$$\begin{gathered} A hemispherical bowl of radius r cm is initially empty. water is poured into it at a cons\tan t rate of k c{m}^3 per minute. \\ When the depth of the water in the at a cons\tan t rate volume, V c{m}^3, of the water in the bowl is given by \\ V=\frac{\mathrm{\pi }}{3}{x}^2(3r-x) \\ ¡) show that \frac{dx}{dt}=\frac{k}{\mathrm{\pi x}(2\mathrm{r}-\mathrm{x})} \\ ¡¡) find an expression for t as a function of x \end{gathered}$$

$$\begin{gathered} A function is defined as f \left(\mathcal{x}\right)=10x{\mathcal{x}}^2-{\mathcal{x}}^4-9 \\ \left(i\right) Find all stationary points and determine their nature. \\ \left(ii\right) Sketch this function. \\ A particle moves on a straight line such that its dis\tan ce \\ from the origin at time t seconds is \mathcal{x} metres. \\ \left(i\right) Prove that \frac{d^2\mathcal{x}}{d{t}^2}=\frac{d}{d\mathcal{x}}\left(\frac{1}{2}{v}^2\right)where v is the velocity of the particle. \\ \left(ii\right) If \frac{d^2\mathcal{x}}{d{t}^2}=10{\mathcal{x}}^2-2{\mathcal{x}}^3, find v=0 in terms of \mathcal{x}=1. \\ \left(iii\right) U\sin g part \left(b\right), or otherwise, determine the maximum velocity of this particle. \end{gathered}$$

$Which one of following is equivalent to \tan (\frac{\mathrm{\pi }}{4}+x)?$

$$\begin{gathered} The expression \sin x-\sqrt{3}\cos x can be written in the form 2\sin (x+a) \\ What is the value of a? \end{gathered}$$

$$\begin{gathered} Consider two functions \\ f\left(x\right)=a-x^2 \\ g\left(x\right)=x^4-a \\ For precisely which values of a > 0 is the area of the region bounded by the x-axis and the curve y=f\left(x\right) \\ bigger than the area of the region bounded by the x-axis and the curve y=g\left(x\right) ? \end{gathered}$$

$$\begin{gathered} The function f\left(x\right) is defined by f\left(x\right)=3+\sqrt{x}. \\ \left(i\right) Find an expression for f^{-1}\left(x\right) in terms of x \\ \left(ii\right) Find any points of intersection of the graphs y=f\left(x\right) and y=f^{-1}\left(x\right) \end{gathered}$$

$c) Fully factorise 6x^3+ 17x^2-4x-3.$

$$\begin{gathered} Evaluate \underset{\mathcal{x}\to 0}{\lim }\frac{\sin 3\mathcal{x}}{3\mathcal{x}} \\ Find \frac{d}{d\mathcal{x}}\left[\ln \sqrt{\frac{1+\mathcal{x}}{1-\mathcal{x}}}\right] \\ Evaluate {\int }_{-3}^3\frac{d\mathcal{x}}{{\mathcal{x}}^2+9} \\ Use the substitution \sqrt{\mathcal{x}}=u to evaluate {\int }_1^4\frac{d\mathcal{x}}{\mathcal{x}+\sqrt{\mathcal{x}}} \end{gathered}$$

$$\begin{gathered} After t years the number N of individual is a population is given by N=400+100e^{-0.1t} \\ What is the difference between the initial population size and the limiting population size? \end{gathered}$$

$$\begin{gathered} A Mathematics department consists of 5 female and 5 male teachers. How many committees of 3 teachers can be chosen which contain at least one female and one male? \\ \left(A\right) 100 \\ \left(B\right) 120 \\ \left(C\right) 200 \\ \left(D\right) 2500 \end{gathered}$$

$$\begin{gathered} what is the value of {\int }_0^{\frac{\mathrm{\pi }}{4}}\left(se{c}^2\mathcal{x}-\mathcal{x}\right)d\mathcal{x}? \\ \left(A\right) 1-\frac{{\mathrm{\pi }}^2}{32} \\ \left(B\right) 1-\frac{{\mathrm{\pi }}^2}{16} \\ \left(C\right) 1-\frac{\mathrm{\pi }}{8} \\ \left(D\right) 1-\frac{\mathrm{\pi }}{4} \end{gathered}$$

Shade the following regions bounded by the curves:

$\mathcal{y}<\sqrt{4-{\left(\mathcal{x}-2\right)}^2} and \mathcal{y}>\frac{{\mathcal{x}}^2}{2}$

$Let AB be a \tan gent to the circle as shown. FInd \mathcal{x} in exact from where the dis\tan ces CD and DB are both equal to \mathcal{x}, while the dis\tan ce AB is \mathcal{x}+1$

$$\begin{gathered} Two circles C_{1}and C_{2}intersect at P and Q as shown in the diagram.The \tan gent TP to C_2 at P meets C_{1}at K . \\ The line MP meets C_1 at L. \\ copy or trace the diagram into your writing booklet. \\ prove that ∆PKL is isosceles \end{gathered}$$

$$\begin{gathered} in the diagram above, AD\left|\right|BC and the line EC is a \tan gent to the circule at D. \\ copy or trace the diagram \\ prove that B{D}^2=AD.BC \end{gathered}$$

$What is the tenn independent of \mathcal{x} in the expansion of {(\mathcal{x}+\frac{2}{\mathcal{x}})}^{20}?$

(b) $$\begin{gathered} Consider the letters which form the word DESCARTES. \\ \left(i\right) How many distinct arrangements of the letters are possible? \\ \left(ii\right) How many distinct arrangements are possible if the two E\text{'}s are to be together. \end{gathered}$$

(c) $$\begin{gathered} Roger and Mirka agree to play 6 sets of tennis. Based on past experience, \\ Roger has a 0.8 probability of winning any one set played between them. \\ What is the probability that Roger will win at least four ofthe six sets. \end{gathered}$$

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that}: \\ {\left(1+\mathcal{x}\right)}^{2\mathrm{n}}=\sum _{\mathrm{k}=0}^{2\mathrm{n}}\left(\begin{array}{c}2\mathrm{n}\\ \mathrm{k}\end{array}\right){\mathcal{x}}^{\mathrm{k}} \\ \left(\mathrm{i}\right) \mathrm{Show} \mathrm{that} \sum _{\mathrm{k}=0}^{2\mathrm{n}}\left(\begin{array}{c}2\mathrm{n}\\ \mathrm{k}\end{array}\right)=4^{\mathrm{n}} \\ \left(\mathrm{ii}\right) \sum _{\mathrm{k}=0}^{2\mathrm{n}}\left(\begin{array}{c}2\mathrm{n}\\ \mathrm{k}\end{array}\right)\frac{1}{\mathrm{k}+1}=\frac{4^{\mathrm{n}+1}-2}{4\mathrm{n}+2} \end{gathered}$$

$$\begin{gathered} The coefficient of {\mathcal{x}}^5 in {\left({\mathcal{x}}^2-\frac{2}{\mathcal{x}}\right)}^7 is \\ A. {}^{7}C_3{\left(-2\right)}^3 \\ B. {}^{7}C_4{\left(-2\right)}^4 \\ C. {}^{7}C_5{\left(-2\right)}^5 \\ D. {}^{7}C_4{\left(-2\right)}^3 \end{gathered}$$