Problems

A regular polygon of \(2n+1\) sides is inscribed in a circle of radius a. From one corner perpendiculars are drawn to the sides; prove that their sum is \[ (2n+1)a \cos \frac{\pi}{2n+1}. \]

A rhombus of smoothly jointed rods rests with two sides in contact with a smooth circular disc all in the same vertical plane. Shew that, if the diameter of the disc be one-fifth of the length of a rod, the reactions at the highest and lowest joints are in the ratio 15:1.

A uniform lamina of any shape is suspended from a point O by three strings OA, OB, OC attached to any three non-collinear points A, B, C of the lamina so that it hangs with its plane inclined to the vertical. Shew that, if G is the centre of gravity of the lamina, the tensions in OA, OB, OC are in the proportion \[ \text{OA} \cdot \text{area GBC} : \text{OB} \cdot \text{area GCA} : \text{OC} \cdot \text{area GAB}. \]

A smooth wire is bent into the form \(y=\sin x\) and placed in a vertical plane with the axis of \(x\) horizontal. A bead of mass \(m\) slides down the wire starting from rest at \(x=\frac{\pi}{2}\). Shew that the pressure on the wire as the bead passes through the origin is \(mg/\sqrt{2}\), and find the pressure as it passes through \(x=-\frac{\pi}{2}\).

A body is suspended from a fixed point by a light elastic string of natural length \(l\) whose modulus of elasticity is equal to the weight of the body and makes vertical oscillations of amplitude \(a\). Shew that, if as the body rises through its equilibrium position it picks up another body of equal weight, the amplitude of the oscillation becomes \((l^2+\frac{1}{2}a^2)^{\frac{1}{2}}\).

A machine gun of mass M contains a mass M' of bullets which it discharges at the rate \(m\) units of mass per unit time, V being the velocity of the bullets relative to the ground. Shew that, if \(\mu\) be the coefficient of friction between the gun and the ground, the whole time of recoil of the gun will be \[ (2mV - \mu g M')M' / 2\mu g m M. \]

Solve the equations:

1. [(i)] \(\tan x + \tan 2x = \tan 3x\),
2. [(ii)] \(\cos^{-1}x+\cos^{-1}(1-x)=\cos^{-1}a\).

Find the sum of \(n\) terms of the series \[ \cos\alpha + \cos(\alpha+\beta) + \cos(\alpha+2\beta) + \dots. \] Prove that the sum of the squares of the distances of a point P from the angular points of a regular polygon of \(n\) sides inscribed in a circle of radius \(a\) is \(n(a^2+c^2)\), where \(c\) is the distance of P from the centre of the circle.

In the case of a triangle with the usual notation, prove that

1. [(i)] \(r_1+r_2+r_3-r = 4R\),
2. [(ii)] \(2R\cos A = 2R+r-r_1\).

If in a triangle \(a, b, B\) are given and the conditions of ambiguity are satisfied, prove that the difference of the radii of the circles inscribed in the two triangles is \[ \tan\frac{B}{2}\left\{b^2-a^2\sin^2 B\right\}^{\frac{1}{2}}. \]

Find all the values of \[ (\cos\theta+i\sin\theta)^{\frac{1}{n}} \] where \(n\) is an integer. Find the sum to infinity of the series \[ \cos^2 x - \frac{1}{2}\sin^2 2x + \frac{1}{3}\cos^2 3x - \frac{1}{4}\sin^2 4x + \dots. \]