Edexcel C2 (Core Mathematics 2)

1. During one day, a biological culure is allowed to grow under controlled conditions.

At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times ( 1.06 ) ^ { t } .$$ Using this model,

1. find the number of bacteria present at 11 a.m.,
2. find, to the nearest minute, the time when the initial number of bacteria will have doubled.

2. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.

3. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 1\).

1. Use the trapezium rule with four intervals of equal width to find an estimate for the area of the shaded region.
2. State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.

4. The first three terms in the expansion in descending powers of \(x\) of $$\left( x + \frac { k } { x ^ { 2 } } \right) ^ { 15 } ,$$ where \(k\) is a constant, are $$x ^ { 15 } + 30 x ^ { 12 } + A x ^ { 9 } .$$

1. Find the values of \(k\) and \(A\).
2. Find the value of the term independent of \(x\) in the expansion.

5. \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).

1. Show that \(a = 8\).
2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.

6. $$f ( x ) = \cos 2 x , \quad 0 \leq x \leq \pi .$$

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).

7. The points \(P\) and \(Q\) have coordinates \(( - 2,6 )\) and \(( 4 , - 1 )\) respectively. Given that \(P Q\) is a diameter of circle \(C\),

1. find the coordinates of the centre of \(C\),
2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0 .$$ The point \(R\) has coordinates (2, 7).
3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.

8. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.

1. Find the common ratio of the series.
2. Show that the first term of the series is \(\log _ { 3 } 2\).
3. Find, to 3 significant figures, the sum of the first six terms of the series.

9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).

1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Fully factorise \(\mathrm { f } ( x )\).
3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).