6.6 FM Progress Homework due 14.10.15

Total: 43 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. The third term of an AP is 24 whilst the 7th is -12. Find the sum of the first 20 terms.

   [3 marks]

2. Find \[ \sum_{r=5}^{30} \frac{5-2r}{3} \]

   [3 marks]

3. Find n s.t. \[ \sum_{r=7}^{n} \frac{4r+1}{3} = 600\]

   [4 marks]

4. Find $u_n$ if $S_n = 2n^2+3n+1$

   [3 marks]

5. Complete the square: $y=5-4x-2x^2$ and use your answer to sketch y, indicating clearly any points where the curve crosses the axes

   [4 marks]

6. Sketch $y=(3-x)^2(x+5)^3$ indicating any intersections with the axes

   [4 marks]

7. \[ 4y^2+x^2=36 \\ y+kx=3 \\ \]Find the value(s) of k such that the line is a tangent to the ellipse.

   [4 marks]

8. Find the values of k s.t. $2x^2-x+8=kx$ has no solutions.

   [3 marks]

9. Find the centre and radius of the circle that passes through A(7,11), B(-10,-6) and C(15,-1)

   [5 marks]

10. Simplify as far as possible: \[ \frac{4}{\sqrt 3} - \frac{6}{\sqrt 5 - \sqrt 3} \]

    [3 marks]

11. Describe the series of transformations which transform $y= \frac{1}{x}$ to the curve \[y=5-\frac{2}{x-2}\]

    [3 marks]

12. $y=3x-4$ is a tangent to a circle at the point (2,2). Given the centre has coords (k,1) find k and hence the equation of the circle.

    [4 marks]