OCR Further Pure Core 1 (Further Pure Core 1) 2021 June

The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_1}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
   2. Find the value of \(r\) at the point \(P\). [1]
   3. Mark the point \(P\) on a copy of the graph. [1]
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]

You are given that \(f(x) = \ln(2 + x)\).

1. Determine the exact value of \(f'(0)\). [2]
2. Show that \(f''(0) = -\frac{1}{4}\). [2]
3. Hence write down the first three terms of the Maclaurin series for \(f(x)\). [3]

You are given that \(\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & -2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 0 & 1 \\ -8 & 4 & 0 \\ 19 & -8 & -1 \end{pmatrix}\).

1. Find \(\mathbf{AB}\). [1]
2. Hence write down \(\mathbf{A}^{-1}\). [1]
3. You are given three simultaneous equations $$x + 2y + z = 0$$ $$2x + 5y + 2z = 1$$ $$3x - 2y - z = 4$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations. [2]
   2. Find this unique solution. [2]

1. Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that $$x = \frac{1}{2}\ln\left(\frac{1+u}{1-u}\right).$$ [4]
2. Solve the equation \(4\tanh^2 x + \tanh x - 3 = 0\), giving the solution in the form \(a\ln b\) where \(a\) and \(b\) are rational numbers to be determined. [4]
3. Explain why the equation in part (b) has only one root. [1]

In this question you must show detailed reasoning. Find \(\int_{-1}^{11} \frac{1}{\sqrt{x^2 + 6x + 13}} dx\) giving your answer in the form \(\ln(p + q\sqrt{2})\) where \(p\) and \(q\) are integers to be determined. [7]