U6FM Hwk

Total: 27 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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1. Express $\cos 6\theta$ in terms of $\cos^n \theta$

   [3 marks]

2. Express $\cos^6 \theta$ in terms of $\cos (n\theta)$

   [4 marks]

3. Solve, giving your answers in the form $re^{i\theta}$ with $ -\pi < \theta \leq \pi$ , \[ z^6=1-\sqrt3 i \] Represent your solutions on an Argand diagram

   [4 marks]

4. Represent on an Argand diagram the locus of $z$ s.t.
   1. $\arg{(z-3-2i)} = -\frac{5\pi}{6}$

      [3 marks]

   2. $|z+1|=|z-2i|$ also give the cartesian equation

      [3 marks]

   3. $|z+1|=|4z-8i|$ also give the cartesian equation

      [4 marks]

   4. Give the cartesian equation and sketch the locus \[ \arg (\frac{z+3}{z-3}) = 3\pi /4 \]

      [6 marks]

L6FM Hwk 3 Due Mon 26.9.16

Total: 26 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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1. Write in the from $a(x+b)^2 + c$
   i.e. complete the square: \[2x^2-5x-3\]

   [3 marks]

2. Simplify \[ \frac{2}{x^2-3x-10} - \frac{3}{x^2-25} \]

   [4 marks]

3. Simplify as far as possible: \[ \frac{15}{\sqrt {3}} - \frac{11}{2\sqrt{7} + 3\sqrt{3}} \]

   [4 marks]

4. Solve $-3-4x-5x^2>0$ showing how you arrived at your answer.

   [3 marks]

5. \[ f(x)=\frac{2\sqrt x-x^2}{3x\sqrt x} \]
   1. Find the equation of the tangent at $x=4$, giving your answer in the form $ax+by+c=0$ where a,b,c are integers

      [5 marks]

   2. Find the equation of the normal at $x=4$, giving your answer in the form $ax+by+c=0$ where a,b,c are integers

      [4 marks]

   3. Find $f''(4)$

      [3 marks]

L6FM Homework 2 Due Mon 19.9.16

Total: 20 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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1. Write in the from $a(x+b)^2 + c$
   i.e. complete the square: \[3x^2-7x-20\]

   [3 marks]

2. Simplify \[ \frac{3}{x^2-5x +4} - \frac{5}{x^2-16} \]

   [4 marks]

3. Simplify as far as possible: \[ \frac{15}{\sqrt {5}} - \frac{46}{2\sqrt{7} - \sqrt{5}} \]

   [4 marks]

4. Solve $6x^2+x-12>0$

   [3 marks]

5. Solve $5+x+x^2>0$

   [3 marks]

6. Solve \[ x^2 = 4+5xy \\ x=5y +1 \]

   [3 marks]

L6 FM Homework

Total: 10 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Factorise fully $6x^2-7x-20$

   [2 marks]

2. Simplify as far as possible: \[ \frac{8}{4^{5x-1}}=\sqrt{32^{3+x}} \]

   [4 marks]

3. Simplify as far as possible: \[ \frac{26}{\sqrt {13}} - \frac{8}{\sqrt{13} - \sqrt{11}} \]

   [4 marks]

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]