1. Solve the inequality
$$ \ 2 \ – \ 5x \ > \ 2|x \ – \ 3|. $$ $$\tag*{[4]} $$
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities
$$ |z| \ \geq 2 \ {\small\textrm{ and} } \ |z \ – \ 1 + i| \ \leq \ 1. $$ $$\tag*{[4]} $$
3. The parametric equations of a curve are
\begin{multline}
\shoveleft \hspace{2em} x \ = \ 3 \ – \ \cos 2\theta, \\
\shoveleft \hspace{2em} y \ = \ 2\theta + \sin 2\theta, \\
\shoveleft {\small\textrm{ for } \ 0 < \theta < \frac{1}{2}\pi }\\[5pt]
\shoveleft {\small\textrm{ Show that } \ \frac{\text{d}y}{\text{d}x} \ = \ \cot \theta. } \\
\end{multline}
$$\tag*{[5]}$$ Check here for my solution
4. Solve the equation
$$ \hspace{1em}\small \log_{10} (2x+1) \ = \ 2 \log_{10} (x+1) \ – \ 1. $$
Give your answers correct to 3 decimal places. $$ \tag*{[6]} $$
5. (a) By sketching a suitable pair of graphs, show that the equation $ \ \textrm{cosec} {\small \ x \ = \ 1 + e^{-\frac{1}{2} x}} \ $ has exactly two roots in the interval $ \ \small 0 < x < \pi. $ $$ \tag*{[2]} $$ (b) The sequence of values given by the iterative formula $$ x_{n+1} \ = \ \pi \ - \ \sin^{-1} \Big( \frac{1}{e^{- \frac{1}{2} x_{n} } + 1 } \Big), $$ with initial value $ \small \ x_{1} \ = \ 2 $, converges to one of these roots. $$ $$ Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $$ \tag*{[3]} $$ Check here for my solution
6. (a) Express $ \sqrt{6} \ \cos \theta + 3 \sin \theta \ $ in the form $ R \cos (\theta \ – \ \alpha) $, where $ R > 0 $ and $ 0^{\circ} < \alpha < 90^{\circ}.$ State the exact value of \(R\) and give \( \alpha \) correct to 2 decimal places. $$ \tag*{[3]} $$ (b) Hence solve the equation \( \sqrt{6} \ \cos \frac{1}{3}x + 3 \sin \frac{1}{3}x \ = \ 2.5 \), for $ \ 0^{\circ} < \alpha < 360^{\circ}. $ $$ \tag*{[4]} $$ Check here for my solution
7. (a) Verify that $ \ -1 + \sqrt{5} i \ $ is a root of the equation $ \ 2x^{3}+x^{2}+6{x} \ – \ 18 \ = \ 0. $ $$ \tag*{[3]} $$
(b) Find the other roots of this equation. $$ \tag*{[4]} $$
8. The coordinates ($x$, $y$) of a general point of a curve satisfy the differential equation
$$ x \frac{\textrm{d}y}{\textrm{d}x} \ = \ (1 \ – \ 2x^{2} )y, \\ $$
for $ {\small \ x > 0 }$. It is given that $ {\small y \ = \ 1 } $ when $ {\small x \ = \ 1. } $
Solve the differential equation, obtaining an expression for $y$ in terms of $x$. $$\tag*{[6]}$$
9. Let $ f(x) \ = \ \frac{8+5x+12x^2}{(1-x) (2+3x)^{2} } . \\ $
(a) Express f(x) in partial fractions. $$\tag*{[5]}$$
(b) Hence obtain the expansion of $ {\small f(x) }$ in ascending powers of $ x $, up to and including the term in $x^{2}$. $$\tag*{[5]}$$
10. The diagram shows the curve $ \ y \ = \ (2 \ – \ x) e^{-\frac{1}{2} x}, $ and its minimum point $M$.
(a) Find the exact coordinates of $M$. $$\tag*{[5]}$$
(b) Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of $\textrm{e}$. $$\tag*{[5]}$$
11. Two lines have equations $ \ \mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda (a\mathbf{i}+2\mathbf{j} \ – \ \mathbf{k}) \ $ and $ \ \mathbf{r} = 2\mathbf{i} + \mathbf{j} \ – \ \mathbf{k} + \mu (2\mathbf{i} \ – \ \mathbf{j}+\mathbf{k}), $ where $a$ is a constant.
(a) Given that the two lines intersect, find the value of $a$ and the position vector of the point of intersection. $$\tag*{[5]}$$
(b) Given instead that the acute angle between the directions of the two lines is $ \ \cos^{-1} \big( \frac{1}{6} \big), $ find the two possible values of $a$.
$$\tag*{[6]}$$