Problems

A triangle \(ABC\) formed of uniform rods of the same material and thickness rests in a vertical plane with each rod in contact with a smooth circular cylinder whose axis is horizontal and whose section is the inscribed circle of the triangle. Shew that in equilibrium the intersection of the medians of the triangle \(ABC\) must be in the vertical plane through the axis of the cylinder.

Prove that if \(x + y + z = a\), where \(a\) is a given positive number, the function \[ u = x^2 + y^2 + z^2 - 2yz - 2zx - 2xy \] has the minimum value \(-\frac{1}{2}a^2\) and no maximum. Prove also that if \(x, y, z\) are further restricted to be not negative, the maximum value of \(u\) is \(a^2\).

Obtain a cubic equation whose roots are the values of \(x, y, z\) given by \begin{align*} x+y+z &= 3, \\ x^2 + y^2 + z^2 &= 5, \\ x^3 + y^3 + z^3 &= 7. \end{align*} Prove that \[ x^4 + y^4 + z^4 = 9. \]

Assuming that if \(f'(x)\) is positive \(f(x)\) increases with \(x\), and that if \(f'(x)\) is negative \(f(x)\) decreases as \(x\) increases, prove that for \(p>q\) (\(p,q\) positive or negative) \[ \frac{x^p-1}{p} > \frac{x^q-1}{q}, \] when \(x > 1\); and obtain the corresponding inequality for \(0 < x < 1\). Prove further, without assuming the logarithmic series, that for \(x\) positive but otherwise unrestricted, \[ \log(1+x) \geq x - \frac{x^2}{2} + \dots + \frac{(-1)^{n} x^n}{n} \quad (\substack{ > \text{ when } n \text{ is even} \\ < \text{ when } n \text{ is odd} }), \] and that for \(-1 < x < 0\), \[ x - \frac{x^2}{2} + \dots + \frac{(-1)^{n} x^n}{n} > \log(1+x) > x - \frac{x^2}{2} + \dots - \frac{(-1)^{n} x^n}{n(1+x)}. \] Deduce the logarithmic series \[ \log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \] for \(-1 < x \le 1\).

A rectangular block of height \(2h\) rests with two faces vertical and its base in contact with a fixed rough circular cylinder of radius \(a\) whose axis is horizontal, the base of the block making an angle \(\alpha\) with the horizontal plane. Find the change in the potential energy when the block is rolled on the cylinder through a small angle \(\theta\); and shew that if \(h = a \cos \alpha\), the block is in neutral equilibrium to a first approximation, but is actually in unstable equilibrium.

Four straight lines are given; prove that a system of three circles can be found in an infinite number of ways such that the 6 intersections of the 4 lines are the centres of similitude of the three circles taken in pairs.

If \[ (1 + x)^n = a_0 + a_1 x + a_2 x^2 + \dots\dots, \] where \(n\) is a positive integer, shew by considering the product of \((1 + x)^n\) and \(\left(1 + \frac{1}{x}\right)^n\), or otherwise, that the sum of \[ a_0 a_1 + a_1 a_2 + a_2 a_3 + \dots\dots \] is \[ \frac{(2n)!}{(n + 1)! (n - 1)!}, \] and find the value of \[ a_0 a_1 - a_1 a_2 + a_2 a_3 - \dots\dots \] where \(n\) is the odd integer \(2p + 1\).

In connection with the tracing of an algebraic curve \(f(x,y)=0\) explain

1. how to find the approximate form of the curve in regions far distant from the origin (asymptotes, etc.);
2. how to test whether a given point on the curve is a multiple point;
3. the utility of the information derivable from the differential coefficient \(\dfrac{dy}{dx}\).

Two equal uniform beams \(AB, BC\) of length \(a\) and of the same weight per unit length \(w\) are smoothly hinged at \(B\) and supported in a horizontal line by props at \(A\) and \(D\), where \(BD = \frac{1}{2}DC\). Find expressions for the shearing force and bending moment at any point of each beam and draw graphs to represent the variations in their values.

Four circles are such that the three pairs of points in which one of the circles is cut by the other three form an involution on that circle; prove that the same property holds for each of the circles; prove also that the two points common to any two of the circles are concyclic with the two points common to the other two.