10. (a) On the axes below sketch the graphs of (i) $y = x(4 - x)$ (ii) $y = x^2(7 - x)$ showing clearly the coordinates of the points where the curves cross the coordinate axes - Edexcel - A-Level Maths Pure - Question 11 - 2010 - Paper 1

To sketch the graph of the equation $y = x^2(7 - x)$:

1. Identify points where the curve crosses the axes:

   • Set $y = 0$:
     • $x^2(7 - x) = 0$ gives $x = 0$ and $x = 7$. So, the points are $(0,0)$ and $(7,0)$.
   • Set $x = 0$:
     • $y = 0^2(7 - 0) = 0$. So, the point is $(0,0)$, already mentioned.

2. Determine the turning points:

   • The vertex can be found with the same method. Here, $b = -7$ and $a = 1$ leads to: x = rac{7}{2} = 3.5
   • Substitute $x = 3.5$ in the equation: $y = (3.5)^2(7 - 3.5) = 12.25(3.5) = 42.875$, so the vertex is approximately $(3.5, 42.875)$.

3. Sketch the curve:

   • The curve starts at $(0,0)$, increases to its maximum, then decreases back to $(7,0)$.

To find the x-coordinates of intersection:

1. Set the equations equal:
   $x(4 - x) = x^2(7 - x)$

2. Rearranging leads to:
   $x(4 - x) - x^2(7 - x) = 0$

3. Factor out $x$:
   $x(4 - x - x(7 - x)) = 0$
   $x(4 - 7x + x^2) = 0$.

4. Which simplifies to:
   $x(4 - 8x + x^2) = 0$ or $x(x^2 - 8x + 4) = 0$. Hence, the solutions are confirmed.

To find the coordinates of point A:

1. Substituting x back:

   • We need to find the positive solutions for the earlier derived equations and simplify the coordinates in the required format.

2. Solving the quadratic $x^2 - 8x + 4 = 0$: x = rac{8 imes 1 ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{-}rac{ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{-} ext{ } ext{ } ext{ } ext{ } {16 - 4}}}{2 ext{ }} = $4 + rac{ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{-}rac{ ext{ }}{4}}{2} ext{ would be considered positive after confirming lies on both curves} ext{ } ext{ } ext{ } ext{ } ext{ }\

   • This can be expressed in the form $ext{ } (p + q/3) ext{ } ext{ and } (r + s/3)$.