Problems

Give a rough sketch of the curve \[ y^2 - 2x^3 y + x^7 = 0, \] and find the area of the loop.

1. [(i)] If \(a_1, a_2, a_3\) are the roots of \[ x^3+px+q=0, \] where \(p+q+1 \ne 0\), find the equation with roots \[ 1/(1-a_i) \quad (i=1, 2, 3). \]
2. [(ii)] Solve the equations \begin{align*} x+y+z &= 7, \\ x^2+y^2+z^2-4z &= 5, \\ xyz-2xy &= 4. \end{align*}

If \(x_1, \dots, x_n\) are real numbers, prove that \[ n(x_1^2+\dots+x_n^2) \ge (x_1+\dots+x_n)^2 \] and determine when equality occurs. Prove that \[ \sum_{r=1}^n \frac{1}{r^2+r} \le \frac{n}{(n+1)}. \]

A sequence \(u_0, u_1, \dots\) is defined by \(u_0=3\), \(u_{n+1}=(2u_n+4)/u_n\). Prove that

1. [(i)] \(3 \le u_n \le \frac{10}{3}\) for all \(n\);
2. [(ii)] \(|u_n-a| \le (\frac{2}{3})^n (a-3)\) for all \(n\), where \(a\) is the positive root of \(x^2-2x-4=0\).

Two polynomials \(f_0(x), f_1(x)\) are given and a sequence of polynomials \(f_2(x), f_3(x), \dots, f_r(x)\) are defined by the rule that \(f_{i+1}(x)\) is the remainder when \(f_{i-1}(x)\) is divided by \(f_i(x)\), \(f_r(x)\) being the last such remainder that is different from zero. Prove, by induction on \(i\), that

1. [(i)] each of the polynomials \(f_i(x)\) can be expressed in the form \(a_i(x)f_0(x)+b_i(x)f_1(x)\), where \(a_i(x)\) and \(b_i(x)\) are polynomials;
2. [(ii)] \(f_r(x)\) divides \(f_i(x)\) for \(i=1, \dots, r\).

If \(f_r(x)\) is a constant and \(f_0(x)\) divides \(f_1(x)g(x)\), prove that \(f_0(x)\) divides \(g(x)\).

If \(u_0 = \sinh\alpha\), \(u_1=\sinh(\alpha+\beta)\) and \(u_{n+2}-2u_{n+1}\cosh\beta+u_n=0\) for all \(n \ge 0\), prove that \(u_n = \sinh(\alpha+n\beta)\). Sum the series \[ \sum_{r=0}^n \sinh(\alpha+r\beta) \] for all values of \(\beta\).

Find

1. [(i)] \(\int_0^1 \cos^{-1}\sqrt{1-x^2} dx\),
2. [(ii)] \(\int_0^1 \frac{dx}{1+x^2+x^4}\),
3. [(iii)] \(\int \frac{dx}{(x^2+1)^{\frac{1}{2}}+(x^2-1)^{\frac{1}{2}}}\).

If \(y=\sin^{-1}x\), show that \(y''(1-x^2)=xy'\). By Leibniz' Theorem or otherwise, find \(y^{(n)}\) at \(x=0\) and hence expand \(\sin^{-1}x\) as a power series in \(x\).

Sketch the curve whose equation, in Cartesian coordinates, is \[ x^4 - 2xy^2 + y^4 = 0. \]

If \[ I_{m,n} = \int_0^\pi \sin^m x \sin nx dx, \] obtain a relation between \(I_{m,n}\) and \(I_{m-2,n}\) for \(m \ge 2\). Hence evaluate \(I_{4,5}\).