Processing math: 100%

Friday, 5 February 2016

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]