6.6 FM Progress answers 5.10.15

Total: 25 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Factorise fully $6x^2-7x-20 = (3x+4)(2x-5)$

   [2 marks]

2. Complete the square: \[ y=5-2x-3x^2 = -3(x+\frac{1}{3})^2 + \frac{16}{3} \]

   

   Has max at (-1/3 , 16/3)
   y intercept is 5
   x intercepts are -5/3 and 1

   [4 marks]

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

   

   
   y intercept is 75
   cross at x=3 with touch x=-5/2
   negative cubic shape

   [4 marks]

4. \[ y^2=x^2+x \\ y=2x+k \\ 4x^2+4kx+k^2 = x^2+x \\ 3x^2+x(4k-1)+k^2=0 \\ \therefore (4k-1)^2-12k^2<0 \\4k^2-8k+1<0 \\ \frac{2-\sqrt 3}{2} < k < \frac{2+\sqrt 3}{2} \]Find the values of k such that the hyperbola and the line do not meet.

   [4 marks]

5. Find the centre and radius of the circle that passes through A(9,3), B(13,-5) and C(-5,-11) \[ (x-3)^2 + (y+5)^2 = 100 \]

   [5 marks]

6. Solve: \[ \frac{3}{27^{5-x}}=\sqrt{81^{5-2x}} \\ \\ 3^{1-3(5-x)}=3^{\frac{4(5-2x)}{2}} \\ \\ 1-15+3x=10-4x \\ x= \frac{24}{7}\]

   [3 marks]

7. Simplify as far as possible: \[ \frac{15}{\sqrt 5} - \frac{6}{\sqrt 7 - \sqrt 5} = -3\sqrt 7 \]

   [3 marks]