$$\begin{gathered} \left(i\right) Show that \frac{d}{du} {\log }_e (u+\sqrt{a^2+u^2)}= \frac{1}{\sqrt{a^2+u^2}} , where a is a cons\tan t. \\ (ii) Hence evaluate{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\cos \mathcal{x}}{\sqrt{4+{\sin }^2 \mathcal{x}}}d\mathcal{x}. \\ Use the substitution t = \tan \frac{\mathcal{x}}{2} to find \int \frac{1}{1+\cos \mathcal{x}+\sin \mathcal{x}} d\mathcal{x}. \\ A complex number satisfies \left|\mathcal{z}-4\right| \le 2 and Im(\mathcal{z})\le 0. \\ (i) Sketch the locus of \mathcal{z}. \\ (ii) Show that - \frac{\mathrm{\pi }}{6} \le arg \mathcal{z}\le 0. \\ By rationali\sin g the numerator of the integrand, evaluate {\int }_{\frac{1}{2}}^1 \sqrt{\frac{\mathcal{x}}{2-\mathcal{x}}} d\mathcal{x}. \\ Use a suitable substitution to evaluate {\int }_3^9 \frac{3}{(9+\mathcal{x})\sqrt{\mathcal{x}}} d\mathcal{x}. \end{gathered}$$

$$\begin{gathered} \left(i\right) Find a general solution to the equation \cos 3\mathcal{x}= \sin 2\mathcal{x}. \\ \left(ii\right) Hence, or otherwise, find the smallest positive solution of the equation \\ \cos 3\mathcal{x}= \sin 2\mathcal{x}. \end{gathered}$$

$Solve the quadratic equation{\mathcal{z}}^2 - 2i\mathcal{z} + 3 = 0.$

$$\begin{gathered} Given that \mathcal{z}=\cos \theta +\mathcal{i}\sin \theta \\ \left(i\right){\mathcal{z}}^n+\frac{1}{{\mathcal{z}}^n}=2\cos n\theta \\ \left(ii\right) Express {\mathcal{x}}^5-1 as the product of three factors each containing real \\ coefficents. \\ \left(iii\right) Prove that \left(1-\cos \frac{2\mathrm{\pi }}{5}\right)\left(1-\cos \frac{4\mathrm{\pi }}{5}\right)=\frac{5}{4} \end{gathered}$$

$$\begin{gathered} The complex number w is given by w = -1 + \sqrt{3} i. \\ Show that w2 = 2\overline{w} . \\ Evaluate \left|w\right| and arg w \\ Show that w is a root of z^3 - 8 = 0 . \end{gathered}$$

$$\begin{gathered} The points O, I, Z and P on the Argand Plane represent the complex numbers 0, 1,\mathcal{z} and \mathcal{z} + 1 \\ respectively, where \mathcal{z}= \cos \theta + i \sin \theta is any complex number of modulus 1, with 0 <\theta <\pi . \\ Explain why OIPZ is a rhombus \\ Show that \frac{\mathcal{z}-1}{\mathcal{z}+1} is purely imaginary. \\ Find the modulus of \mathcal{z} + 1 in terms of \theta . \end{gathered}$$

A particle is projected from a point on a straight line with velocity $u m{s}^{-1}$ and moves in such a way that when it has travelled a distance of $\mathcal{x}$ meters it has velocity of $v=\frac{u}{4+u\mathcal{x}} m{s}^{-1}$. Prove that the acceleration of the particle is $-v^3 m{s}^{-2}$.

One of the largest Pyramids in Egypt is approximately 150m high and has a square base with a base area of approximately 50,000$m^2$. The diagram below shows a square based pyramid with a base area A and height h. The thickness of the cross-section at height $\mathcal{y} is △\mathcal{y}$.

i) Show that the area of the cross section at height $\mathcal{y}$ can be represented as: $A\times {\left(\frac{h-\mathcal{y}}{h}\right)}^2$

ii) Find the volume of the pyramid by using slicing technique.

Find, by the method of cylindrical shells, the volume of the solid generated when the region bounded by the curve $\mathcal{y}={\mathcal{x}}^2+1$, the line $\mathcal{x}=2$ and the co-ordinates axes is rotated about the line $\mathcal{x}=3$.

