Edexcel C4 (Core Mathematics 4) 2015 June

1. Find the binomial expansion of $$(4 + 5x)^{\frac{1}{2}}, \quad |x| < \frac{4}{5}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient in its simplest form. [5]
2. Find the exact value of \((4 + 5x)^{\frac{1}{2}}\) when \(x = \frac{1}{10}\) Give your answer in the form \(k\sqrt{2}\), where \(k\) is a constant to be determined. [1]
3. Substitute \(x = \frac{1}{10}\) into your binomial expansion from part (a) and hence find an approximate value for \(\sqrt{2}\) Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers. [2]

The curve \(C\) has equation $$x^2 - 3xy - 4y^2 + 64 = 0$$

1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
2. Find the coordinates of the points on \(C\) where \(\frac{dy}{dx} = 0\) (Solutions based entirely on graphical or numerical methods are not acceptable.) [6]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = 4x - xe^{\frac{1}{x}}, x \geqslant 0\) The curve meets the \(x\)-axis at the origin \(O\) and cuts the \(x\)-axis at the point \(A\).

1. Find, in terms of \(\ln 2\), the \(x\) coordinate of the point \(A\). [2]
2. Find $$\int xe^{\frac{1}{x}} dx$$ [3]
3. Find, by integration, the exact value for the area of \(R\). Give your answer in terms of \(\ln 2\) [3]

The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis and the curve with equation $$y = 4x - xe^{\frac{1}{x}}, x \geqslant 0$$

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = \begin{pmatrix} 5 \\ -3 \\ p \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ -3 \end{pmatrix}, \quad l_2: \mathbf{r} = \begin{pmatrix} 8 \\ 5 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Find the coordinates of \(A\). [2]
2. Find the value of the constant \(p\). [3]
3. Find the acute angle between \(l_1\) and \(l_2\), giving your answer in degrees to 2 decimal places. [3]

The point \(B\) lies on \(l_2\) where \(\mu = 1\)

1. Find the shortest distance from the point \(B\) to the line \(l_1\), giving your answer to 3 significant figures. [3]

A curve \(C\) has parametric equations $$x = 4t + 3, \quad y = 4t + 8 + \frac{5}{2t}, \quad t \neq 0$$

1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) where \(t = 2\), giving your answer as a fraction in its simplest form. [3]
2. Show that the cartesian equation of the curve \(C\) can be written in the form $$y = \frac{x^2 + ax + b}{x - 3}, \quad x \neq 3$$ where \(a\) and \(b\) are integers to be determined. [3]

\includegraphics{figure_2} Figure 2 shows a sketch of the curve with equation \(y = \sqrt{(3-x)(x+1)}\), \(0 \leqslant x \leqslant 3\) The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis, and the \(y\)-axis.

1. Use the substitution \(x = 1 + 2\sin\theta\) to show that $$\int_0^3 \sqrt{(3-x)(x+1)} dx = k \int_{-\frac{\pi}{6}}^{\frac{\pi}{2}} \cos^2\theta d\theta$$ where \(k\) is a constant to be determined. [5]
2. Hence find, by integration, the exact area of \(R\). [3]

1. Express \(\frac{2}{P(P-2)}\) in partial fractions. [3]

A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{2}P(P-2)\cos 2t, \quad t \geqslant 0$$ where \(P\) is the population in thousands, and \(t\) is the time measured in years since the start of the study. Given that \(P = 3\) when \(t = 0\),

1. solve this differential equation to show that $$P = \frac{6}{3 - e^{\frac{1}{2}\sin 2t}}$$ [7]
2. find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures. [3]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3^x$$ The point \(P\) lies on \(C\) and has coordinates \((2, 9)\). The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).

1. Find the exact value of the \(x\) coordinate of \(Q\). [4]

The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360°\) about the \(x\)-axis.

1. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are exact constants. [You may assume the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] [6]