OCR MEI C2 (Core Mathematics 2) 2010 June

You are given that $$u_1 = 1,$$ $$u_{n+1} = \frac{u_n}{1 + u_n}.$$ Find the values of \(u_2\), \(u_3\) and \(u_4\). Give your answers as fractions. [2]

1. Evaluate \(\sum_{r=2}^{5} \frac{1}{r-1}\). [2]
2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum_{r=2}^{a} f(r)\) where \(f(r)\) and \(a\) are to be determined. [2]

1. Differentiate \(x^3 - 6x^2 - 15x + 50\). [2]
2. Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x^3 - 6x^2 - 15x + 50\). [3]

In this question, \(f(x) = x^2 - 5x\). Fig. 4 shows a sketch of the graph of \(y = f(x)\). \includegraphics{figure_4} On separate diagrams, sketch the curves \(y = f(2x)\) and \(y = 3f(x)\), labelling the coordinates of their intersections with the axes and their turning points. [4]

Find \(\int_{2}^{5} \left(1 - \frac{6}{x^3}\right) dx\). [4]

The gradient of a curve is \(6x^2 + 12x^{\frac{1}{2}}\). The curve passes through the point \((4, 10)\). Find the equation of the curve. [5]

Express \(\log_a x^3 + \log_a \sqrt{x}\) in the form \(k \log_a x\). [2]

Showing your method clearly, solve the equation \(4 \sin^2 \theta = 3 + \cos^2 \theta\), for values of \(\theta\) between \(0°\) and \(360°\). [5]

The points \((2, 6)\) and \((3, 18)\) lie on the curve \(y = ax^n\). Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places. [5]

1. Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
2. Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
3. 1. Expand \((2 + h)^4\). [3]
   2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
   3. Show how your result in part (iii) \((B)\) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]

1. \includegraphics{figure_11_1} A boat travels from P to Q and then to R. As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045°\). R is 9.2 km from P on a bearing of \(113°\), so that angle QPR is \(68°\). Calculate the distance and bearing of R from Q. [5]
2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \includegraphics{figure_11_2} BC is an arc of a circle with centre A and radius 80 cm. Angle CAB = \(\frac{2\pi}{3}\) radians. EC is an arc of a circle with centre D and radius \(r\) cm. Angle CDE is a right angle.
   1. Calculate the area of sector ABC. [2]
   2. Show that \(r = 40\sqrt{3}\) and calculate the area of triangle CDA. [3]
   3. Hence calculate the area of cross-section of the rudder. [3]

\includegraphics{figure_12} A branching plant has stems, nodes, leaves and buds. • There are 7 leaves at each node. • From each node, 2 new stems grow. • At the end of each final stem, there is a bud. Fig. 12 shows one such plant with 3 stages of nodes. It has 15 stems, 7 nodes, 49 leaves and 8 buds.

1. One of these plants has 10 stages of nodes.
   1. How many buds does it have? [2]
   2. How many stems does it have? [2]
2. 1. Show that the number of leaves on one of these plants with \(n\) stages of nodes is $$7(2^n - 1).$$ [2]
   2. One of these plants has \(n\) stages of nodes and more than 200000 leaves. Show that \(n\) satisfies the inequality \(n > \frac{\log_{10} 200007 - \log_{10} 7}{\log_{10} 2}\). Hence find the least possible value of \(n\). [4]