OCR MEI C4 (Core Mathematics 4) 2014 June

Express \(\frac{3x}{(2-x)(4+x^2)}\) in partial fractions. [5]

Find the first three terms in the binomial expansion of \((4+x)^{\frac{1}{2}}\). State the set of values of \(x\) for which the expansion is valid. [5]

Fig. 3 shows the curve \(y = x^3 + \sqrt{(\sin x)}\) for \(0 \leqslant x \leqslant \frac{\pi}{4}\). \includegraphics{figure_3}

1. Use the trapezium rule with 4 strips to estimate the area of the region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{4}\), giving your answer to 3 decimal places. [4]
2. Suppose the number of strips in the trapezium rule is increased. Without doing further calculations, state, with a reason, whether the area estimate increases, decreases, or it is not possible to say. [1]

1. Show that \(\cos(\alpha + \beta) = \frac{1 - \tan \alpha \tan \beta}{\sec \alpha \sec \beta}\). [3]
2. Hence show that \(\cos 2\alpha = \frac{1 - \tan^2 \alpha}{1 + \tan^2 \alpha}\). [2]
3. Hence or otherwise solve the equation \(\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \frac{1}{2}\) for \(0° \leqslant \theta \leqslant 180°\). [3]

A curve has parametric equations \(x = e^{2t}, y = te^{2t}\).

1. Find \(\frac{dy}{dx}\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\). [4]
2. Find the cartesian equation of the curve in the form \(y = ax^b \ln x\), where \(a\) and \(b\) are constants to be determined. [3]

Fig. 6 shows the region enclosed by the curve \(y = (1 + 2x^2)^{\frac{1}{2}}\) and the line \(y = 2\). \includegraphics{figure_6} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\). [6]

Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}

1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
2. 1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
   2. Hence find the equation of this plane. [2]
3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]

The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).

1. Find the volume of the tetrahedron ABCD. [2]

Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel. \includegraphics{figure_8.1} The height of the water surface above the hole \(t\) seconds after opening the hole is \(h\) metres, where $$\frac{dh}{dt} = -A\sqrt{h}$$ and where \(A\) is a positive constant. Initially the water surface is 1 metre above the hole.

1. Verify that the solution to this differential equation is $$h = \left(1 - \frac{1}{2}At\right)^2.$$ [3]

The water stops leaking when \(h = 0\). This occurs after 20 seconds.

1. Find the value of \(A\), and the time when the height of the water surface above the hole is 0.5 m. [4]

Fig. 8.2 shows a similar situation with a different barrel; \(h\) is in metres. \includegraphics{figure_8.2} For this barrel, $$\frac{dh}{dt} = -B\frac{\sqrt{h}}{(1+h)^2},$$ where \(B\) is a positive constant. When \(t = 0\), \(h = 1\).

1. Solve this differential equation, and hence show that $$h^{\frac{1}{2}}(30 + 20h + 6h^2) = 56 - 15Bt.$$ [7]
2. Given that \(h = 0\) when \(t = 20\), find \(B\). Find also the time when the height of the water surface above the hole is 0.5 m. [4]