Roots of polynomials, series and Maclaurin – further maths test to download as PDF

The roots of \(2x^3-3x^2+4x-5=0\) are \(\alpha\), \(\beta\) and \(\gamma\).

1. Write down \(\sum\alpha\), \(\sum\alpha\beta\) and \(\alpha\beta\gamma\). [2]
2. Find \(\sum\alpha^2\) and explain what it tells you about the roots. [2]

The roots of \(x^3-2x^2+3x-4=0\) are \(\alpha\), \(\beta\), \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(3\alpha-1\), \(3\beta-1\) and \(3\gamma-1\). [4]

1. Show that \(\displaystyle\sum_{r=1}^{n}(r^3-r)=\tfrac14(n-1)n(n+1)(n+2)\). [3]
2. Evaluate \(\displaystyle\sum_{r=1}^{10}(r-1)r(r+1)\). [1]

1. Find the Maclaurin series of \(\ln(1+2x)\) up to the term in \(x^3\), and state the values of \(x\) for which it is valid. [3]
2. Use your series with \(x=0.1\) to estimate \(\ln1.2\). [1]

A charity appeal raises \(\dfrac{60}{r(r+1)}\) pounds (in thousands) in week \(r\).

1. Use the method of differences to show that the total raised in the first \(n\) weeks is \(\dfrac{60n}{n+1}\) thousand pounds. [2]
2. Explain why the total never reaches £60 000, and find the first week after which the total exceeds £55 000. [2]