Question
Let \(\mathrm{f}(x)=\ln (\cos x)\). Using the standard series, find the Maclaurin series for \(\mathrm{f}(x)\), up to and including the term in \(x^4\). Hence, find the Maclaurin series of \(\tan x\) up to and including the term in \(x^3\).[6]
Question
(a) Sketch on the same axes, the graphs of \(y=1-x-\frac{1}{x-1}\) and \(y=|a x-a|\), where \(-1<a<1\). State clearly the equation of any asymptotes and coordinates of any turning points and axial intercepts.[5]
(b) Suppose that \(a\) can be any real number, state the largest possible set of solution for the inequality \(1-x-\frac{1}{x-1}>|a x-a|\), and the corresponding range of values of \(a\).[2]
Question
The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors. The angle \(\theta\) between \(\mathbf{u}\) and \(\mathbf{v}\) is an acute angle.
(i) Show that \((\mathbf{u} \cdot \mathbf{v}+|\mathbf{u} \times \mathbf{v}|)^2=1+\sin 2 \theta\).[3]
(ii) The vector \(\mathbf{w}\) is such that \(\mathbf{u}+\mathbf{w}=k \mathbf{v}\), where \(k\) is a constant in terms of \(\theta\). By considering \((\mathbf{u}+\mathbf{w}) \cdot \mathbf{v}\) or otherwise, find the value of \(k\) if \(\mathbf{w}\) is also a unit vector.[3]
(iii) Deduce the range of possible values of \(k\).[1]
Question
(i) State a pair of transformations that will transform the graph of \(y=\mathrm{f}(x)\) onto the graph of \(y=\mathrm{f}(2 x)-1\).
It is given that
$
\mathrm{f}(x)=\left\{\begin{array}{cl}
x^2 & , x \leq 2 \\
2-\frac{1}{x-2} & , x>2 .
\end{array}\right.
$
(ii) Sketch on separate diagrams, the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}(2 x)-1\).
(iii) It is proposed that the following graph is the graph of \(y=\mathrm{f}^{\prime}(x)\).
Give two reasons to explain why the proposed graph is not suitable to be the graph of \(y=\mathrm{f}^{\prime}(x)\).
Question
(a) The curve \(C\) has parametric equations
$
x=\cos t, y=\sec t+\sin t,
$
where \(0<t<\frac{\pi}{2}\).
Find \(\int y \mathrm{~d} x\) in terms of \(x\).
(b) Find the exact value of \(\int_{-\sqrt{2}}^1\left|x^3+1\right| \mathrm{d} x\).
Question
An astroid is a curve with parametric equations
$
x=a \cos ^3 t, y=a \sin ^3 t .
$
(i) Sketch an astroid with \(a=2\) and \(0 \leq t \leq \frac{\pi}{2}\), indicating clearly the coordinates of all axial intercepts.
For the rest of the question, take \(a=1\).
(ii) Find the equation of the tangent to the curve at the point \(P\) where \(t=p\), simplifying your answer.
(iii) The tangent to the curve at \(P\) intersects the \(x\) – and \(y\)-axes at the points \(Q\) and \(R\) respectively. Find the cartesian equation for the locus of the midpoint of \(Q R\).
(i) Given that \(\mathrm{f}(r)=\ln (3 r+1)\), show that \(\mathrm{f}(r)-\mathrm{f}(r-1)=\ln \left(\frac{3 r+1}{3 r-2}\right)\).
(ii) Hence evaluate \(\sum_{r=1}^n \ln \sqrt{\frac{3 r+1}{3 r-2}}\).
(iii) Show that \(\sum_{r=4}^{n-5} \ln \sqrt{\frac{3 r+16}{3 r+13}}=\sum_{r=9}^n \ln \sqrt{\frac{3 r+1}{3 r-2}}\). Hence find \(\sum_{r=4}^{n-5} \ln \sqrt{\frac{3 r+16}{3 r+13}}\) in terms of \(n\).
Question
(i) Given that \(\mathrm{f}(r)=\ln (3 r+1)\), show that \(\mathrm{f}(r)-\mathrm{f}(r-1)=\ln \left(\frac{3 r+1}{3 r-2}\right)\).
(ii) Hence evaluate \(\sum_{r=1}^n \ln \sqrt{\frac{3 r+1}{3 r-2}}\).
(iii) Show that \(\sum_{r=4}^{n-5} \ln \sqrt{\frac{3 r+16}{3 r+13}}=\sum_{r=9}^n \ln \sqrt{\frac{3 r+1}{3 r-2}}\). Hence find \(\sum_{r=4}^{n-5} \ln \sqrt{\frac{3 r+16}{3 r+13}}\) in terms of \(n\).
Question
The curves \(C_1\) and \(C_2\) are given by the equations \(4 x^2+y^2=16\) and \(y=x^2-x-2\) respectively. The region \(R\) is bounded by \(C_1\) and \(C_2\), and \(x \geq 0\).
(a) (i) Using the substitution \(x=2 \sin u\), find \(\int \sqrt{16-4 x^2} \mathrm{~d} x\).
(ii) Hence find the exact area of \(R\).
(b) The region \(S\) is bounded by \(C_2\), the lines \(x=-1\) and \(x=1\), and the \(x\)-axis. Find the volume generated when \(S\) is rotated \(2 \pi\) radians about the \(x\)-axis.
Question
(a) Solve the simultaneous equations
$
\begin{gathered}
z-2 w^*=\mathrm{i}, \\
\mathrm{i} z-w=\mathrm{i} .
\end{gathered}
$
(b) \(u\) and \(v\) are complex numbers such that \(u=-1-\sqrt{3} \mathbf{i},|v|=\sqrt{2}\) and \(\arg (v)=\frac{7 \pi}{12}\).
(i) Find the modulus and argument of \(u\).
[2]
(ii) Express \(v u^*\) in the form \(r \mathrm{e}^{\mathrm{i} \theta}\), where \(-\pi<\theta \leq \pi\).
(iii) Find the least positive integer \(n\) such that \(\left(v u^*\right)^n\) is purely imaginary.[2]
Question
Sales agent \(A\) started work on \(1^{\text {st }}\) June 2020 . He plans to acquire 2 clients in his first month of work. For subsequent months, the number of clients he plans to acquire per month is 3 more than that in the previous month.
(i) Find the number of clients agent \(A\) would acquire in June 2021.[2]
(ii) Find the number of months needed for agent \(A\) to acquire at least 500 clients in total.[3]
Sales agent \(B\) has a total 500 clients on \(1^{\text {st }}\) June 2020 . Unlike agent \(A\), he acquires 3 new clients every month. However, due to strong competition from other sales agents, agent \(B\) loses \(1 \%\) of his clientele at the end of each month when the clients choose not to continue with his service from the following month.
(iii) Show that the clientele size of agent \(B\) at the end of \(n^{\text {th }}\) month is given by
$
297+203(0.99)^n .
$
(iv) The clientele size of agent \(B\) will eventually stabilise. State the value of this clientele size.[1]
(v) Agent \(B\) wishes to keep a clientele size of at least 350 . Find the maximum percentage loss of clients he can have at the end of each month in order to meet this target.[3]