OCR MEI C2 (Core Mathematics 2) 2013 January

1 Find \(\int 30 x ^ { \frac { 3 } { 2 } } \mathrm {~d} x\).

2 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.

1. \(3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots\)
2. \(3,7,11,15 , \ldots\)
3. \(3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots\)

3

1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).

4 \begin{figure}[h] \end{figure} Fig. 4 shows sector OAB with sector angle 1.2 radians and arc length 4.2 cm . It also shows chord AB .

1. Find the radius of this sector.
2. Calculate the perpendicular distance of the chord AB from O .
   \(5 \quad \mathrm {~A}\) and B are points on the curve \(y = 4 \sqrt { x }\). Point A has coordinates \(( 9,12 )\) and point B has \(x\)-coordinate 9.5. Find the gradient of the chord AB . The gradient of AB is an approximation to the gradient of the curve at A . State the \(x\)-coordinate of a point C on the curve such that the gradient of AC is a closer approximation.

6 Differentiate \(2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\). Hence find the set of values of \(x\) for which the function \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\) is increasing.

7 Fig. 7 shows a sketch of a village green ABC which is bounded by three straight roads. \(\mathrm { AB } = 92 \mathrm {~m}\), \(\mathrm { BC } = 75 \mathrm {~m}\) and \(\mathrm { AC } = 105 \mathrm {~m}\). Fig. 7 Calculate the area of the village green.

8

1. Sketch the graph of \(y = 3 ^ { x }\).
2. Solve the equation \(3 ^ { 5 x - 1 } = 500000\).

9

1. Show that the equation \(\frac { \tan \theta } { \cos \theta } = 1\) may be rewritten as \(\sin \theta = 1 - \sin ^ { 2 } \theta\).
2. Hence solve the equation \(\frac { \tan \theta } { \cos \theta } = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

10 Fig. 10 shows a sketch of the curve \(y = x ^ { 2 } - 4 x + 3\). The point A on the curve has \(x\)-coordinate 4 . At point B the curve crosses the \(x\)-axis. \begin{figure}[h] \end{figure}

1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at \(\mathrm { C } ( 16,0 )\).
2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis.

11

1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250 .
   (A) Find the values of \(A\) and \(D\).
   (B) Find the sum of the 21st to 50th terms inclusive of this sequence.
2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250 . Use the formula for the sum of a geometric progression to show that \(\frac { r ^ { 4 } - 1 } { r ^ { 2 } - 1 } = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\).

12 The table shows population data for a country. The data may be represented by an exponential model of growth. Using \(t\) as the number of years after 1960, a suitable model is \(p = a \times 10 ^ { k t }\).

1. Derive an equation for \(\log _ { 10 } p\) in terms of \(a , k\) and \(t\).
2. Complete the table and draw the graph of \(\log _ { 10 } p\) against \(t\), drawing a line of best fit by eye.
3. Use your line of best fit to express \(\log _ { 10 } p\) in terms of \(t\) and hence find \(p\) in terms of \(t\).
4. According to the model, what was the population in 1960 ?
5. According to the model, when will the population reach 200 million?