The equation \[ x^3+px^2+qx+r=0 \] has roots \(\alpha, \beta, \gamma\). Find the equations with roots (i) \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\), (ii) \(\beta\gamma, \gamma\alpha, \alpha\beta\). Hence or otherwise determine necessary and sufficient conditions for the equation \[ x^3+px^2+qx+r=0 \] to have two roots (i) whose sum is \(a\), (ii) whose product is \(b\).
Explain how complex numbers are represented in the Argand diagram. If \(P_1, P_2\) are the points representing \(z_1, z_2\), give constructions for the points representing \(z_1+z_2, z_1z_2\). Prove that the points representing \(z_1, z_2, z_3, z_4\) are concyclic if and only if \[ \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \] is real. (A straight line is considered as a special case of a circle.)
Prove that, if \(a_1, \dots, a_n\) are positive, \[ \frac{1}{n}(a_1+\dots+a_n) \ge (a_1a_2\dots a_n)^{\frac{1}{n}}. \] Deduce, using the binomial theorem, that \[ (n+1)! \le 2^n\{(1!)(2!)\dots(n!)\}^{\frac{2}{n+1}}. \]
If \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), prove that \[ \sum_{n=1}^N \binom{n+r-1}{r} = \binom{N+r}{r+1}. \] Hence express \(\sum_{n=1}^N n^4\) as a polynomial in \(N\).
Prove that \[ \int_0^\pi \left( f(\theta) - \sum_{r=1}^n a_r \sin r\theta \right)^2 d\theta \ge \int_0^\pi \{f(\theta)\}^2 d\theta - \frac{\pi}{2}\sum_{r=1}^n a_r^2, \] for all values of \(a_1, \dots, a_n\), where \[ a_r = \frac{2}{\pi} \int_0^\pi f(\theta) \sin r\theta d\theta. \] By taking \(f(\theta)=1\), prove that, for any value of \(m\), \(\sum_{r=1}^m \frac{1}{(2r-1)^2} \le \frac{\pi^2}{8}\).
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\(\alpha\) is a real number and \[ \frac{\alpha x - x^3}{1+x^2} \] is increasing for all real \(x\). Show that \[ \alpha \ge \frac{9}{8}. \]
Show that \[ \frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}. \] By using the series expansion of \(\arctan x\), or otherwise, evaluate \(\pi\) with an error of less than \(10^{-3}\).
\(P\) and \(Q\) are the points on the curve \(y=f(x)\) corresponding to \(x=a, x=b\) where \(b>a\). The function \(f(x)\) is increasing and the curve between \(P\) and \(Q\) lies above the chord \(PQ\). Prove that \[ (b-a)f(b) > \int_a^b f(x)dx > \frac{1}{2}(b-a)(f(a)+f(b)). \] By splitting the range of integration of \(\int_1^n \log x dx\) into suitable parts, prove that \[ n^{n+1/2}e^{-n+1} \ge n! \ge n^n e^{-n+1}. \]
Let \(q_n\) (\(n=1,2,\dots,N\)) be a set of positive numbers, not necessarily in ascending order of magnitude. For any real \(x\) denote by \(Q(x)\) the number of \(n\) for which \(q_n \le x\). Show that \[ \sum_{n=1}^N \frac{1}{nq_n} \le 1 \] provided that \(Q(x) \le x-1\) for all \(x \ge 1\).