$Find the equation of the normal to y = e^{\cos \mathcal{x}} the point where \mathcal{x}=\frac{\mathrm{\pi }}{2}$

$Find the exact value of {\int }_{\frac{1}{3}}^{\frac{1}{2}}se{c}^2\frac{\mathrm{\pi }\mathcal{x}}{2}d\mathcal{x}.$

$$\begin{gathered} \text{The quadratic equation }{x}^2-3x+5=0\text{ has roots }\alpha \text{ and }\beta {\text{. What is the value of 2α}}^2\text{β+2αβ?} \\ \left(A\right) 15 \\ \left(B\right) -15 \\ \left(C\right) 30 \\ \left(D\right) -30 \end{gathered}$$

$$\begin{gathered} Expand and simplify {(\tan \theta -1)}^2 \\ \left(A\right)se{c}^2\theta \\ \left(B\right)\cos e{c}^2\theta -2 \tan \theta \\ \left(C\right)co{t}^2\theta -2 \tan \theta \\ \left(D\right)se{c}^2\theta - 2 \tan \theta \end{gathered}$$

$$\begin{gathered} Differentiate {({\mathcal{x}}^2 + ln 2)}^3. \\ \left(A\right) 3{({\mathcal{x}}^2+ln 2)}^2 \\ \left(B\right) 3*2\mathcal{x} *\frac{1}{2} {({\mathcal{x}}^2+ln 2)}^2 \\ \left(C\right) 3*2\mathcal{x}{({\mathcal{x}}^2 + ln 2)}^2 \\ \left(D\right) 3*( 2\mathcal{x} + \frac{1}{2}{({\mathcal{x}}^2 + ln 2)}^2 \end{gathered}$$

$$\begin{gathered} For \lambda >1, what is the limiting value of{\int }_0^n\frac{1}{\lambda } e^{-\lambda x} as n \to \infty ? \\ \left(A\right) 0 \\ \left(B\right) \frac{1}{{\lambda }^2} \\ \left(C\right) \frac{1}{\lambda } \\ \left(D\right) 1 \end{gathered}$$

Evaluate

i) ${\int }_0^{\frac{1}{3}}\frac{1}{3x-1}dx$

ii) ${\int }_0^2\frac{e^x-1}{e^x}dx$

iii) if $y=e^{3x}$ the  show  $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y=0$ 

$$\begin{gathered} The first three terms of a geometric series are \sqrt{2}, 2 and 2\sqrt{2.} \\ What is 21st term of series? \\ \left(A\right) 21\sqrt{2} \\ \left(B\right) 22\sqrt{2} \\ \left(C\right) 1024\sqrt{2} \\ \left(D\right) 2048 \end{gathered}$$

$$\begin{gathered} \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{derivatives} \mathrm{of} \frac{{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^2}? \\ \left(\mathrm{A}\right) \frac{{\mathrm{e}}^{\mathrm{x}}}{2\mathrm{x}} \\ \left(\mathrm{B}\right) \frac{3{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^3} \\ \left(\mathrm{C}\right) \frac{{\mathrm{e}}^{\mathrm{x}}\left(\mathrm{x}-2\right)}{{\mathrm{x}}^3} \\ \left(\mathrm{D}\right) \frac{{\mathrm{e}}^{\mathrm{x}}\left({\mathrm{x}}^2-2\mathrm{x}\right)}{{\mathrm{x}}^2} \end{gathered}$$

$$\begin{gathered} \text{What are the solutions of }\cos 2x=\frac{1}{2}\text{ for }-\pi \le x\le \pi \text{ ? } \\ \text{(A) }x=\frac{\pi }{6},\frac{5\pi }{6},-\frac{5\pi }{6},-\frac{\pi }{6} \\ \text{(B) }x=\frac{\pi }{12},\frac{11\pi }{12},-\frac{11\pi }{12},-\frac{\pi }{12} \\ \text{(C) }x=\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6} \\ \text{(D) }x=\frac{\pi }{12},\frac{11\pi }{12},\frac{13\pi }{12},\frac{23\pi }{12} \end{gathered}$$

$\text{Solve }|2x-3|\le 7\text{.}$

$\mathrm{Solve} {\log }_{4}32=\mathrm{x}.$