$$\begin{gathered} Which of the following statements is FALSE? \\ A) {\cos }^{-1}(-\theta )=-{\cos }^{-1}\theta \\ B) {\sin }^{-1}(-\theta )=-{\sin }^{-1}\theta \\ C) {\tan }^{-1}(-\theta )=-{\tan }^{-1}\theta \\ D) {\cos }^{-1}(-\theta )=\mathrm{\pi }-{\cos }^{-1}\theta \end{gathered}$$

$The gradient of the \tan gent to the curve y={\tan }^{-1}\left(\sin x\right) at x=0 is:$

$Find the derivative of e^{\sin 3x}$

$$\begin{gathered} The rise and fall of the tide is assumed toi be simple harmonic, with the time between \\ low and high tide begin six hours. \\ The water depth at a harbour entrance at high and low tide are 14 metre, \\ respectively. \\ If t is the number of hours after low tide, and y the water depth in meter, \\ which equation model this informatiom? \end{gathered}$$

$$\begin{gathered} \left(i\right)For what values of x is si{n}^{-1}xdefined? \\ \left(ii\right)Find the maximum value of 2x(1-x). \\ \left(iii\right)Find the range of the function f givenby f\left(x\right)=si{n}^{-1}\left\{2x(1-x)\right\} with domain0\le x\le 1. \end{gathered}$$

$$\begin{gathered} A goose is flying horizontally along at a speed of S m/s and at an altitude of H m when it passes a goose \\ hunter below lying at (0,0) with a goose rifle. Simul\tan eously, the hunter shoots a bullet up at an angle \theta from \\ the horizontal with a velocity V m/s hoping to shoot the goose. Assume the acceleration due to gravity is g m/s^2. \\ \left(i\right) Explain why x = St describes the dis\tan ce the goose has flown after t seconds. You may assume the equations \\ of motion x =Vt \cos \theta and y =Vt \sin \theta - \frac{g{t}^2}{2} describe the horizontal and vertical dis\tan ces of the bullet after t seconds. \\ Assume that the bullet hits the goose. \\ \left(ii\right) Explain why S =V \cos \theta . \\ \left(iii\right) Hence show that 2 \tan 2 H = - \frac{g{t}^2}{2} + St \theta . \\ \left(iv\right) Hence show that there are two possible occasions when the bullet can shoot the goose if S^2 {\tan }^2\theta > 2gH . \end{gathered}$$

$$\begin{gathered} i) Find the maximum value 2\mathcal{x}(1-\mathcal{x}\left) \\ ii\right) Hence, or otherwise, find the range of the function given by: \\ f\left(\mathcal{x}\right)={\sin }^{-1}\left\{2\mathcal{x}\right(1-\mathcal{x}\left)\right\} in the interval 0\le \mathcal{x}\le 1. \end{gathered}$$

$Evaluate \tan \left\{{\cos }^{-1}\right(\frac{-1}{3}\left)\right\}$

$Find \int \frac{1}{\sqrt{16-9x^2}}dx$

$What is the natural domain of f\left(x\right)={\log }_c\left({\cos }^{-1}x\right)?$