Integrate \[ \int \frac{dx}{\sin^3 x}, \quad \int \frac{dx}{1+e\cos x} \quad (e<1). \] Find a reduction formula for \(\int (x^2+a^2)^n dx\), and evaluate \(\int_0^1 (x^2+4)^{\frac{5}{2}} dx\).
Obtain the formula \[ r = 4R \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2}. \] Three squares are inscribed in the triangle having bases on \(BC, CA, AB\). If their sides be of lengths \(x,y,z\), shew that \[ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{r}. \]
Find the equation of the \(n\)th degree whose roots are \(\tan\left(\alpha + \frac{r\pi}{n}\right)\) where \(r=0, 1, 2, \dots, n-1\). Hence or otherwise shew that \[ \sum_{r=0}^{n-1} \sec^2\left(\alpha+\frac{r\pi}{n}\right) \] takes the values \(n^2\sec^2 n\alpha\) or \(n^2\csc^2 n\alpha\) according as \(n\) is odd or even.
An aeroplane moving at a constant height above the ground describes a circle. Observations made at equal intervals of time \(t\) give angles \(\alpha, \beta, \gamma\) for its altitude, while later it is noticed that the aeroplane passes directly over the point of observation. Shew that it completes a circle in time \(\pi t/\cos^{-1}\left(\frac{\cot\gamma+\cot\alpha}{2\cot\beta}\right)\).
Write down the expressions for \(\cos x, \sin x\) in terms of exponential functions. If \[ \sin x = h \sin(x+\alpha), \quad \text{where } -1 < h < 1, \] shew that \[ x = n\pi + h\sin\alpha + \frac{h^2}{2}\sin 2\alpha + \frac{h^3}{3}\sin 3\alpha + \dots, \] \(n\) being an integer.
State the general conditions of equilibrium of coplanar forces. How may these conditions be modified if the system consists only of (a) one, (b) two, (c) three forces? Shew that any such system of forces may be replaced by a force of given magnitude \(P\) together with a force of given line of action \(AB\); provided that \(P\) is not less than the sum of the resolved parts of the forces perpendicular to \(AB\).
Explain the term ``coefficient of friction.'' The seat of a chair is a square of side 18 inches. The back and legs are vertical, the latter being 18 inches long. The centre of gravity is 6 inches from the back. The chair is drawn forward by a horizontal string attached to a point of the back distant 3' 6'' from the ground. Shew that the chair will slide or tilt according as \(\mu\) is less or greater than \(\frac{4}{7}\). If the chair just slides in this position shew that when an additional mass, equal to that of the chair, rests in the centre of the seat, it will still slide if the string is lowered through any angle less than \(\tan^{-1}\frac{1}{70}\).
What is meant by ``conservation of momentum''? A battleship of symmetrical form and mass 30,000 tons is moving at 10 miles per hour and fires a salvo of all its eight guns in a direction perpendicular to its motion. If the shells weigh 15 cwt. each, have a muzzle velocity of 2000 feet per second, and are fired at an elevation of 30\(^\circ\), shew that the motion immediately after firing makes an angle of about 1\(^\circ\) 21' with that before.
State the laws which determine the motion of elastic bodies after impact. A ball is projected on a pocketless billiard table. Shew that if the effect of friction and rotation be neglected, it will travel always parallel to one of two fixed directions so long as it strikes the four cushions in order: and that the velocity is decreased in the ratio \(e^2:1\) after each complete circuit, \(e\) being the coefficient of restitution.
A car weighing 3 tons will just run down a slope of angle \(\alpha (=\sin^{-1}\frac{1}{30})\) under its own weight. Assuming that the forces resisting its motion remain constant, and that the engine exerts a constant tractive force, find to the nearest unit the horse-power of its engine if it can attain a velocity of 30 miles per hour in 4 minutes on the level.