i} in the MOD-ARG forms and hence evaluate \frac{Z^{20}}{Z^{30}} in the form a+b\mathcal{i}.$

$The square roots of (-3-4i) in the form of a+ib is$

$$\begin{gathered} The polynomial P\left(z\right) has real coefficients. Four of the roots of the equation P\left(z\right)=0 are \\ z=0 , z=1+2i , z=1-2i and z=3i The minimum number of roots that the \\ equation P\left(z\right)=0 could have is \end{gathered}$$

$$\begin{gathered} If w is a complex cube root of unity, show that 1+w+w^2=0 \\ Find the cubic equation whose roots are 1,1+w,1+w^2. \end{gathered}$$

$$\begin{gathered} The following solution demonstrates that \sqrt{2} is an irrational number \\ Let \sqrt{2}=\frac{p}{q}, where p and q are positive integers and have no common factors \\ 2=\frac{p^2}{q^2} on squaring both sides \\ {p}^2=2q^2 \\ {p}^2 is even, p is even \\ Let p=2k, where k is an integer \\ 4k^2=2q^2 \\ {q}^2=2k^2 \\ {q}^2 is even, q is even \\ Which method was used in this proof? \\ A. Direct proof. \\ B. Proof by contradiction \\ C. Proof by contrapositive \\ D. Proof by counter-examples \end{gathered}$$

$$\begin{gathered} Given that {\omega }^5=1 and \omega is a complex number what is the value of \\ 1+\omega +{\omega }^2+{\omega }^3+{\omega }^4+{\omega }^5? \\ A. 1 \\ B. 0 \\ C. \omega \\ D. -\omega \end{gathered}$$

$$\begin{gathered} Let arg\left(\mathcal{z}\right)=\frac{\pi }{5} for a certain complex number \mathcal{z}. What is arg\left({\mathcal{z}}^7\right) \\ A. \frac{-7\pi }{5} \\ B. \frac{-3\pi }{5} \\ C. \frac{2\pi }{5} \\ D. \frac{3\pi }{5} \end{gathered}$$