Further Maths Progress Check due Wed 10.2.16

Total: 51 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Solve \[ \sin (2x-\frac{\pi}{2}) = -\frac{1}{2}\] for $-\pi \leq x \leq \pi$

   [5 marks]

2. By expressing $\cos x + \sin x$ in the form $R\cos(x- \alpha)$ with $0 \leq \alpha \leq \frac{\pi}{2}$, find the maximum value of \[2 - \cos x - \sin x\] State the smallest positive value of x for which this occurs.

   [5 marks]

3. Prove that \[ \sin 4A + \sin 2A \equiv 2\sin 3A \cos A \]

   [4 marks]

4. Solve \[ \cos \theta + 1 = 2 \sec \theta\] for $-\pi \leq x \leq \pi$

   [4 marks]

5. A is acute and B is obtuse. \[ \text{cosec} A = \frac{5}{3} \\ \sec B = -\frac{13}{5} \] Find $\tan (A+B)$ without a calculator

   [4 marks]

6. \[ f(x) = x^3 - ax^2 + x + b\] $(x-2)$ is a factor of $f(x)$ and the remainder is 5 when $f(x)$ is divided by $(2x+1)$. Find $f(3)$.

   [5 marks]

7. \[ \frac{x^4-x-1}{x^2+2} \equiv ax^2 +bx+c + \frac{dx+e}{x^2+2} \]

   [4 marks]

8. Simply as far as possible \[ 1+ \frac{2x}{x^2-2x-8} - \frac{6}{x^2-16} \]

   [4 marks]

9. Express in partial fractions:
   1. \[ \frac{2x}{(x^2-4)(x+1)} \]
   2. \[ \frac{2-x}{(x^2-4)(x+2)} \]
   3. \[ \frac{3x+2}{(x^2+4)(x+1)} \]
   4. \[ \frac{x^3}{(x^2-1)(x+1)} \]

   [16 marks]