Edexcel C4 (Core Mathematics 4)

1. Expand \((1 + 4x)^5\) in ascending powers of \(x\) up to and including the term in \(x^5\), simplifying each coefficient. [4]
2. State the set of values of \(x\) for which your expansion is valid. [1]

Use the substitution \(u = 1 + \sin x\) to find the value of $$\int_0^{\frac{\pi}{4}} \cos x (1 + \sin x)^3 \, dx.$$ [6]

1. Express \(\frac{x+11}{(x+4)(x-3)}\) as a sum of partial fractions. [3]
2. Evaluate $$\int_0^2 \frac{x+11}{(x+4)(x-3)} \, dx,$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction. [5]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 2\sin x + \cosec x\), \(0 < x < \pi\). The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\) is rotated through \(360°\) about the \(x\)-axis. Show that the volume of the solid formed is \(\frac{1}{2}\pi(4\pi + 3\sqrt{3})\). [8]

A curve has the equation $$x^2 - 3xy - y^2 = 12.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
2. Find an equation for the tangent to the curve at the point \((2, -2)\). [3]

Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} 1 \\ 5 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle AOB\), [4]
2. the area of triangle \(OAB\), [4]
3. the shortest distance from \(A\) to the line \(OB\). [2]

A curve has parametric equations $$x = t(t - 1), \quad y = \frac{4t}{1-t}, \quad t \neq 1.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [4]

The point \(P\) on the curve has parameter \(t = -1\).

1. Show that the tangent to the curve at \(P\) has the equation $$x + 3y + 4 = 0.$$ [3]

The tangent to the curve at \(P\) meets the curve again at the point \(Q\).

1. Find the coordinates of \(Q\). [7]

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 model. [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. [4]
2. Compare the suitability of the two models. [3]