OCR C4 (Core Mathematics 4)

Express $$\frac{5x}{(x-4)(x+1)} + \frac{3}{(x-2)(x+1)}$$ as a single fraction in its simplest form. [4]

A curve has the equation $$x^2 + 2xy^2 + y = 4.$$ Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]

Evaluate $$\int_0^{\frac{\pi}{4}} \sin 2x \cos x \, dx.$$ [5]

A curve has parametric equations $$x = \cos 2t, \quad y = \cosec t, \quad 0 < t < \frac{\pi}{2}.$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the value of the parameter \(t\) at \(P\). [2]
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2x + 1.$$ [5]

1. Express \(\frac{2 + 20x}{1 + 2x - 8x^2}\) as a sum of partial fractions. [3]
2. Hence find the series expansion of \(\frac{2 + 20x}{1 + 2x - 8x^2}\), \(|x| < \frac{1}{4}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]

Use the substitution \(x = 2 \tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]

A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.

1. Find the position vector of \(C\). [5]

Given that 1 unit on each coordinate axis represents 200 metres,

1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]

1. Find \(\int \tan^2 x \, dx\). [3]
2. Show that $$\int \tan x \, dx = \ln |\sec x| + c,$$ where \(c\) is an arbitrary constant. [4]

\includegraphics{figure_8} The diagram shows part of the curve with equation \(y = x^{\frac{1}{2}} \tan x\). The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{3}\) is rotated through \(360°\) about the \(x\)-axis.

1. Show that the volume of the solid formed is \(\frac{1}{18}\pi^2(6\sqrt{3} - \pi) - \pi \ln 2\). [5]

An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]

Given that after one week there are 360 insects in the colony,

1. find the value of \(k\) to 3 significant figures. [2]

Given also that after two and three weeks there are 440 and 600 insects respectively,

1. comment on suitability of the modelling assumption. [2]

An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25 \cos 0.5t).$$

1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [3]
2. Compare the suitability of the two models. [2]