A uniform cylinder, whose normal cross-section is an ellipse with eccentricity \(e\), is placed with its generators horizontal on a perfectly rough plane inclined at an angle \(\alpha\) to the horizontal. Show that, if it is released from rest, complete revolutions will occur, whatever the initial position, provided that \[ \tan\alpha > \frac{e^2}{2\sqrt{(1-e^2)}}. \]
Show that an algebraical equation \(f(x)=0\) can at most have only one more real root than the derived equation \(f'(x)=0\). In the case when \(f(x)=x^4+(a-b)x^3+(a+b)x-1\), where \(a\) and \(b\) are both real and non-negative, prove that the equation \(f(x)=0\) has at least two real roots, and that \[ 4(a+b) > (b-a)^3 \] is a sufficient condition for it to have only two real roots.
Prove that the total number of ways in which three non-zero positive integers can be chosen to have as their sum a given even integer \(N\) which is also a multiple of three is \(\frac{1}{12}N^2\).
Three complex numbers whose product is unity, are such that their sum \(p\) is equal to the sum of their reciprocals. Prove that one of the numbers must be unity, and calculate the values of the other two in terms of \(p\). If \(p\) is real, determine the restrictions on its value to ensure that all three original numbers are real. In the case of four complex numbers having the same general property, show that they must consist of two reciprocal pairs.
An infinite series of positive finite real quantities \(C_1, C_2, \dots, C_n, \dots\) is such that, except for the first term, each term is the same fixed positive multiple \(\lambda\) of the harmonic mean of the two adjacent terms. Prove that \(\lambda\) cannot be less than unity. Show that in the special case when \(C_2=\lambda C_1\) the general term is given by \[ C_n = C_1 \operatorname{sech}\,(n-1)\theta, \] where \(\lambda = \cosh\theta\).
Obtain an explicit formula for \((\frac{d}{dx})^n \tan^{-1}x\). Show that for \(x=0\) its value is zero for \(n\) even, and \(\pm (n-1)!\) if \(n\) is odd and of the form \(4p\pm 1\) where \(p\) is an integer. Hence write down the power series for \(\tan^{-1}x\).
The series of polynomials \(f_n(x)\) for \(n=0, 1, 2, \dots\) are defined by \[ f_n(x) = x^{2n+2}e^{1/x}(d/dx)^{n+1}e^{-1/x}. \] Prove that for \(n \ge 1\) \[ f_n(x) = -(2nx-1)f_{n-1}(x)+x^2f_{n-1}'(x). \] Hence show by induction that \(f_n(x)\) is a polynomial in \(x\) of degree \(n\), and find the coefficient of the highest term. Prove further that the equation \(f_n(x)=0\) has \(n\) distinct real roots.
A tank in the form of a rectangular parallelepiped but open at the top is to be made of uniform thin sheet metal to contain a given volume of water. Find what ratios the depth must bear to the length and breadth in order that the amount of metal used shall be least. If instead a given amount of metal were to be used to construct a tank of the same form and of greatest cubic capacity, what would be the appropriate proportions?
Show that if \(p>q>0\) and \(x\) is positive then \[ \frac{1}{p}(x^p-1) > \frac{1}{q}(x^q-1). \] Hence, or otherwise, show that for \(s>0\) and \(n\) a positive integer \[ \frac{1}{p}\left( \frac{x^p}{(p+s)^n} - \frac{1}{s^n} \right) > \frac{1}{q}\left( \frac{x^q}{(q+s)^n} - \frac{1}{s^n} \right). \]
Evaluate the integral: \[ \int_0^{\pi/3} (\cos 2x - \cos 4x)^{\frac{1}{2}} dx. \]