$$\begin{gathered} What is the domain and range of \mathcal{y}=2{\cos }^{-1}(\mathcal{x}-1)? \\ A) Domain 0\le \mathcal{x}\le 2, Range 0\le \mathcal{y}\le \mathrm{\pi } \\ \mathrm{B}) \mathrm{Domain} -1\le \mathcal{x}\le 1, \mathrm{Range} 0\le \mathcal{y}\le \mathrm{\pi } \\ \mathrm{C}) \mathrm{Domain} 0\le \mathcal{x}\le 2, \mathrm{Range} 0\le \mathcal{y}\le 2\mathrm{\pi } \\ \mathrm{D}) \mathrm{Domain} -1\le \mathcal{x}\le 1 \mathrm{Range} 0\le \mathcal{y}\le 2\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} What is the value of {\int }_e^{e^2}\frac{1}{\mathcal{x}{\log }_e\mathcal{x}}dx ? Use the substitution u= {\log }_e\mathcal{x}. \\ A) {\log }_{e}0.5 \\ B) {\log }_{e}2 \\ C) {\log }_{e}4 \\ D) 1 \end{gathered}$$

$$\begin{gathered} A polynomial is defined by P\left(\mathcal{x}\right) = a{\mathcal{x}}^4 + 2b{\mathcal{x}}^3+4c{\mathcal{x}}^2+ 8d\mathcal{x}+16e for \\ cons\tan ts a, b, c, d and e. It is known that \mathcal{x}-2 is a factor of P\left(\mathcal{x}\right), and \\ when P\left(\mathcal{x}\right) is divided by \mathcal{x}+2 the remainder is 32. \\ What is the value of b + d? \\ \left(A\right)1 \\ \left(B\right)-1 \\ \left(C\right)16 \\ \left(D\right)-16 \end{gathered}$$

$$\begin{gathered} Factorise 8{\mathcal{x}}^3-{\mathcal{y}}^3 \\ Solve the inequation \frac{6}{\mathcal{x}+2}\le 1 \end{gathered}$$

$$\begin{gathered} Determine li{m}_{\mathcal{x}\to 0 } \frac{\sin \mathcal{x}}{ 3 \tan \mathcal{x}} . \\ \left(A\right) 0 \\ \left(B\right) \frac{1}{3} \\ \left(C\right) 1 \\ \left(D\right) 3 \end{gathered}$$

$Wllich of the following is a simplification of cot2\mathcal{x}+\tan \mathcal{x}?$

$$\begin{gathered} After 𝑡 years, the number of individuals in a population is given by 𝑁 where N = 300+100e^{-0.2t} . \\ What is the difference between the initial population and the limiting population size? \end{gathered}$$

$$\begin{gathered} Find the exact value of {\int }_0^2\frac{1}{4+{\mathcal{x}}^2}d\mathcal{x}. \\ Use the substitution u =2\mathcal{x}-1 to find \int 4\mathcal{x}\sqrt{2\mathcal{x}-1}d\mathcal{x}. \end{gathered}$$

$$\begin{gathered} By integrating the expansion of {\left(1-\mathcal{x}\right)}^n show that \\ 1-\frac{{}^{n}C_1}{2}+\frac{{}^{n}C_1}{3}...{\left(-1\right)}^n\frac{{}^{n}C_n}{+1}=\frac{1}{n+1} \end{gathered}$$

$$\begin{gathered} A ball is thrown into the air from a point O, where \mathcal{x} = 0, with an initial velocity of 25 m/s at an angle \theta = {\tan }^{-1}\frac{3}{4} to the horizontal. If air resis\tan ce is neglected and the acceleration due to gravity is taken as -10 m/s^2 , then the ball reaches its greatest height after: \\ \left(A\right) 1.5 seconds \\ \left(B\right) 15 seconds \\ \left(C\right) \frac{2}{3} seconds. \\ \left(D\right) 3 seconds \end{gathered}$$

