OCR MEI C2 (Core Mathematics 2) 2012 January

1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).

2 Find \(\int \left( x ^ { 5 } + 10 x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).

3 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.

4 Given that \(a > 0\), state the values of

1. \(\log _ { a } 1\),
2. \(\log _ { a } \left( a ^ { 3 } \right) ^ { 6 }\),
3. \(\log _ { a } \sqrt { a }\).

5 Figs. 5.1 and 5.2 show the graph of \(y = \sin x\) for values of \(x\) from \(0 ^ { \circ }\) to \(360 ^ { \circ }\) and two transformations of this graph. State the equation of each graph after it has been transformed.

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-2_506_926_1324_571} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
   \end{figure}
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-2_513_936_2003_561} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}

6 Use logarithms to solve the equation \(235 \times 5 ^ { x } = 987\), giving your answer correct to 3 decimal places.

7 Given that \(y = a + x ^ { b }\), find \(\log _ { 10 } x\) in terms of \(y\), \(a\) and \(b\).

8 Show that the equation \(4 \cos ^ { 2 } \theta = 1 + \sin \theta\) can be expressed as $$4 \sin ^ { 2 } \theta + \sin \theta - 3 = 0 .$$ Hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

9 A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5.
Find the value of \(b\) and find also the sum of the first 15 terms of the progression.

10 In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030 . Find the first term and the common difference.

11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.

12 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).

1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).

13 \begin{figure}[h] \end{figure} In a concert hall, seats are arranged along arcs of concentric circles, as shown in Fig. 13.1. As shown in Fig. 13.2, the stage is part of a sector ABO of radius 11 m . Fig. 13.2 also gives the dimensions of the stage.

1. Show that angle \(\mathrm { COD } = 1.55\) radians, correct to 2 decimal places. Hence find the area of the stage.
2. There are four rows of seats, with their backs along arcs, with centre O, of radii \(7.4 \mathrm {~m} , 8.6 \mathrm {~m} , 9.8 \mathrm {~m}\) and 11 m . Each seat takes up 80 cm of the arc.
   (A) Calculate how many seats can fit in the front row.
   (B) Calculate how many more seats can fit in the back row than the front row.