Paper 1
a) Differentiate \(f(x)=(x^2+1)\sin(3x)\), and evaluate \(f'\left(\frac{\pi}{6}\right)\).
[4 marks]b) Evaluate \(\displaystyle \lim_{x\to1}\frac{x^2+5x-6}{x-1}\).
[3 marks]c) Evaluate \(\displaystyle \lim_{t\to\infty}\frac{\sqrt{9t^2+2t}}{4t-1}\).
[3 marks]d) Using first principles, find the derivative of \(g(x)=\frac{2x-1}{x+3}\), where \(x\ne-3\).
[5 marks]a) Evaluate \(\displaystyle \int \sin^3 x\cos^2 x\,dx\).
[5 marks]b) Evaluate \(\displaystyle \int \frac{3x+1}{x^2+x-2}\,dx\).
[5 marks]c) Using \(\sin z=\frac{e^{iz}-e^{-iz}}{2i}\), show that \(\sin(a+ib)=\sin a\cosh b+i\cos a\sinh b\). Hence find \(\sin(ib)\).
[5 marks]a) Find the modulus, principal argument, and rectangular form of \(z=e^{1-i\frac{2\pi}{3}}\).
[3 marks]b) State De Moivre's theorem and use it to evaluate \(\left(\cos\frac{\pi}{10}+i\sin\frac{\pi}{10}\right)^5\).
[3 marks]c) Use De Moivre's theorem to express \(\cos(4\theta)\) as a polynomial in \(\cos\theta\).
[5 marks]d) Solve \(z^4+4z^2+16=0\), giving all roots in exponential form and rectangular form.
[4 marks]a) Use the ratio test to determine whether \(\displaystyle \sum_{n=1}^{\infty}\frac{3^n}{n!}\) converges.
[4 marks]b) Use the root test to determine whether \(\displaystyle \sum_{n=1}^{\infty}\left(\frac{2n+1}{5n-4}\right)^n\) converges.
[4 marks]c) Use the alternating series test, and then test absolute convergence, for \(\displaystyle \sum_{n=2}^{\infty}(-1)^n\frac{1}{\sqrt n}\).
[4 marks]d) Find the interval of convergence of \(\displaystyle \sum_{n=1}^{\infty}\frac{(x-2)^n}{n4^n}\).
[3 marks]a) Find the cubic Taylor polynomial for \(f(x)=\ln(1+x)\) about \(0\), and use it to approximate \(\ln(1.2)\).
[5 marks]b) Use the Lagrange remainder to bound the error in the approximation from part (a).
[4 marks]c) Find the value of \(k\) for which \((1,-2,3)\) and \((2,k,6)\) are linearly dependent.
[3 marks]d) For \(A=\begin{pmatrix}1&a&0\\2&1&1\\0&3&a\end{pmatrix}\), find the real values of \(a\) for which \(A\) is singular.
[3 marks]a) Find the values of \(a\) for which the system has a unique solution.
[4 marks]b) Classify the non-unique cases as no solution or infinitely many solutions.
[5 marks]c) For the unique case \(a=5\), find the inverse of the coefficient matrix.
[4 marks]d) Use your inverse to solve the system when \(a=5\), leaving the answer in terms of \(b\).
[2 marks]a) Verify the eigenvalue belonging to each supplied eigenvector.
[4 marks]b) Construct matrices \(P\) and \(D\) such that \(A=PDP^{-1}\), and justify that this is a valid diagonalisation.
[4 marks]c) Use the diagonalisation to write a formula for \(A^n\), where \(n\) is a positive integer.
[3 marks]d) Decide whether each statement is true or false, giving a proof or counterexample. (i) Every matrix with a repeated eigenvalue is diagonalizable. (ii) Every \(n imes n\) matrix with \(n\) linearly independent eigenvectors is diagonalizable.
[4 marks]