$$\begin{gathered} What are the values of p such that \frac{p+1}{p}\le 1? \\ \left(A\right) p>0 \\ \left(B\right) p<0 \\ \left(C\right) p\le 0 \\ \left(D\right) -1\le p\le 0 \end{gathered}$$

$$\begin{gathered} An equilateral triangle of side 3 units is shown below. Vectors \tilde{u} and \tilde{v} are represented in the diagram below. \\ What is the value of \tilde{u}. \tilde{v} ? \end{gathered}$$

$$\begin{gathered} \mathrm{i}) \mathrm{Show} \mathrm{that} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{tangent} \mathrm{to} \mathrm{the} \mathrm{parabola} {\mathcal{x}}^2=16\mathcal{y} \mathrm{at} \mathrm{any} \mathrm{point} \mathrm{P}(8\mathrm{t}, 4{\mathrm{t}}^2) \mathrm{on} \mathrm{it} \mathrm{is} \mathcal{y}=\mathrm{t}\mathcal{x}-4{\mathrm{t}}^2. \\ \mathrm{ii}) \mathrm{Show} \mathrm{that} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{line} \mathrm{r} \mathrm{through} \mathrm{the} \mathrm{focus} \mathrm{S} \mathrm{of} \mathrm{the} \mathrm{parabola} \mathrm{which} \mathrm{is} \mathrm{perpendicular} \mathrm{to} \mathrm{the} \mathrm{focal} \\ \mathrm{chord} \mathrm{through} \mathrm{P} \mathrm{is} \left({\mathrm{t}}^2-1\right)\mathcal{y}+2\mathrm{t}\mathcal{x}=4\left({\mathrm{t}}^2-1\right) \\ \mathrm{iii}) \mathrm{Prove} \mathrm{that} \mathrm{the} \mathrm{locus} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} \mathrm{the} \mathrm{line} \mathrm{r} \mathrm{and} \mathrm{the} \mathrm{tangent} \mathrm{at} \mathrm{P} \mathrm{is} \mathrm{a} \mathrm{horizontal} \mathrm{line}. \end{gathered}$$

$$\begin{gathered} Which of the following is the vector projection of p onto q , \\ where p =\left(\underset{3}{4}\right) and q=\left(\underset{2}{-1}\right) \end{gathered}$$

$Evaluate \sin \left({\tan }^{-1}\frac{1}{2}\right) in exact form.$

$$\begin{gathered} Prove by mathematical induction that for any positive integer n \ge 1 \\ \frac{1}{1\times 5}+\frac{1}{5\times 9}+\frac{1}{9\times 13}+.....+\frac{1}{\left(4n-3\right)\left(4n+1\right)}=\frac{n}{4n+1} \end{gathered}$$

$$\begin{gathered} i)Prove by mathematical induction that for all integers n\ge 1. \\ 1^2+2^2+3^3+....+n^2=\frac{1}{6}n\left(n+1\right)\left(2n+1\right). \\ ii)Use this result to show that 2^2+4^2+6^2+....+{100}^2=171700 \\ iii)Hence evaluate 1^2+3^2+5^2+.....+{99}^2. \end{gathered}$$

$$\begin{gathered} The population, P, of animals in an environment in which there are scarce resources is increa\sin g such that \\ \frac{dP}{dt}=P\left(100-P\right), where t is time. The initial population is 20 animals. Which of the following is true? \\ \left(A\right) P=100-80e^{100t} \\ \left(B\right) The population is increa\sin g most rapidly when P=50. \\ \left(C\right) The population is increa\sin g most rapidly when t=50. \\ \left(D\right) The maximum population is P=50. \end{gathered}$$

$$\begin{gathered} Consider the expansion of {\left(2{\mathcal{x}}^2-\frac{1}{\mathcal{x}}\right)}^9 \\ \left(i\right) Find the coefficient of {\mathcal{x}}^6. \\ \left(ii\right) Determine the size of the greatest coefficient. \end{gathered}$$