CAIE Further Paper 2 (Further Paper 2) 2024 November

Find the value of \(\int_6^7 \frac{1}{\sqrt{(x-5)^2-1}} \, dx\), giving your answer in the form \(\ln(a + \sqrt{b})\), where \(a\) and \(b\) are integers to be determined. [4]

The curve \(C\) has equation $$4y^2 + 4\ln(xy) = 1.$$

1. Show that, at the point \(\left(2, \frac{1}{2}\right)\) on \(C\), \(\frac{dy}{dx} = -\frac{1}{6}\). [3]
2. Find the value of \(\frac{d^2y}{dx^2}\) at the point \(\left(2, \frac{1}{2}\right)\). [4]

The curve \(C\) has parametric equations $$x = \frac{1}{2}e^{2t} - \frac{1}{3}t^3 - \frac{1}{2}, \quad y = 2e^t(t-1), \quad \text{for } 0 \leqslant t \leqslant 1.$$ Find the exact length of \(C\). [7]

1. Use de Moivre's theorem to show that $$\cot 6\theta = \frac{\cot^4 \theta - 15\cot^4 \theta + 15\cot^2 \theta - 1}{6\cot^5 \theta - 20\cot^3 \theta + 6\cot \theta}.$$ [6]
2. Hence obtain the roots of the equation $$x^6 - 6x^5 - 15x^4 + 20x^3 + 15x^2 - 6x - 1 = 0$$ in the form \(\cot(q\pi)\), where \(q\) is a rational number. [4]

Find the particular solution of the differential equation $$3\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = x^2,$$ given that, when \(x = 0\), \(y = \frac{dy}{dx} = 0\). [10]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = e^{1-x}\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that \(\int_0^1 e^{1-x} \, dx < U_n\), where $$U_n = \frac{e-1}{n(1-e^{-1})}.$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound \(L_n\) for \(\int_0^1 e^{1-x} \, dx\). [4]
3. Show that \(\lim_{n \to \infty}(U_n - L_n) = 0\). [2]
4. Use the Maclaurin's series for \(e^x\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z(1-e^{-z})\), in ascending powers of \(z\), and deduce the value of \(\lim_{n \to \infty}(U_n)\). [3]

1. Show that \(\frac{d}{dx}(\ln(\tanh x)) = 2\cosh 2x\). [3]
2. Find the solution of the differential equation $$\sinh 2x \frac{dy}{dx} + 2y = \sinh 2x$$ for which \(y = 5\) when \(x = \ln 2\). Give your answer in an exact form. [7]

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} -2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{pmatrix}.$$

1. Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^3 - 12\lambda^2 + 124 + 80 = 0\) and find the eigenvalues of \(\mathbf{A}\). [4]
2. Use the characteristic equation of \(\mathbf{A}\) to show that $$\mathbf{A}^4 = p\mathbf{A}^2 + q\mathbf{A} + r\mathbf{I},$$ where \(p\), \(q\) and \(r\) are integers to be determined. [4]
3. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} - 3\mathbf{I})^4 = \mathbf{PDP}^{-1}\). [6]