Edexcel C4 (Core Mathematics 4)

1. The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation

$$\frac { \mathrm { d } n } { \mathrm {~d} t } = \mathrm { e } ^ { 0.5 t } - 5 , \quad t \geq 0$$

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

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

3. (a) Use the substitution \(u = 2 - x ^ { 2 }\) to find $$\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 3 x \cos x d x$$

1. continued

\begin{figure}[h] \end{figure} 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 \(x = 3\).
(a) Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region. The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(b) Find the exact volume of the solid formed.

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

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

6. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = t + \sin t , \quad y = \sin t , \quad 0 \leq t \leq \pi .$$

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

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).
   7. continued
   7. continued