Edexcel C4 (Core Mathematics 4)

Use integration by parts to find the exact value of \(\int_1^3 x^2 \ln x \, dx\). [6]

Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V\) m\(^3\). The rate at which the fluid flows, in m\(^3\) min\(^{-1}\), is proportional to the square root of \(V\).

1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation $$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$ [3]
2. Show that the general solution of the differential equation may be written as $$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$ [4] Given that at time \(t = 0\) the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
3. find the time, \(T\) minutes, which it takes for the tank to empty. [3]
4. Find the depth of water in the tank at time \(0.5T\) minutes. [2]

1. Use the identity for \(\cos(A + B)\) to prove that \(\cos 2A = 2\cos^2 A - 1\). [2]
2. Use the substitution \(x = 2\sqrt{2} \sin \theta\) to prove that $$\int_2^{\sqrt{6}} \sqrt{(8 - x^2)} \, dx = \frac{1}{3}(\pi + 3\sqrt{3} - 6).$$ [7]

A curve is given by the parametric equations $$x = \sec \theta, \quad y = \ln(1 + \cos 2\theta), \quad 0 \leq \theta < \frac{\pi}{2}.$$

1. Find an equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{3}\). [5]

\includegraphics{figure_2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{4}{x - 3}\), \(x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [1]

The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through 360° about the \(y\)-axis to form a solid shape \(S\).

1. Find the volume of \(S\), giving your answer in the form \(\pi(a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]

The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,

1. show that the volume of the tower is approximately 15500 m\(^3\). [2]

Relative to a fixed origin \(O\), the point \(A\) has position vector \(3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), the point \(B\) has position vector \(5\mathbf{i} + \mathbf{j} + \mathbf{k}\), and the point \(C\) has position vector \(7\mathbf{i} - \mathbf{j}\).

1. Find the cosine of angle \(ABC\). [4]
2. Find the exact value of the area of triangle \(ABC\). [3]

The point \(D\) has position vector \(7\mathbf{i} + 3\mathbf{k}\).

1. Show that \(AC\) is perpendicular to \(CD\). [2]
2. Find the ratio \(AD:DB\). [2]

\includegraphics{figure_2} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8\sqrt{\sin \frac{\pi x}{10}}\), in the interval \(0 \leq x \leq 10\). The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = -2\), \(x = 12\) and \(y = 10\). The units on both axes are metres.

1. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.

   \(x\)	0	2	4	6	8	10
   \(y\)	0	6.13				0

   [2]

The area of the cross-section of the tunnel is given by \(\int_0^{10} y \, dx\).

1. Estimate this area, using the trapezium rule with all the values from your table. [4]
2. Deduce an estimate of the cross-sectional area of the concrete surround. [1]
3. State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value. [2]

$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$

1. Express f(x) as a sum of partial fractions. [4]
2. Hence find \(\int \text{f}(x) \, dx\). [5]
3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]