Problems

The points \(A\) and \(B\), at a distance \(a\) apart on a horizontal plane, are in line with the base \(C\) of a tower. The elevations of the top of the tower from \(A\) and \(B\) are observed to be \(\alpha\) and \(\beta\). Find the distance \(BC\). If the observations of elevation are uncertain by \(\pm\epsilon\), shew that the maximum possible percentage error in the calculated value of \(BC\) is \[ \frac{100\epsilon\sin(\beta+\alpha)}{\sin\alpha\cos\beta\tan(\beta-\alpha)}. \]

If \[ S_n(\theta) = \sum_{r=1}^n \cos^r\theta \sin r\theta \] prove (by induction or otherwise) that, if \(0<\theta<\pi\), \[ S_n(\theta) = \cot\theta(1-\cos^n\theta \cos n\theta). \] Prove also that \(S_n(0) = 0\). If \(n\) is fixed, what is the limit of \(S_n(\theta)\) as \(\theta\to 0\)? Shew that the series converges as \(n\to\infty\) for all values of \(\theta\), but that its sum is not continuous.

Water is poured gently into a bowl having the form of a surface of revolution with its axis vertical. If the rate of increase of volume is \(w\) shew that the rate of increase of depth is \(w/a\), and the rate of increase of the area of the wetted surface is \(2\pi rw/a\), where \(a\) denotes the area of the water-surface, and \(r\) the intercept between the meridian curve and the axis on a normal to the curve through a point on the water-surface. If the meridian curve is \[ y = \frac{ax^2}{a^2+x^2}, \] and the axis of rotation is \(x=0\), find \(r\) when \(a=\pi a^2\).

State (without proof) Rolle's theorem, and deduce that there is a number \(\xi\) between \(a\) and \(b\) such that \[ f(b) - f(a) = (b-a)f'(\xi), \quad (1) \] explaining what conditions must be satisfied by the function \(f(x)\) in order that the theorem may be valid. If \(f(x) = \sin x\), find all the values of \(\xi\) between \(a\) and \(b\) which satisfy the equation (1) when \(a=0\) and \(b=3\pi/2\). Illustrate the result with reference to the graph of \(\sin x\).

If \[ \left(\frac{d}{dx}\right)^n e^{-x^2} = \phi_n(x)e^{-x^2}, \] shew that \[ \phi_n + 2x\phi_{n-1} + 2(n-1)\phi_{n-2} = 0, \] and that \[ \phi_n'' - 2x\phi_n' + 2n\phi_n = 0. \] Shew also that \[ \phi_{2n}(0) = 1.3.5 \dots (2n-1).(-2)^n, \] and \[ \phi_{2n}'(0)=0. \]

(i) Evaluate \[ \int_1^e \left(\log \frac{e}{x}\right)^2 dx, \quad \int_0^\pi \frac{dx}{a+b\cos x}, \quad (a>|b|). \] (ii) If \[ I_n = \int_0^\pi \frac{\sin^2 nx}{\sin^2 x} dx, \] where \(n\) is a positive integer, shew that \(I_n - I_{n-1} = I_{n-1} - I_{n-2}\), and hence evaluate \(I_n\).

Determine \(A, B, C\) and \(D\) such that \[ \frac{x^2}{(x^2+1)^4} = \frac{d}{dx} \frac{Ax^5+Bx^3+Cx}{(x^2+1)^3} + \frac{D}{x^2+1}, \] and shew that \[ \int_0^1 \frac{x^2}{(x^2+1)^4}dx = \frac{1}{48} + \frac{\pi}{64}. \]

Any point \(P\) is taken in the plane of a triangle \(ABC\). Through the mid-points of \(BC, CA, AB\) lines are drawn parallel to \(PA, PB, PC\) respectively. Prove that these lines are concurrent.

Two circles \(S, S'\) meet in \(A\) and \(B\), and the centre \(O\) of \(S\) lies on the circumference of \(S'\). From an arbitrary point \(P\) of \(S'\) tangents \(PL, PM\) are drawn to \(S\), meeting \(S'\) in \(X,Y\) respectively. \(PO\) meets \(AB\) in \(T\). Prove that

1. [(i)] \(XY\) is fixed in direction;
2. [(ii)] \(L, M, T\) are collinear.

Describe the process of reciprocation with respect to a circle. Reciprocate the following theorem with respect to a circle centre \(S\), and prove either the original theorem or its reciprocal: If two conics \(\Gamma, \Gamma'\) have a common focus \(S\), and a line through \(S\) meets \(\Gamma\) and \(\Gamma'\) in \(P,Q\) and \(P',Q'\) respectively, then the tangent at \(P\) or \(Q\) meets those at \(P',Q'\) in points lying on two fixed straight lines \(a,b\) through the intersection of the corresponding directrices. Shew further that, if \(\Gamma, \Gamma'\) have the same eccentricity, \(a\) and \(b\) are at right angles.