Show that the radius of curvature of a plane curve \(C\) at the point \(P\) is \(r \frac{dr}{dp}\), where \(r\) is the distance from \(P\) to a fixed point \(O\), and \(p\) is the perpendicular distance from \(O\) to the tangent to \(C\) at \(P\). Find the polar equation of the evolute (locus of centres of curvature) of the curve \(r=ae^{k\theta}\).
A cylindrical hole of radius \(r\) is bored through a solid sphere of radius \(a\), the axis of the hole being along a diameter of the sphere. Find the volume and total surface area of the remaining portion of the sphere, and show that, for fixed \(a\), its surface area is maximum when \(r=a/2\).
The function \(u \equiv f(x_1, x_2, \dots, x_n)\) satisfies the identity \[ f(kx_1, k^2 x_2, \dots, k^n x_n) = k^a f(x_1, x_2, \dots, x_n) \] for fixed \(a\), all \(x_1, x_2, \dots, x_n\), and all positive values of \(k\). Show that \[ x_1 \frac{\partial u}{\partial x_1} + 2x_2 \frac{\partial u}{\partial x_2} + \dots + nx_n \frac{\partial u}{\partial x_n} = au. \] If, further, \(u\) is defined as a function of \(\xi\) and \(\eta\) by the substitutions \[ x_r = \xi^r + \eta^r \quad (r=1, 2, \dots, n), \] show that \[ \xi\frac{\partial u}{\partial \xi} + 2\eta\frac{\partial u}{\partial \eta} = -au. \]
In the system of equations \begin{align*} -ny+mz &= a, \\ nx - lz &= b, \\ -mx+ly &= c, \\ lx+my+nz &= p, \end{align*} \(l, m, n, a, b, c, p\) are given real numbers, and \(l,m,n\) are not all zero. Prove that a necessary and sufficient condition for the equations to have a solution is that \[ la+mb+nc=0; \] and solve the equations when this condition is satisfied.
If \(x_i\) (\(i=1, 2, 3, \dots n\)) are the \(n\) roots of the equation \(f(x)=0\), when \(f(x)\) is a polynomial of degree \(n\), show that \[ \frac{f'(x)}{f(x)} = \sum_{i=1}^n \frac{1}{x-x_i}. \] If \(S_k\) is the sum of the \(k\)th powers of the roots of the equation \(x^4-4x^3-2x^2+1=0\), prove that, for any integer \(k\) (positive, zero, or negative), \(S_k\) is an integer. Find \(S_3, S_4, S_{-4}\).
Denoting by \(c_\nu\) the coefficient of \(x^\nu y^{n-\nu}\) in the expansion of \((x+y)^n\), where \(n\) is a positive integer, evaluate the sum \[ T = \sum_{\nu=0}^n (2\nu-n)^2 c_\nu. \] Hence, or otherwise, prove that those terms of the sum \[ S = \sum_{\nu=0}^n c_\nu \] for which \[ |2\nu-n| \ge k, \] where \(k\) is a positive integer less than \(n\), contribute less than a fraction \(n/k^2\) of the whole sum. Show that, if \(n=1000\), more than nine-tenths of the sum \(S\) is contributed by fewer than one-tenth of its terms.
Prove by the use of complex numbers, or otherwise, that, if \(n\) is a positive integer, \(\cos n\theta\) can be expressed as a polynomial in \(\cos\theta\) of the form \[ \cos n\theta = p_0 \cos^n\theta - p_1 \cos^{n-2}\theta + p_2\cos^{n-4}\theta - \dots, \] where \(p_0, p_1, p_2, \dots\) are positive integers. (It is to be understood that the summation continues so long as the indices remain non-negative.) Show that \[ p_0+p_1+p_2+\dots = \tfrac{1}{2} \{ (1+\sqrt{2})^n + (1-\sqrt{2})^n \}. \]
Show that a plane is divided by \(n\) straight lines, of which no two are parallel and no three meet in a point, into \(\frac{1}{2}(n^2+n+2)\) regions. Consider the same problem with the plane replaced by the surface of a sphere and the lines by great circles, of which no three meet in a point. Into how many regions is space divided by \(n\) planes, of which no two are parallel, no three meet in a line, and no four meet in a point?
Five points \(A, B, C, D, P\), no three collinear, are given in a plane. Prove that the polars of \(P\) with respect to all conics through \(A, B, C, D\) pass through a point \(P'\). Give a geometrical construction for \(P'\) using straight lines only.
Two confocal central conics \(U\) and \(V\) are given, and a variable point \(P\) in their plane is such that two perpendicular tangents can be drawn from \(P\), one to \(U\) and one to \(V\). Prove that \(P\) lies on a fixed circle \(C\). Show that \(C\) passes through the points of intersection of \(U\) and \(V\).