OCR Further Pure Core 2 (Further Pure Core 2) 2018 March

Plane \(\Pi\) has equation \(3x - y + 2z = 33\). Line \(l\) has the following vector equation. $$l: \quad \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$$

1. Find the acute angle between \(\Pi\) and \(l\). [3]
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\). [3]
3. \(S\) is the point \((4, 5, -5)\). Find the shortest distance from \(S\) to \(\Pi\). [2]

The complex number \(2 + i\) is denoted by \(z\).

1. Show that \(z^2 = 3 + 4i\). [2]
2. Plot the following on the Argand diagram in the Printed Answer Booklet.
   [1]
3. State the relationship between \(|z^2|\) and \(|z|\). [1]
4. State the relationship between \(\arg(z^2)\) and \(\arg(z)\). [1]

In this question you must show detailed reasoning. Use the formula \(\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1)\) to evaluate \(121^2 + 122^2 + 123^2 + \ldots + 300^2\). [3]

You are given that the cubic equation \(2x^3 - 3x^2 + x + 4 = 0\) has three roots, \(\alpha\), \(\beta\) and \(\gamma\). By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]

In this question you must show detailed reasoning. An ant starts from a fixed point \(O\) and walks in a straight line for \(1.5\) s. Its velocity, \(v\) cms\(^{-1}\), can be modelled by \(v = \frac{1}{\sqrt{9-t^2}}\). By finding the mean value of \(v\) in \(0 \leq t \leq 1.5\), deduce the average velocity of the ant. [5]

In this question you must show detailed reasoning.

1. Find the coordinates of all stationary points on the graph of \(y = 6\sinh^2 x - 13\cosh x\), giving your answers in an exact, simplified form. [9]
2. By finding the second derivative, classify the stationary points found in part (i). [3]

In the following set of simultaneous equations, \(a\) and \(b\) are constants. \begin{align} 3x + 2y - z &= 5
2x - 4y + 7z &= 60
ax + 20y - 25z &= b \end{align}

1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\). [3]
2. Determine the value of \(a\) for which there is no unique solution for \(x\), \(y\) and \(z\). [3]
3. 1. Find the values of \(\alpha\) and \(\beta\) for which \(\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z\) for any \(y\) and \(z\). [3]
   2. Hence, for the case where there is no unique solution for \(x\), \(y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions. [2]
   3. When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]

In this question you must show detailed reasoning. Show that \(\int_0^2 \frac{2x^2 + 3x - 1}{x^3 - 3x^2 + 4x - 12} dx = \frac{3}{8}\pi - \ln 9\). [12]

In this question you must show detailed reasoning.

1. Show that \(e^{i\theta} - e^{-i\theta} = 2i\sin\theta\). [1]
2. Hence, show that \(\frac{2}{e^{2i\theta} - 1} = -(1 + i\cot\theta)\). [3]
3. Two series, \(C\) and \(S\), are defined as follows. $$C = 2 + 2\cos\frac{\pi}{10} + 2\cos\frac{\pi}{5} + 2\cos\frac{3\pi}{10} + 2\cos\frac{2\pi}{5}$$ $$S = 2\sin\frac{\pi}{10} + 2\sin\frac{\pi}{5} + 2\sin\frac{3\pi}{10} + 2\sin\frac{2\pi}{5}$$ By considering \(C + iS\), find a simplified expression for \(C\) in terms of only integers and \(\cot\frac{\pi}{10}\). [8]
4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. [2]