OCR Further Pure Core 2 (Further Pure Core 2) 2021 June

1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]

In this question you must show detailed reasoning. Show that \(\int_5^{\infty} (x-1)^{-2} dx = 1\). [5]

\(A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after \(0.2\) seconds at point \(B\) whose displacement is \(0.2\) m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x\) m.

1. Sketch a graph of \(x\) against \(t\) for \(0 \leq t \leq 0.4\). [4]
2. Find the displacement of \(P\) from \(M\) at \(0.75\) seconds after release. [2]

In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square \(S\).

1. Find the area of \(S\). [3]
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]

In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin\theta}e^{\cos\theta}\) for \(0 \leq \theta < \pi\). \includegraphics{figure_5}

1. Find the exact area enclosed by the curve. [4]
2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}}e^{\frac{1}{2}}\). [7]