1. (a) Sketch the graph of $ y \ = \ |x \ – \ 2|. $ $$\tag*{[1]} $$

(b) Solve the inequality $ |x \ – \ 2| \ < \ 3x \ - \ 4. $ $$\tag*{[3]} $$ Check here for my solution

2. Solve the equation $$ \ln 3 + \ln (2x+5) \ = \ 2 \ln (x+2). $$

Give your answer in a simplified exact form.

$$\tag*{[4]} $$

3. (a) By sketching a suitable pair of graphs, show that the equation $ {\small \ \sec x \ = \ 2 \ – \ \frac{1}{2} x} \ $ has exactly one root in the interval ${\small \ 0 \ \leq x \ < \frac{1}{2} \pi }.$ $$\tag*{[2]} $$ (b) Verify by calculation that this root lies between 0.8 and 1. $$\tag*{[2]} $$ (c) Use the iterative formula $ \ x_{n+1} \ = \ \cos^{-1} \big( \frac{2}{4 \ - \ x_{n} }\big) \ $ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $$\tag*{[3]} $$ Check here for my solution

4. Find $$ \ \displaystyle \int_{\frac{1}{6} π}^{\frac{1}{3} π} x \sec^{2} x \ \textrm{d}x. $$

Give your answer in a simplified exact form.

$$ \tag*{[7]} $$

5. (a) Show that $$ \frac{\cos 3x}{\sin x} + \frac{\sin 3x}{\cos x} \ = \ 2 \cot 2x $$

$$ \tag*{[4]} $$

(b) Hence solve the equation $$ \frac{\cos 3x}{\sin x} + \frac{\sin 3x}{\cos x} \ = \ 4, $$

for $ {\small \ 0 < x < \pi }.$ $$ \tag*{[3]} $$ Check here for my solution

6. The variables $x$ and $y$ satisfy the differential equation

$$ \frac{\textrm{d}y}{\textrm{d}x} \ = \ \frac{1+4y^2}{e^{x}}. $$

It is given that $ \ {\small y \ = \ 0 \ }$ when $ \ {\small x \ = \ 1 \ }$.

(a) Solve the differential equation, obtaining an expression for $y$ in terms of $x$.

$$ \tag*{[7]} $$

(b) State what happens to the value of $y$ as $x$ tends to infinity.

$$ \tag*{[1]} $$

7. The equation of a curve is $$ x^{3} + 3 x{y}^{2} \ – \ y^{3} \ = \ 5. $$

(a) Show that $ \ \frac{\textrm{d}y}{\textrm{d}x} \ = \ \frac{x^2 + y^2}{y^2 \ – \ 2xy}. $

$$ \tag*{[4]} $$

(b) Find the coordinates of the points on the curve where the tangent is parallel to the $y$-axis.

$$ \tag*{[5]} $$

8. In the diagram, $ \ {\small OABCDEFG} \ $ is a cuboid in which $ \ {\small \overrightarrow{OA} = 2 } \ $ units, $ \ {\small \overrightarrow{OC} = 3 } \ $ units and $ \ {\small \overrightarrow{OD} = 2 } \ $ units. Unit vectors $ \ {\small \mathbf{i}} $, $ \ {\small \mathbf{j}} \ $ and $ \ {\small \mathbf{k}} \ $ are parallel to $ \ {\small \overrightarrow{OA}, \overrightarrow{OC} }\ $ and $ \ {\small \overrightarrow{OD} } \ $ respectively. The point $M$ on $AB \ $ is such that $ \ MB \ = \ 2 AM$. The midpoint of $FG$ is $N$.

(a) Express the vectors $ \ {\small \overrightarrow{OM}} \ $ and $ \ {\small \overrightarrow{MN}} \ $ in terms of $ \ {\small \mathbf{i}} $, $ \ {\small \mathbf{j}} \ $ and $ \ {\small \mathbf{k}} \ $.

$$ \tag*{[3]} $$

(b) Find a vector equation for the line through $M$ and $N$.

$$ \tag*{[2]} $$

(c) Find the position vector of $P$, the foot of the perpendicular from $D$ to the line through $M$ and $N$.

$$ \tag*{[4]} $$

9. Let $ f(x) \ = \ \frac{2 + 11x \ – \ 10x^2}{(1+2x) (1 \ – \ 2x) (2+x) } . \\ $

(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. (a) The complex numbers $v$ and $w$ satisfy the equations

$$ v+iw \ = \ 5 \ {\small \textrm{and}} \ (1+2i)v \ – \ w \ = \ 3i. $$

Solve the equations for $v$ and $w$, giving your answers in the form $x +iy$, where $x$ and $y$ are real.

$$\tag*{[6]}$$

(b) (i) On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $|z \ – \ 2 \ – \ 3i| \ = \ 1.$

$$\tag*{[2]}$$

(ii) Calculate the least value of arg $z$ for points on this locus.

$$\tag*{[2]}$$