Edexcel C4 (Core Mathematics 4)

The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$

1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
3. Explain why this model would not be appropriate for large values of \(t\). [1]

A curve has the equation $$3x^2 + xy - 2y^2 + 25 = 0.$$ Find an equation for the normal to the curve at the point with coordinates \((1, 4)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]

1. Use the substitution \(u = 2 - x^2\) to find $$\int \frac{x}{2 - x^2} \, dx.$$ [4]
2. Evaluate $$\int_0^{\frac{1}{4}} \sin 3x \cos x \, dx.$$ [6]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x\sqrt{\ln x}\), \(x \geq 1\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region. [4]

The shaded region is rotated through \(360°\) about the \(x\)-axis.

1. Find the exact volume of the solid formed. [7]

$$f(x) = \frac{5 - 8x}{(1 + 2x)(1 - x)^2}.$$

1. Express \(f(x)\) in partial fractions. [5]
2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying each coefficient. [6]
3. State the set of values of \(x\) for which your expansion is valid. [1]

\includegraphics{figure_2} Figure 2 shows the curve with parametric equations $$x = t + \sin t, \quad y = \sin t, \quad 0 \leq t \leq \pi.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis. [3]
3. Show that the region bounded by the curve and the \(x\)-axis has area 2. [6]

The line \(l_1\) passes through the points \(A\) and \(B\) with position vectors \((\mathbf{3i} + \mathbf{6j} - \mathbf{8k})\) and \((\mathbf{8j} - \mathbf{6k})\) respectively, relative to a fixed origin.

1. Find a vector equation for \(l_1\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = (-\mathbf{2i} + \mathbf{10j} + \mathbf{6k}) + \mu(\mathbf{7i} - \mathbf{4j} + \mathbf{6k}),$$ where \(\mu\) is a scalar parameter.

1. Show that lines \(l_1\) and \(l_2\) intersect. [4]
2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [2]

The point \(C\) lies on \(l_2\) and is such that \(AC\) is perpendicular to \(AB\).

1. Find the position vector of \(C\). [6]