$Find the value/s of m required for the line \mathcal{y} = m\mathcal{x} - 12 to be a \tan gent to the parabola \mathcal{y} = 2{\mathcal{x}}^2 - \mathcal{x} - 10$

$$\begin{gathered} What is the dis\tan ce betweeen A(-2,3) and B(-5,-1)? \\ A\left)\sqrt{53} \\ B\right)5 \\ C\left)\sqrt{18} \\ D\right)7 \\ The line which is perpendicular to 2x+y-5=0 with a y intercept at 3 has the equation: \\ A)y=2x+3 \\ B)y=-\frac{1}{2}x+3 \\ C)y=\frac{1}{2}x+3 \\ D)7=-2x+3 \\ What is the derivative of e^{-\sin 2x}? \\ A)-2\cos 2x{e}^{-\sin 2x} \\ B)2\cos 2x{e}^{-\sin 2x} \\ C)-2x{e}^{-\sin 2x} \\ D)-2\sin 2x{e}^{-\sin 2x} \\ A geometric series with a common ratio r will have a limiting sum if: \\ A)r<1 \\ B)r>1 \\ C\left)\right|r|<1 \\ D)\left|r\right|>1 \end{gathered}$$

The graphs of $y=-x^2+9$ and y = 3x + 5 intersect at the points A and B.

(i) Find the x values of the points A and B. 

(ii) Find the area bounded by the curve $y=-x^2+9$ and the line y = 3x + 5.

$$\begin{gathered} Differentiate {(2x+\tan 7x)}^4. \\ Given the points A(2,1) and B( -3, -2), find the equation of the locus of \\ point P (x, y) as it moves so that the dis\tan ce PA is always twice the dis\tan ce PB. \\ Solve for x, 5^x=20 \\ Find\int {\tan }^{2}2xdx \end{gathered}$$

$$\begin{gathered} \left(i\right) Show that \frac{d\left({\tan }^{2}x\right)}{dx}=2\tan x+2{\tan }^{3}x \\ \left(ii\right) Hence or otherwise find \frac{d\left(se{c}^{2}x\right)}{dx} \\ \left(iii\right) Use part \left(i\right) to show that {\int }_0^{\frac{\mathrm{\pi }}{3}}2{\tan }^{3}xdx=3-2\ln 2 \end{gathered}$$

$$\begin{gathered} The value of the limit \underset{\mathcal{x}\to 10}{\lim }\frac{{\mathcal{x}}^2-100}{\mathcal{x}-10} is : \\ \left(A\right)Undefined \\ \left(B\right) 0 \\ \left(C\right)8 \\ \left(D\right)0 \end{gathered}$$

$$\begin{gathered} Show that \frac{d \left({\tan }^{2}x\right)}{dx}= 2 \tan x+2{\tan }^{2}x \\ Hence or otherwise find \frac{d\left(se{c}^{2}x\right)}{dx}. \\ Use part \left(i\right) to show that {\int }_0^{\frac{\mathrm{\pi }}{3}}2{\tan }^{3}xdx=3-2\ln 2 \end{gathered}$$

$$\begin{gathered} Solve {({\log }_3 )}^2- {\log }_3{x}^4-12=0, for x>0. \\ On 1st July 2015, Jessica invested \$18 000 in a bank account that paid \\ interest at a rate of 5\% p.a, compounded annually. \\ \left(i\right) How much would be in the account after the payment of interest on 1st July 2025 if no \\ additional deposits were made? \\ \left(ii\right) Consider if Jessica made additional deposits of \$1500 to her account on the 1st July each year, \\ beginning on 1st July 2016. After the payment of interest and her deposit on 1st July 2025, how much was in her account? \\ Consider the function y=x^3+6x^2-135x . \\ \left(i\right) Show that x^3+6x^2-135x=x(x+15)(x-9). \\ \left(ii\right) Find the stationary points and determine their nature. \\ \left(iii\right) Find any point\left(s\right) of inflexion. \\ \left(iv\right) Sketch the curve showing the x and y intercepts, the turning points and point\left(s\right) of inflexion. \end{gathered}$$

$Sketch the graph of y =\left|8- 2x\right| showing any intercepts.$

$$\begin{gathered} Differentiate with respect to \mathcal{x} \\ y={\left(e^{\mathcal{x}}+3\right)}^4 \\ f\left(\mathcal{x}\right)= \frac{x^2}{\tan x} \end{gathered}$$