\(P_1, P_2, \dots, P_N\) are \(N\) points lying on a straight line \(l\). For \(n=1, 2, \dots, N\), the polar coordinates measured from \(P_n\) as vertex and \(l\) as axis are \(r_n, \theta_n\), and \(a_1, a_2, \dots, a_N\) are constants. Show that every curve of the family \[ \sum_{n=1}^N \frac{a_n}{r_n} = \text{constant} \] cuts every curve of the family \[ \sum_{n=1}^N a_n \cos\theta_n = \text{constant} \] orthogonally.
Discuss the behaviour of the function \[ \frac{\log(1+x) - \frac{1}{x}(10-3x-4\cos x)}{x \sin x - x^2} \] as \(x \to 0\) and as \(x \to \infty\).
\(A, B, C\) are the angles of a triangle. Prove the inequalities \[ \sin A + \sin B + \sin C \ge \sin 2A + \sin 2B + \sin 2C > 0, \] and \[ 8 \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2} \le 1. \]
A point \(P\) moves on the quadrant of the circle \(x^2+y^2=1\) for which \(x\ge0, y\ge0\). The circle with centre \(P\) and radius \(\sqrt{5}\) intersects the positive \(x\) axis at \(A\) and the positive \(y\) axis at \(B\). Find the position of \(P\) for which \(AB\) attains its greatest length and give the value of this length.
Define the partial derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\) of a function \(f(x,y)\) of the two variables \(x\) and \(y\). If \(f(kx, ky) = k^n f(x,y)\) for all values of \(k, x\) and \(y\), where \(n\) is a constant independent of \(k\), obtain the relation \[ x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = n f(x,y). \] Verify this result explicitly when \(f(x,y) = (x^3+y^3)^{\frac{1}{3}}\tan^{-1}\frac{y}{x}\) by finding \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).
(i) Express \(\displaystyle \frac{3x^2+12x+8}{(x+1)^5}\) in partial fractions. (ii) Evaluate \(\displaystyle \int_0^2 \frac{x(x+1)}{(x-4)(x^2+4)} \, dx\).
Find the conditions that the roots of the equation \[ x^3+3px^2+3qx+r=0 \] should be (i) in arithmetic progression, (ii) in geometric progression, (iii) in harmonic progression.
The sequence \(u_0, u_1, \dots, u_n, \dots\) is defined by \(u_0=2, u_1=1\), and the recurrence relation \(u_{n+2}=u_{n+1}+u_n\). Show that \(u_n=A\alpha^n+B\beta^n\), where \(A, B, \alpha, \beta\) are independent of \(n\), and find \(A, B, \alpha, \beta\). Prove that
By considering the inequalities \[ \frac{1}{r(r+1)} < \frac{1}{r^2} < \frac{1}{r^2-1}, \] prove that \[ \frac{m}{ (m+1)(2m+1)} < \sum_{r=m+1}^{2m} \frac{1}{r^2} < \frac{m}{(m+1)(2m+1)} + \frac{3m+1}{4m(m+1)(2m+1)}. \] Hence find the value of \(\sum_{r=101}^{200} \frac{1}{r^2}\) with an error of less than \(2.10^{-5}\).
Sum the series \[ \sum_{r=0}^{n-1} \sin^2(\alpha+r\beta). \] Deduce that, if \(0 < \beta < \frac{\pi}{2n}\), \[ n > \frac{\sin n\beta}{\sin\beta} > n \frac{\cos 2(n-1)\beta}{\cos(n-1)\beta}, \] and hence prove that \[ \lim_{\beta\to 0} \frac{\sin n\beta}{\sin \beta} = n. \]