Edexcel C4 (Core Mathematics 4)

Use the substitution \(u = 4 + 3x^2\) to find the exact value of $$\int_0^2 \frac{2x}{(4 + 3x^2)^2} \, dx .$$ [6]

A curve has equation $$x^3 - 2xy - 4x + y^3 - 51 = 0.$$ Find an equation of the normal to the curve at the point \((4, 3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]

$$f(x) = \frac{1 + 14x}{(1 - x)(1 + 2x)}, \quad |x| < \frac{1}{2}.$$

1. Express \(f(x)\) in partial fractions. [3]
2. Hence find the exact value of \(\int_{-\frac{1}{6}}^{\frac{1}{4}} f(x) \, dx\), giving your answer in the form \(\ln p\), where \(p\) is rational. [5]
3. Use the binomial theorem to expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\), simplifying each term. [5]

The line \(l_1\) has vector equation \(\mathbf{r} = \begin{pmatrix} 11 \\ 5 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix}\), where \(\lambda\) is a parameter. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 24 \\ 4 \\ 13 \end{pmatrix} + \mu \begin{pmatrix} 7 \\ 1 \\ 5 \end{pmatrix}\), where \(\mu\) is a parameter.

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

Given that \(\theta\) is the acute angle between \(l_1\) and \(l_2\),

1. find the value of \(\cos \theta\). Give your answer in the form \(k\sqrt{3}\), where \(k\) is a simplified fraction. [4]

\includegraphics{figure_1} The curve shown in Fig. 1 has parametric equations $$x = \cos t, \quad y = \sin 2t, \quad 0 \leq t < 2\pi.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [3]
2. Find the values of the parameter \(t\) at the points where \(\frac{dy}{dx} = 0\). [3]
3. Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the \(x\)-axis. [2]
4. Show that a cartesian equation for the part of the curve where \(0 \leq t < \pi\) is $$y = 2x\sqrt{(1 - x^2)}.$$ [3]
5. Write down a cartesian equation for the part of the curve where \(\pi \leq t < 2\pi\). [1]

\includegraphics{figure_2} Figure 2 shows the curve with equation $$y = x^2 \sin\left(\frac{1}{2}x\right), \quad 0 < x \leq 2\pi.$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.

1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). [7]

The table shows corresponding values of \(x\) and \(y\).

1. Find the value of \(G\). [1]
2. Use the trapezium rule with values of \(x^2 \sin\left(\frac{1}{2}x\right)\)
   1. at \(x = \pi\), \(x = \frac{3\pi}{2}\) and \(x = 2\pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
   2. at \(x = \pi\), \(x = \frac{5\pi}{4}\), \(x = \frac{3\pi}{2}\), \(x = \frac{7\pi}{4}\) and \(x = 2\pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
   [5]

In an experiment a scientist considered the loss of mass of a collection of picked leaves. The mass \(M\) grams of a single leaf was measured at times \(t\) days after the leaf was picked. The scientist attempted to find a relationship between \(M\) and \(t\). In a preliminary model she assumed that the rate of loss of mass was proportional to the mass \(M\) grams of the leaf.

1. Write down a differential equation for the rate of change of mass of the leaf, using this model. [2]
2. Show, by differentiation, that \(M = 10(0.98)^t\) satisfies this differential equation. [2]

Further studies implied that the mass \(M\) grams of a certain leaf satisfied a modified differential equation $$10 \frac{dM}{dt} = -k(10M - 1), \quad (1)$$ where \(k\) is a positive constant and \(t \geq 0\). Given that the mass of this leaf at time \(t = 0\) is 10 grams, and that its mass at time \(t = 10\) is 8.5 grams,

1. solve the modified differential equation (1) to find the mass of this leaf at time \(t = 15\). [9]