The area bounded by the parabola $\mathcal{y}=e^{2\mathcal{x}}$ and the line $\mathcal{y}$ is rotated about the line $\mathcal{y}$. Use the slice (washer) method to find the volume of the solid formed.

$$\begin{gathered} Show that the normal to the ellipse \frac{{\mathcal{x}}^2}{a^2}+\frac{{\mathcal{y}}^2}{b^2}=1 (a^2>b^2) at the point P({\mathcal{x}}_1,{\mathcal{y}}_1) \\ has equation a^2{\mathcal{y}}_1\mathcal{x}+b^2{\mathcal{x}}_1\mathcal{y}=(a^2-b^2){\mathcal{x}}_1{\mathcal{y}}_1 \\ This normal meets the major axis of the ellipse at G Sis one focus of the ellipse. \\ Show that GS=e.PS where e is the eccentricity of the ellipse. \end{gathered}$$

$$\begin{gathered} A mass of 10 kg, centre B is connected by light rods to sleeves A and C which revolve freely about the \\ vertical axis AC but do not move vertically. \\ Given AC = 2 metres, show that the radius of the circular path of rotation of B is \frac{\sqrt{3}}{2} metres. \\ Find the tensions in the rods AB, BC when the mass makes 90 revolutions per minute about the vertical axis. \end{gathered}$$

$$\begin{gathered} Arailway track has been constructed around a circular curve of radius 500 metres. \\ The dis\tan ce across the track between the rails is 1.5 metres and the outer rail \\ G1 metres above the inner rail. \\ Ifthe train travels on the track at a speed {\mathcal{v}}_0 which eliminates any sideways force on the wheels. \\ Draw a diagram showing all the forces acting on the train. \\ Show that {{\mathcal{v}}_0}^2=500g \tan \theta . where \theta is the angle the track makes \\ with the horizontal. Take g=9.8 m{s}^{-2}. Calculate {\mathcal{v}}_0 \\ Ifthe train travels on the track at a speed \mathcal{v}>{\mathcal{v}}_0. \\ State which rail exerts a lateral force on the wheel at the point of contact. \\ Draw a diagram showing all the forces on the train. \\ Show that the lateral force F exerted by the rail on the wheel is \\ given by 3mg \sin \theta , where m is the mass of the train. Deduce that \\ F is one fifth the weight of the train when \mathcal{v}=2{\mathcal{v}}_0. \end{gathered}$$

$$\begin{gathered} For the conic H : \frac{{\mathcal{x}}^2}{25}+\frac{{\mathcal{y}}^2}{16}=1 \\ Calculate the eccentricity e. \\ Sketch H, showing the \mathcal{x} -intercepts, coordinates of the \\ foci, equations of the directrices and equations of the asymptotes. \\ A point P(5sec\theta , 4\tan \theta ) lies on H. \\ By differentiation, find the gradient of the normal to H at P. \\ Show that the equation of the normal to H at P is \\ \frac{5\mathcal{x}}{sec\theta }+\frac{4\mathcal{y}}{\tan \theta }=41 \end{gathered}$$

$$\begin{gathered} P\left(5p,\frac{5}{p}\right),p>0 and Q\left(5q,\frac{5}{q}\right),q>0 are two distinct points on the hyperbola, H, \mathcal{xy} = 25 . \\ Derive the equation of the chord PQ. \\ You are given the equation of the \tan gent at P is \mathcal{x}+p^2\mathcal{y}=2cp.. (DO NOT PROVE THIS) \\ If the \tan gents at P and Q intersect at R, find the co-ordinates of R. \\ If the secant PQ passes through the point A(15,0) , find the locus of R, stating \\ any restrictions on the locus. \end{gathered}$$

$$\begin{gathered} P(a \cos \theta , b \sin \theta ) is a point on the ellipse \frac{{\mathcal{x}}^2}{a^2}+\frac{{\mathcal{y}}^2}{b^2}=1 \\ i. Show that the equation of the normal to P is: \\ a\mathcal{x} \sin \theta - b\mathcal{y} \cos \theta =\left( a^2 - b^2 \right)\sin \theta \cos \theta \\ ii. G is the point where this normal meets the \mathcal{x} axis. N is the foot of \\ the perpendicular from P to the \mathcal{x} axis, O is the origin and e is the \\ eccentricity. \\ Show that \frac{OG}{ON} = e^2. \\ The \tan gent to the hyperbola \mathcal{xy} = c^{2}at the point T \left( ct, \frac{c }{t}\right) \\ meets the \mathcal{x} and \mathcal{y} axes at F and G respectively, \\ and the normal at T meets the line \mathcal{y} = \mathcal{x} at H. \end{gathered}$$

