OCR MEI C2 (Core Mathematics 2) 2014 June

Find \(\int 7x^3 \, dx\). [3]

1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]

The points P\((2, 3.6)\) and Q\((2.2, 2.4)\) lie on the curve \(y = f(x)\). Use P and Q to estimate the gradient of the curve at the point where \(x = 2\). [2]

The point R\((6, -3)\) is on the curve \(y = f(x)\).

1. Find the coordinates of the image of R when the curve is transformed to \(y = \frac{1}{2}f(x)\). [2]
2. Find the coordinates of the image of R when the curve is transformed to \(y = f(3x)\). [2]

\includegraphics{figure_5} Fig. 5 shows triangle ABC, where angle ABC = \(72°\), AB = \(5.9\) cm and BC = \(8.5\) cm. Calculate the length of AC. [3]

\includegraphics{figure_6} A circle with centre O has radius \(12.4\) cm. A segment of the circle is shown shaded in Fig. 6. The segment is bounded by the arc AB and the chord AB, where the angle AOB is \(2.1\) radians. Calculate the area of the segment. [4]

The second term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]

Simplify \(\frac{\sqrt{1 - \cos^2 \theta}}{\tan \theta}\), where \(\theta\) is an acute angle. [3]

Solve the equation \(\tan 2\theta = 3\) for \(0° < \theta < 360°\). [3]

Use logarithms to solve the equation \(3^{x+1} = 5^{2x}\). Give your answer correct to 3 decimal places. [4]

\includegraphics{figure_11} Fig. 11 shows a sketch of the curve with equation \(y = x - \frac{4}{x^2}\).

1. Find \(\frac{dy}{dx}\) and show that \(\frac{d^2y}{dx^2} = -\frac{24}{x^4}\). [3]
2. Hence find the coordinates of the stationary point on the curve. Verify that the stationary point is a maximum. [5]
3. Find the equation of the normal to the curve when \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [5]

Oskar is designing a building. Fig. 12 shows his design for the end wall and the curve of the roof. The units for \(x\) and \(y\) are metres. \includegraphics{figure_12}

1. Use the trapezium rule with 5 strips to estimate the area of the end wall of the building. [4]
2. Oskar now uses the equation \(y = -0.001x^3 - 0.025x^2 + 0.6x + 9\), for \(0 \leq x \leq 15\), to model the curve of the roof.
   1. Calculate the difference between the height of the roof when \(x = 12\) given by this model and the data shown in Fig. 12. [2]
   2. Use integration to find the area of the end wall given by this model. [4]

The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960. An exponential model may be used for these data, assuming that the relationship between \(h\) and \(t\) is of the form \(h = a \times 10^{bt}\), where \(a\) and \(b\) are constants to be determined.

1. Show that this relationship may be expressed in the form \(\log_{10} h = mt + c\), stating the values of \(m\) and \(c\) in terms of \(a\) and \(b\). [2]
2. Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of \(\log_{10} h\) against \(t\), drawing by eye a line of best fit. [4]
3. Use your graph to find \(h\) in terms of \(t\) for this model. [4]
4. Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model. [2]
5. Give one reason why this model will not be suitable in the long term. [1]