$$\begin{gathered} For what values of x, where x\ne 0, does the geometric series \\ 1+\frac{2x}{x+1}+{\left(\frac{2x}{x+1}\right)}^2+.... have a limiting sum? \end{gathered}$$

$$\begin{gathered} e) P divides the interval from (-5, 6) to (4, 3) externally in the ratio 3 : 1. \\ Find the coordinates of P. \end{gathered}$$

$$\begin{gathered} For the polynomial P\left(\mathcal{x}\right)={\mathcal{x}}^3+2{\mathcal{x}}^2-15\mathcal{x}-36 \\ Factorize P\left(\mathcal{x}\right) fully over the real numbers. \\ Hence solve {\mathcal{x}}^3+2{\mathcal{x}}^2-15\mathcal{x}-36<0. \end{gathered}$$

$$\begin{gathered} The function f\left(\mathcal{x}\right)={\log }_e \mathcal{x}+5\mathcal{x} has a zero near \mathcal{x}=0.2. U\sin g \mathcal{x}=0.2 as a first approximation, use one application of Newton\text{'}s method to find a second approximation to the zero. Write your answer to 3 decimal places. \\ \left(i\right) Find the natural domain of the function f\left(\mathcal{x}\right)=\frac{1}{\sqrt{4-{\mathcal{x}}^2}} \\ \left(ii\right) The sketch below shows part of the graph of \mathcal{Y}=f\left(\mathcal{x}\right). The area under the curve for 0\le \mathcal{x}\le 1 is shaded. \\ Find the area of the shaded region. \end{gathered}$$

$$\begin{gathered} Express \sqrt{3} \cos \mathcal{x}-\sin \mathcal{x} in the form R\cos (\mathcal{x}+\alpha ) where \\ 0<\alpha <\frac{\mathrm{\pi }}{2} and R>0. \\ Hence ,solve \sqrt{3} \cos \mathcal{x}-\sin \mathcal{x}=1 for 0\le \mathcal{x}\le \frac{\mathrm{\pi }}{2}. \end{gathered}$$

$$\begin{gathered} The area enclosed by the curves \mathcal{y}=\sin \mathcal{x} and \mathcal{y}=\cos \mathcal{x} is shaded in the diagram. \\ Which expression could be used to calculate this area? \\ A.{\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{5\mathrm{\pi }}{4}}\left(\cos \right(\mathcal{x})-\sin (\mathcal{x}\left)\right) d\mathcal{x} \\ B.{\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{5\mathrm{\pi }}{4}}\left(\sin \right(\mathcal{x})-\cos (\mathcal{x}\left)\right) d\mathcal{x} \\ C.{\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{5\mathrm{\pi }}{4}}\left(\cos \right(\mathcal{x})+\sin (\mathcal{x}\left)\right) d\mathcal{x} \\ D.{\int }_{\frac{\mathrm{\pi }}{3}}^{\frac{4\mathrm{\pi }}{3}}\left(\cos \right(\mathcal{x})-\sin (\mathcal{x}\left)\right) d\mathcal{x} \end{gathered}$$

$$\begin{gathered} What is the indefinite integral for \int ({\cos }^2 \mathcal{x}+se{c}^2 \mathcal{x}) d\mathcal{x}? \\ \left(A\right)\frac{1}{2}\mathcal{x}+\frac{1}{4}\sin 2\mathcal{x} +\frac{1}{2} \tan \mathcal{x}+C \\ \left(B\right)\frac{1}{2}\mathcal{x}-\frac{1}{4}\sin 2\mathcal{x} +\frac{1}{2} \tan \mathcal{x}+C \\ \left(C\right)\frac{1}{2}\mathcal{x}+\frac{1}{4}\sin 2\mathcal{x}+\tan \mathcal{x}+C \\ \left(D\right) \frac{1}{2}\mathcal{x}-\frac{1}{4}\sin 2\mathcal{x}+\tan \mathcal{x}+C \end{gathered}$$

$$\begin{gathered} Evaluate {\int }_{-1}^1\frac{d\mathcal{x}}{\sqrt{2-{\mathcal{x}}^2}}. \\ Differentiate (1-{\mathcal{x}}^2) \ln ({\mathcal{x}}^2-1) with respect to \mathcal{x}. \\ Evaluate {\int }_0^1\frac{2\mathcal{x}}{2\mathcal{x}+1}d\mathcal{x} u\sin g the substitution u=2\mathcal{x}+1. \end{gathered}$$

$$\begin{gathered} The point R divides the interval from A(-4, 1) to B(2, 7) internally in the ratio 5 : 1. What are the coordinates of R? \\ \left(A\right) (6, 1) \\ \left(B\right) (1, 6) \\ \left(C\right) (-3, 2) \\ \left(D\right) (2, -3) \end{gathered}$$

$Which of the following diagrams could represent the graph of y={(x+3)}^2{(2-x)}^3 ?$

$$\begin{gathered} Point A,B and C lie onthe circle .the length of th chord AB is a \mathrm{co}ns\tan t k. \\ the radius measure of \angle ABC and \angle BCA are \theta and \alpha respectively. \end{gathered}$$

$$\begin{gathered} Let l equal the sum of the length of chords CA and CB .show that \\ t=\frac{k}{\sin \alpha }\left[\sin \theta +\sin (\theta +\alpha )\right]. \\ ii) why is \alpha a ccons\tan t? \\ iii) Evaluate \frac{dt}{d\theta } when \theta =\frac{\mathrm{\pi }}{2}-\frac{\alpha }{2}. \end{gathered}$$

$The interval joining the points A(2,1) and B(-3,-4) cuts the \mathcal{x}-axis at C. Find the ratio in which the point C divides AB.$

$Which of the following represents the graph of \mathcal{y}=2{\sin }^{-1}(1-\mathcal{x})?$

$Given that \sin \left(x-\frac{\mathrm{\pi }}{3}\right)=3 \cos \left(x-\frac{\mathrm{\pi }}{4}\right), find the exact value of \tan x.$

$$\begin{gathered} The points P(2ap, a{p}^2) and Q(aq, a{q}^2) lie on the parabola x^2=4ay. \\ \left(i\right) Show that the equation of the chord PQ is (p+q)x-2y-2apq=0 \\ \left(ii\right) Show that the gradient of the \tan gent at P is p. \\ \left(iii\right) Prove that if the \tan gent at P is parallel to the normal at Q then \\ PQ passes through the focus S \end{gathered}$$

$$\begin{gathered} Prove by mathematical induction that n^3+2n is divisble by 3 for all \\ positive integers n. \end{gathered}$$

$$\begin{gathered} The primitive of\frac{1}{\sqrt{4-9{\mathcal{x}}^2}} is \\ \left(A\right) \frac{1}{2}{\sin }^{-1}2\mathcal{x}+C \\ \left(B\right) \frac{1}{2}{\sin }^{-1}3\mathcal{x}+C \\ \left(C\right) \frac{1}{3}{\sin }^{-1}\frac{\mathcal{x}}{2}+C \\ \left(D\right) \frac{1}{3}{\sin }^{-1}\frac{3\mathcal{x}}{2}+C \end{gathered}$$

$U\sin g the substitution u=\frac{1}{2}\mathcal{x}-1, show that {\int }_4^6\frac{\mathcal{x}}{\sqrt{{\displaystyle \frac{1}{2}}\mathcal{x}-1}}d\mathcal{x}=\frac{8}{3}(5\sqrt{2}-4)$