$$\begin{gathered} Express \frac{3\mathcal{x}+1}{(\mathcal{x}+1)({\mathcal{x}}^2+1)} in the form \frac{a}{\mathcal{x}+!}+\frac{b\mathcal{x}+c}{{\mathcal{x}}^2+1}. \\ Hence find \int \frac{3\mathcal{x}+1}{(\mathcal{x}+1)({\mathcal{x}}^2+1)} \\ Use the substitution \mathcal{x}=2\sin \theta , or otherwise , to evaluate {\int }_1^{\sqrt{3}}\frac{{\mathcal{x}}^2}{\sqrt{4-{\mathcal{x}}^2}}d\mathcal{x}. \end{gathered}$$

$$\begin{gathered} Sketch on separate diagrams, the graphs of: \\ \left(i\right) \mathcal{y}={(\mathcal{x}-1)}^2(\mathcal{x}+2) \\ \left(ii\right) {\mathcal{y}}^2={(\mathcal{x}-1)}^2(\mathcal{x}+2) \\ \left(iii\right) \mathcal{y}=\frac{1}{{(\mathcal{x}-1)}^2(\mathcal{x}+2)} \\ \left(iv\right) \mathcal{y}={\log }_e{(\mathcal{x}+1)}^2 \end{gathered}$$

$$\begin{gathered} The diagram above is a sketch of the function \mathcal{y}=f\left(\mathcal{x}\right) \\ On separate diagrams sketch: \\ \mathcal{y}={\left(f\left(\mathcal{x}\right)\right)}^2 \\ \mathcal{y}=\sqrt{f\left(\mathcal{x}\right)} \\ \mathcal{y}=\ln \left[f\left(\mathcal{x}\right)\right] \\ {\mathcal{y}}^2=f\left(\mathcal{x}\right) \\ If f\text{'}\left(\mathcal{x}\right)=\frac{2-\mathcal{x}}{{\mathcal{x}}^2} and f\left(1\right)=0 , find f\text{'}\text{'}\left(\mathcal{x}\right) and f\left(\mathcal{x}\right) \\ Explain why the graph of f\left(\mathcal{x}\right) has only one turning point and find the value \\ of the function at that point, stating whether it is a maximum or a minimum value. \\ Show that f\left(4\right) and f \left(5\right) have opposite signs and draw a sketch of f\left(\mathcal{x}\right). \end{gathered}$$

$$\begin{gathered} The following is a sketch of a function \mathcal{y}=f\left(\mathcal{x}\right) \\ Draw separate one-third page sketches of the following: \\ (clearly showing impor\tan t features) \\ \mathcal{y}=-f\left(\mathcal{x}\right) \\ \mathcal{y}=\sqrt{f\left(\mathcal{x}\right)} \\ \mathcal{y}=f(1-\mathcal{x}) \\ \mathcal{y}={\cos }^{-1}f\left(\mathcal{x}\right) \\ \mathcal{y}=\frac{1}{1-f\left(\mathcal{x}\right)} \end{gathered}$$

$$\begin{gathered} Write down the equation of P\left(x\right) if it is a monic polynomial of degree 3 \\ with integer coefficients, a cons\tan t term of 12 and one root equal to \sqrt{3 }. \\ Leave your answer in factored form. \end{gathered}$$

