In a triangle \(ABC\), \(P\) is the point of contact of \(BC\) with the escribed circle opposite to \(A\); \(Q\) and \(R\) are similarly defined. Prove that \(AP\), \(BQ\), \(CR\) are concurrent, at a point \(N\). \(D\) is the mid-point of \(BC\), and the internal bisector of the angle \(BAC\) meets \(BC\) at \(X\). Calculate the cross-ratio \((B, P; D, X)\), and hence or otherwise prove that \(N\) lies on the line joining the centroid and the in-centre of \(ABC\).
Prove that the pencil of conics passing through four general points \(A\), \(B\), \(C\), \(D\), meets a general line \(l\) in pairs of points in involution. Explain the geometrical significance of the double points, \(X\), \(Y\), of the involution. Prove that the conics through three fixed points \(A\), \(B\), \(C\), and such that two further fixed points \(X\), \(Y\), are conjugate, all meet in a fourth fixed point, \(D\). Show how to construct \(D\) metrically in the case where \(A\), \(B\), \(C\) are real points and \(X\), \(Y\) are the circular points at infinity. Show also that if \(C\), \(X\), \(Y\) are real points and \(A\), \(B\) are taken as the circular points at infinity, then \(D\) is the inverse of \(C\) with respect to the circle with diameter \(XY\).
Obtain the condition for the points with parameters \(t_1\), \(t_2\), \(t_3\), \(t_4\) on the parabola \((at^2, 2at)\) to be concyclic. For a variable point \(P\) of the parabola, \(C\) is defined as the centre of the circle of curvature, and \(Q\) as the remaining intersection of the circle and the parabola. Find (a) the locus of \(C\), and (b) the envelope of \(PQ\).
In a game, three dice are thrown and a player scores the total of the numbers shown on the dice. Calculate (a) the mean, and (b) the standard deviation, of the scores. What are the corresponding results if \(n\) dice are used? [The standard deviation of a set of numbers \(x_1\), \(x_2\), ..., \(x_n\) is defined as \(\sigma\) where $$n\sigma^2 = \sum_{r=1}^{n} (x_r - \bar{x})^2$$ and the mean \(\bar{x}\) is such that $$n\bar{x} = \sum_{r=1}^{n} x_r.]$$
\(f(x)\) is a polynomial of degree \(n\), whose zeros \(z_1\), \(z_2\), ..., \(z_n\) are all different. Obtain the expansion in partial fractions $$\frac{f(x)}{g(x)} = \sum_{r=1}^{n} \frac{f(z_r)}{(x-z_r)g'(z_r)}.$$ \(g(x)\) is a polynomial of degree less than \(n\). Explain how to deal with the case where \(f(x)\) has degree \(n\) or greater. Express $$\frac{(x+1)(x+2)...(x+n)}{(x-1)(x-2)...(x-n)}$$ in partial fractions, and show that $$\sum_{r=1}^{n} \frac{(-1)^{r+1}(n+r)!}{(r!)^2(n-r)!} = 1-(-1)^n.$$
Evaluate $$\int_0^{\pi} \frac{d\theta}{a^2-2a\cos\theta+1} \quad (a \neq 1).$$ A sequence of integrals is defined for \(n = 0, 1, 2, ...\) and \(a > 1\), by $$I_n(a) = \int_0^{\pi} \frac{\cos n\theta d\theta}{a^2-2a\cos\theta+1}.$$ Prove that, for \(n > 1\), $$(a^2+1)I_n = a(I_{n+1}+I_{n-1})$$ and obtain an analogous expression for the case \(n = 0\). Hence show that $$I_n = \frac{\pi}{a^n(a^2-1)}.$$
State and prove the parallel axis theorem for moments of inertia. Two rigid bodies are geometrically similar and are made of the same uniform material. Prove that their moments of inertia about corresponding axes are in proportion to the fifth powers of their linear dimensions. Calculate the moment of inertia of a rectangular block whose edges have lengths \(2a\), \(2b\), \(2c\), about a diagonal.
A uniform plane lamina has a polygonal boundary and rests on a smooth horizontal table. Forces act at the mid-points of the sides, each directly along the inward normal and represented in magnitude by the length of the side on which it acts. Show that the lamina is in equilibrium. The lamina is now turned through an angle \(\alpha\) (less than \(\pi\)) about its centroid, the forces retaining their magnitudes, points of application, and directions in space. If the lamina is now released from rest, show that it will turn about its centroid through an angle \(2\pi - 2\alpha\) before coming again to rest, and will return to its initial position at equal intervals of time.
A particle is projected from the origin with velocity \(u\) in a direction making an angle \(\alpha\) with the horizontal in a medium that resists the motion by a force \(kv\) per unit mass, where \(v\) is the velocity of the particle. Write down the Cartesian equations of motion of the particle, and hence show that the trajectory is given by $$y = \frac{g}{k^2}\log\left[1-\frac{kx}{u\cos\alpha}\right] + \frac{x}{u\cos\alpha}\left[u\sin\alpha + \frac{g}{k}\right].$$ Show further that the maximum height is attained after a time $$\frac{1}{k}\log\left[1+\frac{ku\sin\alpha}{g}\right].$$ Verify that on passing to the limit \(k \to 0\), these results reduce to those obtained by omitting the resistance in the original equations.
A gramophone turntable with radius \(a\) and moment of inertia \(I\) is rotating freely with angular velocity \(\omega_1\) about a vertical frictionless spindle. An insect of mass \(m\) (to be regarded as a particle) flies with velocity \(u\) along a horizontal line tangential to the rim of the turntable, and alights on it, coming to rest relative to it. Find the new angular velocity, \(\omega_2\). The insect then walks with constant velocity \(v\) to the centre of the turntable. Prove that the turntable makes $$\frac{a\omega_2}{2\pi v}(y+\gamma^{-1})\tan^{-1}\gamma$$ revolutions during the walk, \(\gamma\) being defined by \(\gamma^2 = ma^2/I\).