Given that \(\sinh x = \dfrac{3}{4}\), find the exact value of:

(a) \(\cosh x\)    [1]

(b) \( anh x\)    [1]

(c) \(\sinh 2x\)    [2]

The matrix \(\mathbf{A}\) is given by

\(\mathbf{A} = egin{pmatrix} 2 & 1 & 1 \ 1 & 3 & 0 \ 1 & 0 & 3 \end{pmatrix}\).

(a) Show that the eigenvalues of \(\mathbf{A}\) are \(1\), \(3\) and \(4\).    [4]

(b) Find the eigenvectors of \(\mathbf{A}\).    [4]

(c) Hence write down matrices \(\mathbf{Q}\) and \(\mathbf{D}\) such that \(\mathbf{A} = \mathbf{Q}\mathbf{D}\mathbf{Q}^{-1}\), where \(\mathbf{D}\) is a diagonal matrix.    [2]

Solve the differential equation

\(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2\dfrac{\mathrm{d}y}{\mathrm{d}x} + 5y = 10\sin x\),

given that when \(x = 0\), \(y = 0\) and \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\).    [8]

The curve \(C\) is given by \(y = \cosh x\) for \(0 \le x \le 1\).

(a) Find the arc length of \(C\).    [4]

(b) The curve \(C\) is rotated completely about the \(x\)-axis. Find the surface area of the solid generated.    [5]

Let \(I_n = \displaystyle\int_0^{\pi/2} x^n \cos x \,\mathrm{d}x\) for \(n \ge 0\).

(a) Show that \(I_n = \left(\dfrac{\pi}{2} ight)^n - n(n-1)I_{n-2}\) for \(n \ge 2\).    [5]

(b) Find the exact value of \(I_4\).    [5]

(a) Use de Moivre's theorem to express \(\sin^5 heta\) in terms of sines of multiples of \( heta\).    [5]

(b) Hence find \(\displaystyle\int_0^{\pi/2} \sin^5 heta \,\mathrm{d} heta\).    [5]

The matrix \(\mathbf{B}\) is given by

\(\mathbf{B} = egin{pmatrix} 1 & 2 & 0 \ 2 & 1 & 2 \ 0 & 2 & 1 \end{pmatrix}\).

(a) Show that the characteristic equation of \(\mathbf{B}\) is \(\lambda^3 - 3\lambda^2 - 5\lambda + 7 = 0\).    [2]

(b) Verify the Cayley-Hamilton theorem for \(\mathbf{B}\).    [3]

(c) Use the Cayley-Hamilton theorem to find \(\mathbf{B}^{-1}\).    [3]

(d) Hence express \(\mathbf{B}^4\) in the form \(p\mathbf{B}^2 + q\mathbf{B} + r\mathbf{I}\).    [3]

Given that \(f(x) = \sinh^{-1} x\).

(a) Express \(\sinh^{-1} x\) in logarithmic form.    [2]

(b) Differentiate \(f(x)\) and show that \(f'(x) = \dfrac{1}{\sqrt{1+x^2}}\).    [3]

(c) Find the Maclaurin series expansion of \(f(x)\) up to and including the term in \(x^3\).    [5]

(d) Use your series to estimate \(\sinh^{-1}(0.4)\), giving your answer to 4 decimal places.    [3]