$$\begin{gathered} Let \mathcal{z} = -3\mathfrak{i} and w =2+\mathfrak{i} . Express the following in the form \\ \mathcal{x}+\mathfrak{i} \mathcal{y} , where \mathcal{x} and \mathcal{y} are real numbers: \\ \left(i\right) \frac{\mathcal{z}}{w} \\ \left(ii\right) \frac{1}{- 2\mathfrak{i} \mathcal{z}} \\ Let \mathcal{z}=\frac{1}{2}+\frac{\sqrt{3}}{2}\mathfrak{i} \\ \left(i\right) Express \mathcal{z} in modulus-argument form. \\ \left(ii\right) Show that {\mathcal{z}}^6 =1. \\ \left(iii\right) Hence, or otherwise, graph all the roots of {\mathcal{z}}^6-1= 0 on an Argand diagram. \\ The complex numbers \alpha , \beta ,\gamma and \delta are represented on an Argand diagram \\ by the points A, B, C and D respectively. \\ \left(i\right) Describe the point that represents \frac{1}{2}( \alpha +\gamma ) . \\ \left(ii\right) Deduce that if \alpha +\gamma =\beta + \delta then ABCD is a paralle\log ram. \\ Let \mathcal{z}=\mathcal{x}+\mathfrak{i}\mathcal{y} . Find the points of intersection of the curves given by: \\ \left|\mathcal{z}-\mathfrak{i}\right|=1 and Re\left(\mathcal{z}\right)=Im\left(\mathcal{z}\right) . \end{gathered}$$

$$\begin{gathered} Consider w =-\sqrt{ 3} + i \\ Express w in modulus-argument form \\ Hence or otherwise show that w^7 + 64w = 0 \end{gathered}$$

$$\begin{gathered} Sketch the region in the complex plane where the inequalities \\ 1 \le |z - i| \le 2 and Im \left(z\right) \ge 0 hold simul\tan eously. \\ Clearly mark in all x and y intercepts. \end{gathered}$$

$$\begin{gathered} A particle of mass 1 kg moves in a straight line before coming to rest. The resul\tan t force acting on the particle directly opposes its motion and has \\ magnitude m(1+v) where v is its velocity. Initially the particle is at the origin and travelling with velocity Q where Q >0 \end{gathered}$$

$$\begin{gathered} \left(i\right) Show that v is related to the displacement \mathcal{x} by the formula \\ \left(ii\right) Find an expression for v in terms of t. \\ \left(iii\right) Find an expression for \mathcal{x} in terms of t. \\ \left(iv\right) Show that Q=\mathcal{x}+v+t \\ \left(v\right) Find the dis\tan ce travelled and the time taken by the particle in coming to rest. \end{gathered}$$

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

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$$\begin{gathered} Draw separate one-third page sketches of the graphs of the following: \\ \left(i\right) \mathcal{y}=f\left|\left(\mathcal{x}\right)\right| \\ \left(ii\right) \mathcal{y}=f\left(2-\mathcal{x}\right) \\ \left(iii\right) \mathcal{y}={\log }_{e}f\left(\mathcal{x}\right) \\ Sketch the graph of \mathcal{y}=\frac{1}{\mathcal{x}\left(\mathcal{x}-2\right)} , without the use of calculus. \\ \left(i\right) Find the value of g for which p\left(\mathcal{x}\right)=9{\mathcal{x}}^4-25{\mathcal{x}}^2+10g\mathcal{x}-g^2 is divisible by both \mathcal{x} -1 and \mathcal{x} + 2 . \\ \left(ii\right) With this value of g , solve the equation 9{\mathcal{x}}^4-25{\mathcal{x}}^2+10g\mathcal{x}-g^2 =0. \end{gathered}$$

$Diagram A shows the complex number z represented in the Argand plane.$

$$\begin{gathered} Diagram B shows: \\ \left(A\right)\overline{\mathcal{z}} \\ \left(B\right) 2\mathcal{iz} \\ \left(C\right) -2\mathcal{z} \\ \left(D\right) {\mathcal{z}}^2 \end{gathered}$$

$$\begin{gathered} \mathcal{z} and w are two complex numbers. Which of the the following statements is always TRUE? \\ \left(A\right) \left|\mathcal{z}\right|-\left|w\right|\ge |\mathcal{z}+w| \\ \left(B\right) \left|\mathcal{z}\right|+\left|w\right|\ge |\mathcal{z}-w| \\ \left(C\right) \left|\mathcal{z}\right|+\left|w\right|\le |\mathcal{z}+w| \\ \left(D\right) |\mathcal{z}+w|+\left|\mathcal{z}\right|\ge \left|\mathcal{z}\right| \end{gathered}$$

