.

$$\begin{gathered} A function is defined by the following rule: \\ f\left(\mathcal{x}\right) =\left\{{\sin }^{-1}\mathcal{x} for -1\le \mathcal{x}<1\right\} \\ \left\{{\cos }^{-1}\mathcal{x} for 0\le \mathcal{x}\le 1\right\} \\ What is the value of f\left(-\frac{1}{2}\right)+f\left(0\right)? \\ \left(A\right) -\frac{\pi }{6} \\ \left(B\right) \frac{\pi }{6} \\ \left(C\right) \frac{\pi }{3} \\ \left(D\right) \frac{2\pi }{3} \end{gathered}$$

$Which of the following is an expression for \frac{d}{d\mathcal{x}}{\sin }^{-1}(2\mathcal{x}-1)?$

$$\begin{gathered} The exact value of {\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right){\cos }^{-1}\left(\frac{1}{2}\right) in radians is: \\ \left(A\right) 0·822 \\ \left(B\right) 2700 \\ \left(C\right) \frac{{\pi }^2}{12} \\ \left(D\right) \frac{{\pi }^2}{24} \end{gathered}$$

$$\begin{gathered} The graphs of \mathcal{y}=a\mathcal{x} and \mathcal{y}= {\tan }^{-1}\left(b\mathcal{x}\right) intersect exactly three times if: \\ \left(A\right) a = b \\ \left(B\right) a = -b \\ \left(C\right) 0 <b<a \\ \left(D\right) 0 <a<b \end{gathered}$$

$$\begin{gathered} The diagram above shows the graph of: \\ \left(A\right) \mathcal{Y}={\sin }^{-1}(\mathcal{x}+1) \\ \left(B\right) \mathcal{Y}={\sin }^{-1}(\mathcal{x}-1) \\ \left(C\right) \mathcal{Y}={\cos }^{-1}(\mathcal{x}+1)-\frac{\pi }{2} \\ \left(D\right) \mathcal{Y}={\cos }^{-1}(\mathcal{x}-1)-\frac{\pi }{2} \end{gathered}$$

$$\begin{gathered} Consider the function f\left(\mathcal{x}\right)={\cos }^{-1}\left(\mathcal{x}\right)+{\cos }^{-1}(-\mathcal{x}) \\ \left(i\right) Show that f\left(\mathcal{x}\right) is cons\tan t by finding f\text{'}\left(\mathcal{x}\right). \\ \left(ii\right) Find the value of the cons\tan t. \end{gathered}$$

$Given \mathcal{Y}={\tan }^{-1}\left(\frac{a}{\mathcal{x}}\right), show that \frac{d\mathcal{Y}}{d\mathcal{x}}=\frac{-a}{{\mathcal{x}}^2+a^2}$

$\mathrm{Using} \mathrm{the} \mathrm{substitution} \mathrm{t}=\tan \frac{\mathrm{\theta }}{2} \mathrm{or} \mathrm{otherwise}, \mathrm{show} \mathrm{that} \mathrm{cosec\theta }+\mathrm{cot\theta }=\cot \frac{\mathrm{\theta }}{2}$

$$\begin{gathered} Which expression is equivalent to 1 - \cos 2\mathcal{x}? \\ \left(A\right) {\sin }^2\mathcal{x}- {\cos }^2\mathcal{x} \\ \left(B\right) 2 ({\sin }^2\mathcal{x}+ 1) \\ \left(C\right) 2 {\sin }^2\mathcal{x} \\ \left(D\right) 2 (1 + {\cos }^2\mathcal{x}) \end{gathered}$$