$Find \int {\sin }^3 \mathcal{x}{\cos }^3\mathcal{x} d\mathcal{x}$

$\left(i\right) If I_n={\int }_0^1\frac{d\mathcal{x}}{{(1+{\mathcal{x}}^2)}^n} prove that 2n I_{n+1}=2^{-n}+(2n-1) I_n$

$\left(ii\right) Hence evaluate {\int }_0^1\frac{d\mathcal{x}}{{(1+{\mathcal{x}}^2)}^3}$

$\left(i\right) Use the substitution \mathcal{x}=a-\mathcal{y} where a is a cons\tan t to proave that {\int }_0^{a}f\left(\mathcal{x}\right)d\mathcal{x}={\int }_0^{a}f(a-\mathcal{x})d\mathcal{x}$

$\left(ii\right) Hence show that {\int }_0^{\mathrm{\pi }}\frac{\mathcal{x}\sin \mathcal{x}}{1+{\cos }^2\mathcal{x}}d\mathcal{x}=\frac{{\mathrm{\pi }}^2}{4}$

$$\begin{gathered} Find \int \frac{{\mathcal{x}}^2}{{\left(1+{\mathcal{x}}^3\right)}^2} d\mathcal{x} \\ Find \int \frac{{\mathcal{x}}^2+4}{{\mathcal{x}}^2+1} d\mathcal{x} \\ Use integration by parts to evaluate{\int }_0^1\mathcal{x}{e}^{-3\mathcal{x}} d\mathcal{x} \\ \left(i\right) Find real numbers a b , and c such that \\ \frac{\mathcal{x}}{{\left(\mathcal{x}-1\right)}^2\left(\mathcal{x}-2\right)}\equiv \frac{a}{\mathcal{x}-1}+\frac{b}{{\left(\mathcal{x}-1\right)}^2}+\frac{c}{\mathcal{x}-2} \\ \left(ii\right) Evaluate \int \frac{\mathcal{x}}{{\left(\mathcal{x}-1\right)}^2\left(\mathcal{x}-2\right)} d\mathcal{x} \\ Use the substitution \mathcal{x} = \sin \theta to evaluate{\int }_{\frac{1}{2}}^{\frac{1}{2}}\frac{{\mathcal{x}}^2}{\sqrt{1-{\mathcal{x}}^2}} d\mathcal{x} \end{gathered}$$

$$\begin{gathered} Use integration by parts to evaluate{\int }_0^{\ln 2}\mathcal{x}{e}^{-\mathcal{x}} d\mathcal{x}. \\ Give your answer in simplest form. \end{gathered}$$

$$\begin{gathered} \left(i\right) Express {\mathcal{z}}_1=-1+\sqrt{3\mathcal{i}} in modulus argument form. \\ \left(ii\right) Given {\mathcal{z}}_2=3\left(\cos \left(\frac{\mathrm{\pi }}{6}\right)+\mathcal{i}\sin \left(\frac{\mathrm{\pi }}{6}\right)\right), find the value of {\mathcal{z}}_1{\mathcal{z}}_2 in modulus argument form. \\ \left(iii\right) Hence express {\left({\mathcal{z}}_1{\mathcal{z}}_2\right)}^3 in the form \mathcal{x}+\mathcal{iy}, where \mathcal{x} and \mathcal{y} are real numbers. \end{gathered}$$

$$\begin{gathered} \left(i\right) On and Argand diagram sketch arg(\mathcal{z}-2)=arg\mathcal{z}+\frac{\mathrm{\pi }}{2}. \\ \left(ii\right) Describe the locus. \end{gathered}$$

$$\begin{gathered} If \mathcal{z}=1+\mathcal{i} is a root of the equation {\mathcal{z}}^3+p{\mathcal{z}}^2+qz+6 where \\ p and q are real, 3 find p and q . \end{gathered}$$