$$\begin{gathered} Divide the polynomial P\left(\mathcal{x}\right)={\mathcal{x}}^4+3{\mathcal{x}}^3-7{\mathcal{x}}^2+ 11\mathcal{x}-1 by{\mathcal{x}}^2 + 2 and \\ write your result in the form P\left(\mathcal{x}\right)=({\mathcal{x}}^2+2)Q\left(\mathcal{x}\right)+c\mathcal{x}+d . \\ Hence determine the values of a and b for which the polynomial \\ ({\mathcal{x}}^4+ 3{\mathcal{x}}^3-7{\mathcal{x}}^2+ 2\mathcal{x})+a\mathcal{x}+b is exactly divisible by {\mathcal{x}}^2+ 2 . \end{gathered}$$

$$\begin{gathered} The equation |z - 3 |÷ |z + 3 |= 10 corresponds to an ellipse in the Argand diagram. \\ Prove that the equation of the ellipse is \frac{{\mathcal{x}}^2}{25} \frac{{\mathcal{y}}^2}{16}=1 \\ Sketch the ellipse showing all impor\tan t features. \end{gathered}$$

$$\begin{gathered} In the diagram below PQ and RM are parallel chords in a circle. The \tan gent at Q \\ meets RM produced at S and SK is another \tan gent to the circle. PK cuts RM at L . \\ Copy or trace this diagram into your answer booklet. \\ Let \angle SQK = x and prove \angle SQK =\angle SLK 2 \\ Explain why LKSQ is a cyclic quadrilateral. \\ Prove PL= QL \end{gathered}$$

$$\begin{gathered} It is given that \sum _{r-0}^{\text{'}\text{'}}{(-1)}^n\frac{{}^{n}C_r}{\mathcal{x}+n}=\frac{n!}{\mathcal{x}(\mathcal{x}+1)(\mathcal{x}+2).....(\mathcal{x}+n)} . (DO NOT PROVE) \\ Hence prove1-\frac{1}{2}{}^{n}C_1+\frac{1}{3}{}^{n}C_2-........\frac{{(-1)}^n {}^{n}C_n}{n+1}=\frac{1}{n+1} \end{gathered}$$

$$\begin{gathered} The curve \mathcal{y} = 8\mathcal{x} - {\mathcal{x}}^2 and the line \mathcal{y} = 12 is sketched below. . \\ Find the coordinates of the points of intersection A and B 1 \\ The shaded area is rotated around the \mathcal{y} axis. Use the method of \\ cylindrical shells to find the exact volume formed. \\ (You may leave your answer unsimplified in fractional form) \end{gathered}$$

$$\begin{gathered} Six people are divided into three groups of two. The number of different ways this can be done is \\ \left(A\right) 90 \\ \left(B\right) 45 \\ \left(C\right) 30 \\ \left(D\right) 15 \end{gathered}$$

$$\begin{gathered} The expression i^3+ i^6 + i^9 + i^{12} + i^{15} is equal to: \\ \left(A\right) -i \\ \left(B\right) i \\ \left(C\right) 1 \\ (D -1 \end{gathered}$$

$Suppose \mathcal{x}= r (\cos \theta + i\sin \theta ) is a complex number. The complex number {\mathcal{x}}^2 will be:$

$$\begin{gathered} \left(A\right) r^2(\cos {\theta }^2+ i \sin {\theta }^2) \\ \left(B\right) 2r(\cos 2\theta + i \sin 2\theta ) \\ \left(C\right) r^2(\cos 2\theta + i \sin 2\theta ) \\ \left(D\right) 2r(\cos {\theta }^2+ i \sin {\theta }^2) \end{gathered}$$