OCR MEI C2 (Core Mathematics 2)

1 The common ratio of a geometric progression is - 0.5 . The sum of its first three terms is 15 . Find the first term.
Find also the sum to infinity.

2 The diagram shows the graph of \(y = \mathrm { f } ( x )\). The graph passes through the point with coordinates \(( 0,2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-2_524_1350_775_346} On separate diagrams sketch the graphs of the following functions, indicating clearly the point of intersection with the \(y\) axis.

1. \(\quad y = - \mathrm { f } ( x )\)
2. \(y = f ( 3 x )\)

3 Given that \(A\) is the obtuse angle such that \(\sin A = \frac { 1 } { 5 }\), find the exact value of \(\cos A\).

4 You are given that \(y = x ^ { 3 } - 12 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the turning points of the curve.

5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)

1. Write down the first three terms of the sequence.
2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).

6 Find the solution to this equation, correct to 3 significant figures. $$\left( 2 ^ { x } \right) \left( 2 ^ { x + 1 } \right) = 10 .$$

7 The gradient of a curve \(y = \mathrm { f } ( x )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 10 x + 6\). The curve passes through the point \(( 2,3 )\) Find the equation of the curve.

8 In the triangle ABC shown, \(\mathrm { AB } = 8 \mathrm {~cm}\). \(\mathrm { AC } = 12 \mathrm {~cm}\) and angle \(\mathrm { ABC } = 82 ^ { \circ }\). Find \(\theta\) correct to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-3_382_540_1492_718}

9 Fig. 9 shows
\(P \quad\) The line \(y = x\)
\(Q\) The curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\)
\(R \quad\) The curve \(\quad y = \sqrt { x }\). \begin{figure}[h] \end{figure}

1. Write down the area of the triangle formed by the line \(y = x\), the line \(x = 1\) and the \(x\)-axis.
2. Show that the area of the region formed by the curve \(y = \sqrt { x }\), the line \(x = 1\) and the \(x\)-axis is \(\frac { 2 } { 3 }\). An estimate is required of the Area, \(A\), of the region formed by the curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\), the line \(x = 1\) and the \(x\)-axis.
3. Use results to parts (i) and (ii) to complete the statement $$\ldots \ldots \ldots \ldots . . < A < \ldots \ldots \ldots \ldots \ldots . .$$
4. Use the Trapezium Rule with 4 strips to find an estimate for \(A\).
5. Draw a sketch of Fig. 9. Use it to illustrate the area found as the trapezium rule estimate for \(A\).
   Explain how your diagram shows that the trapezium rule estimate must be:
   consistent with the answer to part (iv);
   an under-estimate for A .

10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).

1. Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
2. Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
3. Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
   State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\).
4. Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).

11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).

1. For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
2. For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
   Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\).
3. Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
4. When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).