Problems

Given that \(x\) and \(y\) are functions of \(u\) and \(v\) defined by \(f(x,y,u,v)=0\) and \(\phi(x,y,u,v)=0\), find \(\frac{\partial x}{\partial u}\) in terms of partial derivatives of \(f\) and \(\phi\) with respect to \(x, y, u\) and \(v\). If \begin{align*} x^2+y^2-25uv &= 0, \\ ux+vy-1 &= 0, \end{align*} prove that \(\frac{\partial x}{\partial u} = \pm \frac{1}{14}\) when \(u=v=1\), and give the reason for the ambiguity in sign.

Prove that the sum of the roots of the equation \[ \begin{vmatrix} a_1 - x & b_1 & c_1 \\ a_2 & b_2 - x & c_2 \\ a_3 & b_3 & c_3 - x \end{vmatrix} = 0 \] is \(a_1 + b_2 + c_3\). Express the sum of the squares of the roots in terms of \(a_1, b_1, \dots, c_3\).

If the equation \[ f(x) = x^n + a_1x^{n-1} + \dots + a_n = 0 \] has all its roots real and distinct, prove that the same is true of the equation \[ f(x) - \lambda f'(x) = 0, \] where \(\lambda\) is any real constant. Deduce, or prove otherwise, that the equation \[ f(x) + b_1f'(x) + b_2f''(x) = 0 \] has its roots real and distinct, if \(b_1\) and \(b_2\) are any constants for which the roots of \[ y^2 + b_1y + b_2 = 0 \] are real. Generalise this result.

If \[ f_n(x, q) = \sum_{r=0}^{n-1} \frac{(1-q^{2n-2})(1-q^{2n-4})\dots(1-q^{2n-2r})}{(1-q^2)(1-q^4)\dots(1-q^{2r})} x^r, \] where the term \(r=0\) is to be interpreted as having the value 1, prove that \begin{align*} f_n(x, q) &= x^{n-1} f_n(\frac{1}{x}, q^2); \\ f_n(x, q) - f_{n-1}(x, q) &= x^{n-1} f_{n-1}(\frac{q^2}{x}, q). \end{align*} Deduce simple formulae for \(f_n(q, q)\) and \(f_n(-q, q)\).

Find the sum to \(N\) terms of the series whose \(n\)th term is \[ \frac{1}{1+2+3+\dots+n}. \] Find the sum to infinity of the series whose \(n\)th term is \[ \frac{1+x+x^2+\dots+x^{n-1}}{1+2+3+\dots+n}, \] where \(x\) is numerically less than 1.

Prove that \[ \sin 3\theta = 4 \sin \theta \sin(\theta + \tfrac{1}{3}\pi) \sin(\theta + \tfrac{2}{3}\pi). \] The trisectors of the angles of a triangle ABC meet in \(X, Y, Z\) (\(X\) being the point of intersection of the trisectors of B and C lying nearest to BC, and similarly for \(Y\) and \(Z\)). Express the ratio \(AY/AZ\) as simply as you can in terms of the angles of the triangle ABC, and hence find the angles of the triangle AYZ. Hence, or otherwise, prove that XYZ is an equilateral triangle.

Three coplanar circles \(\alpha, \beta, \gamma\) have a common point \(O\). The common chord \(PO\) of \(\beta\) and \(\gamma\) passes through the centre \(A\) of \(\alpha\); and the common chord \(QO\) of \(\gamma\) and \(\alpha\) passes through the centre \(B\) of \(\beta\). Prove by inversion, or otherwise, that the common chord \(RO\) of \(\alpha\) and \(\beta\) passes through the centre \(C\) of \(\gamma\). Prove also that \(P, Q, R, A, B, C\) lie on a circle.

Prove that, if two pairs of opposite edges of a tetrahedron are at right angles, so is the third pair. Shew that a necessary and sufficient condition for a tetrahedron to be of this form is that the middle points of its edges should lie on a sphere.

The normal at a variable point \(P\) of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] meets the major axis in \(G\), and a point \(Q\) is taken on the normal so that \(PQ = k \cdot PG\), where \(k\) is a constant. Prove that the locus of \(Q\) is an ellipse. For what value or values of \(k\) is the locus a circle?

Find a necessary and sufficient condition that four points with parameters \(t_1, t_2, t_3, t_4\) on the rectangular hyperbola \(x=at, y=a/t\) should be concyclic. The normal at a point \(P\) of a rectangular hyperbola meets the curve again at \(Q\), and the circle of curvature at \(P\) meets the curve again at \(R\). Prove that \(QR\) is a diameter of the hyperbola.