Edexcel C4 (Core Mathematics 4)

A curve has the equation $$2x^2 + xy - y^2 + 18 = 0.$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [8]

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]

1. Show that \((1 + \frac{1}{24})^{-\frac{1}{2}} = k\sqrt{6}\), where \(k\) is rational. [2]
2. Expand \((1 + \frac{1}{4}x)^{-\frac{1}{2}}\), \(|x| < 2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
3. Use your answer to part \((b)\) with \(x = \frac{1}{6}\) to find an approximate value for \(\sqrt{6}\), giving your answer to 5 decimal places. [3]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

A curve has parametric equations $$x = \frac{t}{2-t}, \quad y = \frac{1}{1+t}, \quad -1 < t < 2.$$

1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\left(\frac{2-t}{1+t}\right)^2\). [4]
2. Find an equation for the normal to the curve at the point where \(t = 1\). [3]
3. Show that the cartesian equation of the curve can be written in the form $$y = \frac{1+x}{1+3x}.$$ [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_1} Figure 1 shows part of the curve with equation \(y = x^2 \tan x\). The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{3}\) is rotated through \(2\pi\) radians about the \(x\)-axis.

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

\includegraphics{figure_2} Figure 2 shows a hemispherical bowl of radius 5 cm. The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h\) cm and the volume of water in the bowl is \(V\) cm³, where $$V = \frac{1}{3}\pi h^2(15 - h).$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).

1. Show that $$\frac{dh}{dt} = -\frac{kh(15-h)}{3(10-h)},$$ where \(k\) is a positive constant. [5]
2. Express \(\frac{3(10-h)}{h(15-h)}\) in partial fractions. [3]

Given that when \(t = 0\), \(h = 5\),

1. show that $$h^2(15-h) = 250e^{-kt}.$$ [6]

Given also that when \(t = 2\), \(h = 4\),

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