Problems

If \(u_0, u_1, u_2, \dots\) are numbers connected by the recurrence formula \[ u_n-4u_{n-1}+5u_{n-2}-2u_{n-3}=0, \] find an expression for \(u_n\), given \(u_0=0, u_1=2, u_2=5\).

The roots of the equation \[ x^3+3px+q=0 \] are \(\alpha, \beta, \gamma\). Find the equation whose roots are \((\beta-\gamma)^2, (\gamma-\alpha)^2, (\alpha-\beta)^2\). Deduce that if \[ 4p^3+q^2 > 0, \] the original equation has one real and two imaginary roots. Prove also that if \(a,b,c\) are the roots of the above equation of squared differences, \[ a^2+b^2+c^2 = 2(bc+ca+ab). \]

(a) Prove that \[ \sin nx = 2^{n-1} \sin x \prod_{m=1}^{n-1} \left\{\cos x - \cos\frac{m\pi}{n}\right\}. \] (b) Prove that if \(x_1, x_2, \dots, x_n\) are the roots of the equation \[ x^{n-1}(x-1)+(x^2+1)(a_0x^{n-2}+a_1x^{n-3}+\dots+a_{n-2})=0, \] then \[ \sum_{r=1}^n \tan^{-1} x_r = (m+\tfrac{1}{2})\pi, \] where \(m\) is an integer.

If \(\alpha+\beta+\gamma=2m\pi\), where \(m\) is an integer, prove that \[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma-2\cos\alpha\cos\beta\cos\gamma=1. \] If \(p,q,r\) are the respective distances of the vertices \(A, B, C\) of a triangle from a point \(O\) in the plane of \(ABC\), shew that \[ a^2b^2c^2 - \Sigma a^2p^2(b^2+c^2-a^2) + \Sigma a^2(p^2-q^2)(p^2-r^2)=0, \] where \(a,b,c\) are the sides of the triangle.

Prove that \[ \int_0^\infty \frac{dx}{x^2+2x\cos\alpha+1} = \frac{\alpha}{\sin\alpha} \qquad 0 < \alpha < \pi. \] Evaluate \(\displaystyle\int_0^\infty \frac{dx}{x^4+2x^2\cos\alpha+1}\) and \(\displaystyle\int_0^\infty \frac{(x^2+1)dx}{x^4+2x^2\cos\alpha+1}\).

If \(x\) and \(y\) are functions of \(\xi\) and \(\eta\), and \begin{align*} a &= \left(\frac{\partial x}{\partial\xi}\right)^2 + \left(\frac{\partial y}{\partial\xi}\right)^2, & b &= \left(\frac{\partial x}{\partial\eta}\right)^2 + \left(\frac{\partial y}{\partial\eta}\right)^2, & h &= \frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta} + \frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}, \end{align*} and if \[ H = \frac{\partial x}{\partial\xi}\frac{\partial y}{\partial\eta} - \frac{\partial x}{\partial\eta}\frac{\partial y}{\partial\xi}, \] shew that \[ H\frac{\partial U}{\partial x} = \frac{\partial U}{\partial\xi}\frac{\partial y}{\partial\eta} - \frac{\partial U}{\partial\eta}\frac{\partial y}{\partial\xi}, \quad H\frac{\partial U}{\partial y} = -\frac{\partial U}{\partial\xi}\frac{\partial x}{\partial\eta} + \frac{\partial U}{\partial\eta}\frac{\partial x}{\partial\xi}, \] where \(U\) is any differentiable function of \((x,y)\). Prove also that \[ \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} = \frac{1}{H}\frac{\partial}{\partial\xi}\left\{\frac{b\dfrac{\partial U}{\partial\xi} - h\dfrac{\partial U}{\partial\eta}}{H}\right\} + \frac{1}{H}\frac{\partial}{\partial\eta}\left\{\frac{-h\dfrac{\partial U}{\partial\xi} + a\dfrac{\partial U}{\partial\eta}}{H}\right\}. \]

Polynomials \(f_0(x), f_1(x), f_2(x), \dots\) are defined by the relation \[ f_n(x) = \frac{d^n}{dx^n}(x^2-1)^n. \] Prove that \[ \int_{-1}^{+1} f_n(x) f_m(x) dx = 0 \] if \(m \neq n\), and that \[ \int_{-1}^{+1} \{f_n(x)\}^2 dx = \frac{2(n!)^2 2^{2n+1}}{2n+1}. \] Shew that if \(\phi(x)\) is any polynomial of degree \(m\), \[ \phi(x) = \sum_{n=0}^m a_n f_n(x), \] where \[ a_n = \frac{2n+1}{(n!)^2 2^{2n+1}} \int_{-1}^{+1} \phi(x) f_n(x) dx. \]

If \(b_1, b_2, b_3, \dots, b_n\) are numbers such that \[ b_1 \ge b_2 \ge b_3 \ge \dots \ge b_n, \] and \(a_1, a_2, a_3, \dots, a_n\) are such that \[ a_1+a_2+\dots+a_r \le S \quad (r=1,\dots,n) \] shew that \[ a_1b_1+a_2b_2+\dots+a_nb_n \le Sb_1. \] If \(b_1, b_2, b_3, \dots\) is a decreasing sequence such that \[ \lim_{n\to\infty} b_n = 0, \] shew that \[ \sum_1^\infty b_n\cos nx \] converges if \(x \neq 2m\pi\), where \(m\) is an integer.

If \(Y\) is the foot of the perpendicular from a focus of a central conic on to the tangent to the conic at any point and if \(H\) is the image of the focus in the tangent, prove that the loci of \(Y\) and \(H\) are circles. Obtain the corresponding results in the case of a parabola.

Prove that the inverse of a sphere with respect to any internal point is a sphere. Invert with respect to the orthocentre the theorem that the three perpendiculars of a triangle are concurrent.