$$\begin{gathered} Express {(\sqrt{3} + \mathcal{i})}^8 in the form \mathcal{x} + \mathcal{iy} \\ On an Argand diagram, sketch the region where the inequalities \\ \left|\mathcal{z}\right|\le 3 and \frac{– 2\pi }{3} \le arg (\mathcal{z} + 2) \le \frac{\pi }{6} both hold. \\ Show that \frac{1+ \sin \theta + \mathcal{i}\cos \theta }{1+ \sin \theta - \mathcal{i}\cos \theta }= \sin \theta + \mathcal{i}\cos \theta \end{gathered}$$

$$\begin{gathered} Express \mathcal{z}=\frac{-1+\mathcal{i}}{\sqrt{3}+\mathcal{i}} in modulus-argument form \\ Hence evaluate \frac{\cos 7\pi }{12} in surd form. \\ The Argand diagram below shows the points A and B which \\ represent the complex numbers {\mathcal{z}}_1 and {\mathcal{z}}_2 respectively. \end{gathered}$$

$$\begin{gathered} Given that \Delta BOA is a right-angled \\ isosceles triangle, show that {({\mathcal{z}}_1+{\mathcal{z}}_2)}^2=2{\mathcal{z}}_1{\mathcal{z}}_2 \end{gathered}$$

$$\begin{gathered} A plane of mass M kg on landing, experiences a variable resistive force due to air resis\tan ceof \\ magnitude B{v}^2 newtons, where v is the speed of the plane. That is, M\mathcal{x}=-B{v}^2 . \\ Show that the dis\tan ce \left(D_1\right) travelled in slowing the plane from speed V to speed U \\ under the effect of air resis\tan ce only, is given by: \\ {D}_1=\frac{M}{B}In\left(\frac{V}{U}\right) \\ After the brakes are applied, the plane experiences a cons\tan t resistive force of A Newtons \\ (due to brakes) as well as a variable resistive force, B{v}^2 . That is M\mathcal{x}=-(A+B{v}^2). \\ After the brakes are applied when the plane is travelling at speed U , show that the \\ dis\tan ce D_2 required to come to rest is given by: \\ D_2=\frac{M}{B}In\left(\frac{V}{U}\right) \\ Use the above information to estimate the total stopping dis\tan ce after landing, for a 100 \\ tonne plane if it slows from 90 m/s^2 to 60 m/s^2 under a resistive force of 125v^2 Newtons \\ and is finally brought to rest with the assis\tan ce of a cons\tan t braking force of magnitude \\ 75 000 Newtons. \end{gathered}$$

$$\begin{gathered} \left(i\right) Show that the area enclosed between the parabola {\mathcal{x}}^2=4a\mathcal{y} and its latus rectum is \frac{8a^2}{3} unit{s}^2. \\ A solid figure (as shown below) has the ellipse \frac{{\mathcal{x}}^2}{16}+\frac{{\mathcal{y}}^2}{4}=1 as its base in the \mathcal{xy} plane. \\ Cross-sections perpendicular to the \mathcal{x}-axis are parabolas with latus rectums in the \mathcal{xy} plane. \end{gathered}$$

$$\begin{gathered} \left(ii\right)Show that the area of the cross-section at \mathcal{x}=h is \frac{16-h^2}{6} unit{s}^2. \\ \left(iii\right) Hence, find the volume of this solid. \end{gathered}$$

$$\begin{gathered} The equation of the \tan gent to the rec\tan gular hyperbola \mathcal{xy}=c^2 at P \left(ct,\frac{c}{t}\right) is given by \\ \mathcal{x}+t^2\mathcal{y}=2ct. The \tan gent cuts the \mathcal{x} and \mathcal{y} axis at A and B respectively. \end{gathered}$$

$$\begin{gathered} Which of the following statement is false? \\ \left(A\right) P is the centre of the circle that passes through O, A and B. \\ \left(B\right) The area of △AOB\hspace{0.17em}is 2c^2 square units. \\ \left(C\right) The dis\tan ce AB is \sqrt{4c^2{t}^2+\frac{4c^2}{t^2}} \\ \left(D\right)\hspace{0.17em}AP>BP \end{gathered}$$