.

$$\begin{gathered} i. Show that the \tan gent at T is \\ \mathcal{x}+t^2\mathcal{y}=2ct \\ ii. Show that the normal at T is t \\ 3\mathcal{x} - t\mathcal{y} = c \left( t^4 - 1 \right) \\ iii. Prove that FH \perp H \end{gathered}$$

$The following diagram shows the sketch of the function \mathcal{y} = f\left(\mathcal{x}\right).$

.

$$\begin{gathered} On separate diagrams of \frac{1}{3}page each, carefully sketch: \\ \mathcal{y}=f\left(\left|\mathcal{x}\right|\right) \\ \mathcal{y}=\frac{1}{f\left(\mathcal{x}\right)} \\ \mathcal{y}={\cos }^{-1}\left(f\left(\mathcal{x}\right)\right). \\ Given x \in R, and \lceil \mathcal{x}\rceil be a real number that is the smallest integer that is greater than, \\ or equal to \mathcal{x}. \\ i. Evaluate \lceil 2.5\rceil and {(2.5 + \lceil 2.5\rceil )}^2 . ii. Sketch a graph of \mathcal{y}= \lceil \mathcal{x}\rceil + {(\mathcal{x} + \lceil \mathcal{x}\rceil )}^2 . By u\sin g the substitution \mathcal{x}= 2 \tan \theta , evaluate the definite integral \\ \int \frac{d\mathcal{x}}{{\left(4+{\mathcal{x}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}} \end{gathered}$$

$$\begin{gathered} A parabolic segment has height h and width 2a. \\ Use simpson\text{'}s rule with three function values to show that the exact area of this segment is \frac{4}{3}ah. \end{gathered}$$

$In the diagram below, a tent has a circullar base with centre O and radius a, AOB is a diameter of the base. The shaded area PMQR is a typical cross section of the tent perpendicular to AB, and meets AB at point M dis\tan t \mathcal{x} from O. The curve PRQ is a parabola with axis RM and QM=RM$

$$\begin{gathered} \mathrm{The} \mathrm{region} \mathrm{bounded} \mathrm{by} \mathrm{y}\le 4{\mathrm{x}}^2 -{\mathrm{x}}^4 \mathrm{and} 0\le \mathrm{x}\le 2 \mathrm{is} \mathrm{rotated} \mathrm{about} \mathrm{the} \mathrm{y} \mathrm{axis} \mathrm{to} \mathrm{form} \mathrm{a} \mathrm{solid}. \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{volume} \mathrm{of} \mathrm{this} \mathrm{solid} \mathrm{using} \mathrm{the} \mathrm{method} \mathrm{of} \mathrm{cylindrical} \mathrm{shells}? \\ \left(\mathrm{A}\right) \frac{16\mathrm{\pi }}{3}{\mathrm{units}}^3 \\ \left(\mathrm{B}\right) \frac{8\mathrm{\pi }}{3}{\mathrm{units}}^3 \\ \left(\mathrm{C}\right) \frac{20\mathrm{\pi }}{3}{\mathrm{units}}^3 \\ \left(\mathrm{D}\right) \frac{32\mathrm{\pi }}{3}{\mathrm{units}}^3 \end{gathered}$$

$$\begin{gathered} The region is bounded by the lines \mathcal{x}=1, \mathcal{y}=1, \mathcal{y}= -1 and by the curve \mathcal{x}=-{\mathcal{y}}^2. The region is rotated \\ through 360º about the line \mathcal{x}= 2 to form a solid. What is the correct expression for volume of this solid? \end{gathered}$$