AQA C1 (Core Mathematics 1)

7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

1. Find:
   1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
3. 1. Verify that \(V\) has a stationary value when \(t = 1\).
   2. Determine whether this is a maximum or minimum value.

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\).
\includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).