$$\begin{gathered} On an Argand Diagram, sketch the region where the inequalities \\ 2\le \left|\begin{array}{c}z\end{array}\right|\le 5 and arg\frac{\mathrm{\pi }}{6}<arg z<\frac{2\mathrm{\pi }}{3} hold simul\mathit{tan}eously . \end{gathered}$$

$$\begin{gathered} Let P(a \cos \theta , b \tan \theta ) be a point on the hyperbola \frac{{\mathcal{x}}^2}{a^2}+\frac{{\mathcal{y}}^2}{b^2}=1 where a>0 \\ and b>0 as shown in the diagram. The foci of the hyperbola are S and S\text{'}, l is the \\ \tan gent to the point P. \end{gathered}$$

$$\begin{gathered} The points R and R\text{'} lie on l so that SR and S\text{'}R\text{'} are prependicular to l. \\ The line l has equation b\mathcal{x} sec \theta -a\mathcal{y} \tan \theta -ab=0 \\ \left(i\right) Show that SR=\frac{ab(e sec \theta -1)}{\sqrt{a^2 {\tan }^2\theta +b^{2}se{c}^2\theta }} \\ \left(ii\right) Show that SR\times S\text{'}R\text{'}=b^2 \end{gathered}$$

$$\begin{gathered} When the polynomial P\left(\mathcal{x}\right) is divided by{\mathcal{x}}^2 + 9, the remainder is 2\mathcal{x} - 5. \\ What is the remainder when P\left(\mathcal{x}\right) is divided by\mathcal{x} - 3i ? \\ \left(A\right) -18 + 15i \\ \left(B\right) -18 - 15i \\ \left(C\right) -5 +6i \\ \left(D\right) -5 - 6i \end{gathered}$$

$$\begin{gathered} Which of the following is the correct expansion of \int {\sin }^3\mathcal{x} d\mathcal{x} \\ \left(A\right) \frac{1}{3}{\cos }^3\mathcal{x}-\cos \mathcal{x}+c \\ \left(B\right) \frac{1}{3}{\cos }^3\mathcal{x}+\cos \mathcal{x}+c ܿ \\ \left(C\right) \frac{1}{3}{\sin }^3\mathcal{x}-\sin \mathcal{x}+cܿ \\ \left(D\right) \frac{1}{3}{\sin }^3\mathcal{x}+\sin \mathcal{x}+c \end{gathered}$$

$$\begin{gathered} Let A , B and C be the angles of triangle ABC . \\ i) Prove that \tan A=\frac{\tan B+ \tan C}{1-\tan B\tan C} \\ ii) Hence prove that \tan A + \tan B + \tan C = \tan A \tan B \tan C \end{gathered}$$

$$\begin{gathered} The diagram above shows the graphs of the functions y = f\left(x\right) and y = g\left(x\right). \\ Which of the following could represent the relationship between f\left(x\right) and g\left(x\right)? \\ \left(A\right) g\left(x\right) = \frac{1}{2} \left|f\right(x\left)\right| \\ \left(B\right) g\left(x\right) =\sqrt{ "f\left(x\right) } \\ \left(C\right) g\left(x\right) = \sqrt{"\left|f\right(x\left)\right|} \\ \left(D\right)(g{\left(x\right))}^2 = f(x) \end{gathered}$$

Angie draws the following diagram shows a shape made by 13 points. She designs the diagram so that it has point symmetry about A

There are ${ }^{13}{C}_3$ ways of picking any 3 points. However some of these points are collinear. If 3 of the points are taken from the set of 5 horizontal or vertical then no triangle can be formed.

How many triangles can she make with these points as vertices?

Taryn is dealt a hand consisting of five cards from an SBHS Deck. 

An SBHS deck has fifty-four cards: a numberless silver card, a numberless golden card, and a standard deck of fifty-two playing cards. 

What is the probability that Taryn gets dealt two pairs?

$$\begin{gathered} What is the gradient of the \tan gent to the curve -8\mathcal{x}+\mathcal{y}+2\mathcal{y}=0 at point (1,2)? \\ A) 2 \\ B) \frac{8}{3} \\ C) -1 \\ D) \frac{4}{5} \end{gathered}$$