CAIE Further Paper 2 (Further Paper 2) 2020 June

Find the general solution of the differential equation $$\frac{d^2x}{dt^2} - 8\frac{dx}{dt} - 9x = 9e^{8t}.$$ [6]

Let \(I_n = \int_0^1 (1+3x)^n e^{-3x} dx\), where \(n\) is an integer.

1. Show that \(3I_n = 1 - 4^n e^{-3} + 3nI_{n-1}\). [3]
2. Find the exact value of \(I_2\). [3]

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} 5 & -1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{pmatrix}.$$

1. Find the eigenvalues of \(\mathbf{A}\). [4]
2. Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\). [4]

\includegraphics{figure_4} The diagram shows the curve with equation \(y = \ln x\) for \(x \geqslant 1\), together with a set of \((N-1)\) rectangles of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\ln N! > N \ln N - N + 1.$$ [5]
2. Use a similar method to find, in terms of \(N\), an upper bound for \(\ln N!\). [3]

The curve \(C\) has parametric equations $$x = \frac{1}{2}t^2 - \ln t, \quad y = 2t + 1, \quad \text{for } \frac{1}{2} \leqslant t \leqslant 2.$$

1. Find the exact length of \(C\). [5]
2. Find \(\frac{d^2y}{dx^2}\) in terms of \(t\), simplifying your answer. [4]

1. Starting from the definitions of \(\tanh\) and \(\sech\) in terms of exponentials, prove that $$1 - \tanh^2 \theta = \sech^2 \theta.$$ [3]

The variables \(x\) and \(y\) are such that \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\), for \(-\frac{1}{4}\pi < x < \frac{3}{4}\pi\).

1. By differentiating the equation \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\) with respect to \(x\), show that $$\frac{dy}{dx} = -\operatorname{cosec}\left(x + \frac{1}{4}\pi\right).$$ [4]
2. Hence find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\cos\left(x + \frac{1}{4}\pi\right)\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [5]

1. Show that an appropriate integrating factor for $$(x^2 + 1)\frac{dy}{dx} + y\sqrt{x^2 + 1} = x^2 - x\sqrt{x^2 + 1}$$ is \(x + \sqrt{x^2 + 1}\). [4]
2. Hence find the solution of the differential equation $$(x^2 + 1)\frac{dy}{dx} + y\sqrt{x^2 + 1} = x^2 - x\sqrt{x^2 + 1}$$ for which \(y = \ln 2\) when \(x = 0\). Give your answer in the form \(y = f(x)\). [7]

1. Use de Moivre's theorem to show that \(\sin^6 \theta = -\frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10)\). [6]

It is given that \(\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6\cos 4\theta + 15\cos 2\theta + 10)\).

1. Find the exact value of \(\int_0^{\frac{1}{4}\pi}\left(\cos^6\left(\frac{1}{4}x\right) + \sin^6\left(\frac{1}{4}x\right)\right)dx\). [4]
2. Express each root of the equation \(16c^6 + 16\left(1-c^2\right)^3 - 13 = 0\) in the form \(\cos k\pi\), where \(k\) is a rational number. [5]