OCR MEI C2 (Core Mathematics 2) 2013 June

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when

1. \(y = 2 x ^ { - 5 }\),
2. \(y = \sqrt [ 3 ] { x }\).

3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 18 } { x ^ { 3 } } + 2\). The curve passes through the point \(( 3,6 )\). Find the equation of the curve.

4

1. Starting with an equilateral triangle, prove that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-3_1032_1113_264_466} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} Fig. 5 shows the graph of \(y = 2 ^ { x }\).

1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
   \(6 S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
3. Another geometric progression has first term \(2 a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\).
4. A third geometric progression has first term \(a\) and common ratio \(r ^ { 2 }\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\).

7 Fig. 7 shows a curve and the coordinates of some points on it. \begin{figure}[h] \end{figure} Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve and the positive \(x\) - and \(y\)-axes.

8 Fig. 8 shows the graph of \(y = \mathrm { g } ( x )\). \begin{figure}[h] \end{figure} Draw the graph of

1. \(y = \mathrm { g } ( 2 x )\),
2. \(y = 3 \mathrm {~g} ( x )\). Section B (36 marks)

9 Fig. 9 shows a sketch of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24\) and the line \(y = 6 x + 24\). \begin{figure}[h] \end{figure}

1. Differentiate \(y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places.
2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = - 4\). Find algebraically the \(x\)-coordinate of the other point of intersection.
3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6 x + 24\) for \(- 4 \leqslant x \leqslant 0\), shown shaded on Fig. 9.

10 Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \begin{figure}[h] \end{figure}

1. (A) Calculate AC and angle ACB . Hence calculate AD .
   (B) Calculate the area of the garden.
2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M , of side FG , and sector angle 1.1 radians, as shown. \(\mathrm { FG } = 1.8 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-6_565_986_1528_520} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure} Calculate the area of one of these fence panels.

11 A hot drink when first made has a temperature which is \(65 ^ { \circ } \mathrm { C }\) higher than room temperature. The temperature difference, \(d ^ { \circ } \mathrm { C }\), between the drink and its surroundings decreases by \(1.7 \%\) each minute.

1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures.
2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer.
3. Show that when \(d < 3 , n\) must satisfy the inequality $$n > \frac { \log _ { 10 } 3 - \log _ { 10 } 65 } { \log _ { 10 } 0.983 }$$ Hence find the least integer value of \(n\) for which \(d < 3\).
4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10 ^ { - k t }\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made.