OCR MEI C4 (Core Mathematics 4) 2013 June

1. Express \(\frac{x}{(1 + x)(1 - 2x)}\) in partial fractions. [3]
2. Hence use binomial expansions to show that \(\frac{x}{(1 + x)(1 - 2x)} = ax + bx^2 + ...\), where \(a\) and \(b\) are constants to be determined. State the set of values of \(x\) for which the expansion is valid. [5]

The diagram is a copy of Fig. 4. R is a place with latitude \(45°\) north and longitude \(60°\) west. Show the position of R on the diagram. M is the sub-solar point. It is on the Greenwich meridian and the declination of the sun is \(+20°\). Show the position of M on the diagram. [2] \includegraphics{figure_4}

Show that the equation \(\cos ec x + 5 \cot x = 3 \sin x\) may be rearranged as $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ Hence solve the equation for \(0° \leq x \leq 360°\), giving your answers to 1 decimal place. [7]

Use Fig. 8 to estimate the difference in the length of daylight between places with latitudes of \(30°\) south and \(60°\) south on the day for which the graph applies. [3]

Using appropriate right-angled triangles, show that \(\tan 45° = 1\) and \(\tan 30° = \frac{1}{\sqrt{3}}\). Hence show that \(\tan 75° = 2 + \sqrt{3}\). [7]

The graph is a copy of Fig. 6. The article says that it shows the terminator in the cases where the sun has declination \(10°\) north, \(1°\) north, \(5°\) south and \(15°\) south. Identify which curve (A, B, C or D) relates to which declination. [2] \includegraphics{figure_6}

\(10°\) north:
\(1°\) north:
\(5°\) south:
\(15°\) south:

1. Find a vector equation of the line \(l\) joining the points \((0, 1, 3)\) and \((-2, 2, 5)\). [2]
2. Find the point of intersection of the line \(l\) with the plane \(x + 3y + 2z = 4\). [3]
3. Find the acute angle between the line \(l\) and the normal to the plane. [3]

In lines 94 and 95 the article says "Fig. 8 shows you that at latitude \(60°\) north the terminator passes approximately through time \(+9\) hours and \(-9\) hours so that there are about 18 hours of daylight." Use Equation (4) to check the accuracy of the figure of 18 hours. [4]

The points A, B and C have coordinates \(A(3, 2, -1)\), \(B(-1, 1, 2)\) and \(C(10, 5, -5)\), relative to the origin O. Show that \(\overrightarrow{OC}\) can be written in the form \(\lambda\overrightarrow{OA} + \mu\overrightarrow{OB}\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points O, A, B and C from the fact that \(\overrightarrow{OC}\) can be expressed as a combination of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\)? [6]

1. Use Equation (3) to calculate the declination of the sun on February 2nd. [3]
2. The town of Boston, in Lincolnshire, has latitude \(53°\) north and longitude \(0°\). Calculate the time of sunset in Boston on February 2nd. Give your answer in hours and minutes using the 24-hour clock. [4]

The motion of a particle is modelled by the differential equation $$v \frac{dv}{dt} + 4x = 0,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity. Initially \(x = 1\) and \(v = 4\).

1. Solve the differential equation to show that \(v^2 = 20 - 4x^2\). [4]

Now consider motion for which \(x = \cos 2t + 2 \sin 2t\), where \(x\) is the displacement from a fixed point at time \(t\).

1. Verify that, when \(t = 0\), \(x = 1\). Use the fact that \(v = \frac{dx}{dt}\) to verify that when \(t = 0\), \(v = 4\). [4]
2. Express \(x\) in the form \(R \cos(2t - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v^2 = 20 - 4x^2\). [7]
3. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value. [3]

Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \includegraphics{figure_7}

1. Find the lengths OA, OB and AC. [5]
2. Find \(\frac{dy}{dx}\) in terms of \(u\). Hence find the angle \(\theta\). [6]
3. Show that the cartesian equation of the curve is \(y = e^{x/5} + e^{-x/5}\). [2]

An object is formed by rotating the region OACB through \(360°\) about Ox.

1. Find the volume of the object. [5]