Edexcel C4 (Core Mathematics 4)

The region bounded by the curve \(y = x^2 - 2x\) and the \(x\)-axis is rotated through \(2\pi\) radians about the \(x\)-axis. Find the volume of the solid formed, giving your answer in terms of \(\pi\). [6]

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

A curve has the equation $$2 \sin 2x - \tan y = 0.$$

1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [5]
2. Show that the tangent to the curve at the point \(\left(\frac{\pi}{6}, \frac{\pi}{3}\right)\) has the equation $$y = \frac{1}{2}x + \frac{\pi}{4}.$$ [3]

\includegraphics{figure_1} Figure 1 shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]

The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.

1. Show that the area of triangle \(OAB\) is \(a^2\). [6]

The gradient at any point \((x, y)\) on a curve is proportional to \(\sqrt{y}\). Given that the curve passes through the point with coordinates \((0, 4)\),

1. show that the equation of the curve can be written in the form $$2\sqrt{y} = kx + 4,$$ where \(k\) is a positive constant. [5]

Given also that the curve passes through the point with coordinates \((2, 9)\),

1. find the equation of the curve in the form \(y = \text{f}(x)\). [4]

\includegraphics{figure_2} Figure 2 shows a vertical cross-section of a vase. The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60°\). When the depth of water in the vase is \(h\) cm, the volume of water in the vase is \(V\) cm\(^3\).

1. Show that \(V = \frac{1}{9}\pi h^3\). [3]

The vase is initially empty and water is poured in at a constant rate of 120 cm\(^3\) s\(^{-1}\).

1. Find, to 2 decimal places, the rate at which \(h\) is increasing
   1. when \(h = 6\),
   2. after water has been poured in for 8 seconds. [7]

Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix}\) respectively.

1. Find a vector equation for the line \(l_1\) which passes through \(A\) and \(B\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -7 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.$$

1. Show that lines \(l_1\) and \(l_2\) do not intersect. [5]
2. Find the position vector of the point \(C\) on \(l_2\) such that \(\angle ABC = 90°\). [6]

$$\text{f}(x) = \frac{x(3x-7)}{(1-x)(1-3x)}, \quad |x| < \frac{1}{3}.$$

1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = A + \frac{B}{1-x} + \frac{C}{1-3x}.$$ [4]
2. Evaluate $$\int_0^{\frac{1}{4}} \text{f}(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational. [5]
3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]