CAIE Further Paper 2 (Further Paper 2) 2021 November

It is given that \(y = \sinh(x^2) + \cosh(x^2)\).

1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^4\). [2]
2. Deduce the value of \(\frac{d^4y}{dx^4}\) when \(x = 0\). [1]
3. Use your answer to part (a) to find an approximation to \(\int_0^{\frac{1}{2}} y \, dx\), giving your answer as a rational fraction in its lowest terms. [2]

Find the solution of the differential equation $$\frac{dy}{dx} + \frac{4x^3y}{x^4 + 5} = 6x$$ for which \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f(x)\). [7]

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 1 - x^2\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of the rectangles, show that $$\int_0^1 (1 - x^2) \, dx < \frac{4n^2 + 3n - 1}{6n^2}.$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 (1 - x^2) \, dx\). [4]

1. Write down all the roots of the equation \(x^5 - 1 = 0\). [2]
2. Use de Moivre's theorem to show that \(\cos 4\theta = 8\cos^4 \theta - 8\cos^2 \theta + 1\). [4]
3. Use the results of parts (a) and (b) to express each real root of the equation $$8x^9 - 8x^7 + x^5 - 8x^4 + 8x^2 - 1 = 0$$ in the form \(\cos k\pi\), where \(k\) is a rational number. [4]

The curve \(C\) has parametric equations $$x = 3t + 2t^{-1} + at^3, \quad y = 4t - \frac{3}{2}t^{-1} + bt^3, \quad \text{for } 1 \leq t \leq 2,$$ where \(a\) and \(b\) are constants.

1. It is given that \(a = \frac{2}{3}\) and \(b = -\frac{1}{2}\). Show that \(\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \frac{25}{4}(t^2 + t^{-2})^2\) and find the exact length of \(C\). [6]
2. It is given instead that \(a = b = 0\). Find the value of \(\frac{d^2y}{dx^2}\) when \(t = 1\). [4]

The matrix \(\mathbf{P}\) is given by $$\mathbf{P} = \begin{pmatrix} 1 & 6 & 6 \\ 0 & 2 & 6 \\ 0 & 0 & -3 \end{pmatrix}.$$

1. Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\). [5]
2. Find the matrix \(\mathbf{A}\) such that $$\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{pmatrix}.$$ [4]
3. State the eigenvalues and corresponding eigenvectors of \(\mathbf{A}^3\). [2]

It is given that \(y = x^2w\) and $$x^2\frac{d^2w}{dx^2} + 4x(x + 1)\frac{dw}{dx} + (5x^2 + 8x + 2)w = 5x^2 + 4x + 2.$$

1. Show that $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 5x^2 + 4x + 2.$$ [4]
2. Find the general solution for \(w\) in terms of \(x\). [7]

1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh^2 x = \sech^2 x.$$ [3]
2. Using the substitution \(u = \tanh x\), or otherwise, find \(\int \sech^2 x \tanh^2 x \, dx\). [2]
3. It is given that, for \(n \geq 0\), \(I_n = \int_0^{\ln 3} \sech^n x \tanh^2 x \, dx\). Show that, for \(n \geq 2\), $$(n + 1)I_n = \left(\frac{4}{3}\right)^{\frac{3}{n-2}} + (n - 2)I_{n-2}.$$ [You may use the result that \(\frac{d}{dx}(\sech x) = -\tanh x \sech x\).] [5]
4. Find the value of \(I_4\). [3]