The tangents at the points \(B\), \(C\) on a conic are \(e\), \(f\) respectively; \(x\), \(y\) are the tangents from a point \(A\). Denoting the meet of \(e\), \(f\) by \((gf)\), and so on, prove that
A variable conic through fixed points \(K\), \(L\), \(M\), \(N\) meets a fixed line through \(N\) in \(P\). Prove that the envelope of the tangent at \(P\) is a conic inscribed in the triangle \(KLM\). Interpret this result when \(K\) and \(L\) are the circular points at infinity.
If \(n\) is a positive integer, show that $$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{n+2} & b^{n+2} & c^{n+2} \end{vmatrix}$$ has the value \((b-c)(c-a)(a-b)S\), where $$S = \sum a^r b^s c^t$$ summed over all values \(r\), \(s\), \(t\) satisfying \(r+s+t=n\). Prove a similar result for $$\begin{vmatrix} 1 & 1 & 1 & 1 \\ a & b & c & d \\ a^2 & b^2 & c^2 & d^2 \\ a^{n+3} & b^{n+3} & c^{n+3} & d^{n+3} \end{vmatrix}$$ and generalise the result.
The lines \(AB\) and \(A'B'\) are equal in length and lie in a plane. Show that \(A'B'\) can always be brought into coincidence with \(AB\) by either a rotation about a point in the plane or a translation. Show also that \(A'B'\) can be brought into coincidence with \(AB\) by a reflection in a suitably chosen line followed by translation parallel to the line. Prove that successive reflections of a plane figure in two non-parallel lines in the plane are equivalent to a rotation and that an odd number of reflections is equivalent to a single reflection followed by a translation.
Prove that the series $$\frac{1}{1} + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \ldots$$ is divergent. Prove also the series $$\frac{1}{1} + \frac{1}{4} + \ldots + \frac{1}{81} + \frac{1}{100} + \ldots + \frac{1}{8^2} + \frac{1}{100} + \ldots,$$ derived from the first series by the omission of all terms whose denominators contain the digit 9, is convergent.
Verify that the differential equation $$x^2 y'' + [(n + \frac{1}{2})x + \frac{1}{2}](1-x^2)]y = 0,$$ where \(n\) is a positive integer, has the solution $$y = x^{\frac{1}{2}} e^{-\frac{1}{2}x} L_n(x),$$ where \(L_n(x)\) is the polynomial of degree \(n\) given by $$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}(x^n e^{-x}).$$
Prove that \((\sin x)/x\) is a decreasing function of \(x\) for \(0 < x < \frac{1}{2}\pi\). Assuming that \(F(x) \geq 0\) when \(a \leq x \leq b\) implies that \(\int_a^b F(x)dx \geq 0,\) prove that, if \(m \leq f(x) \leq M\) when \(a \leq x \leq b\), \(m(b-a) \leq \int_a^b f(x)dx \leq M(b-a),\) and deduce that \(I = \int_0^{\pi/3} \frac{\sin x}{x} dx\) lies between \(0.866\) and \(1.048\). Prove further that, if also \(\phi(x) > 0\) when \(a \leq x \leq b\), \(m \int_a^b \phi(x)dx \leq \int_a^b f(x)\phi(x)dx \leq M \int_a^b \phi(x)dx,\) and by making the substitution \(x = 2y\) prove that \(I\) lies between \(0.955\) and \(1\).
The horizontal carriageway of a suspension bridge is suspended from a chain of \(2n+1\) light links by \(2n\) light vertical rods at a constant distance \(a\) apart (so that the links carry in length). The ends of the chain are fixed at points at the same level at a distance \(2na\) apart. If the tension in the \(k\)th link (\(k = 0, 1, 2, \ldots, n-1, n\)) is \(T_k\) and the lengths of the rods attached to its ends are \(y_k\) and \(y_{k+1}\), show that \(y_{k+2} - 2y_{k+1} + y_k = \frac{aW}{T_0},\) and find \(y_k\) and \(T_k\) in terms of \(y_0\) and \(T_0\).
The polar coordinates of a moving particle are \((r, \theta)\). Prove that the radial and transverse components of its acceleration are \(\ddot{r} - r\dot{\theta}^2\) and \(2\dot{r}\dot{\theta} + r\ddot{\theta}\). A particle moves under the action of a force directed towards the origin and of magnitude \(\mu\) per unit mass (\(\mu\) constant). Establish the equations of conservation of energy and moment of momentum: \(\frac{1}{2}(\dot{r}^2 + r^2\dot{\theta}^2) - \frac{\mu}{r} = E, \quad r^2\dot{\theta} = h,\) and prove that the differential equation of the orbit is \(\frac{d^2u}{d\theta^2} + \left(u - \frac{\mu}{h^2}\right) = 0,\) where \(u = 1/r\). If the particle is initially at a point \(A\) at a distance \(c\) from the origin \(O\), and its velocity is at right angles to \(OA\) and of magnitude \(V\), find the conditions that the orbit shall be (i) an ellipse, (ii) an ellipse with its centre between \(O\) and \(A\).
A circular hoop of radius \(a\) rolls along the ground with velocity \(U\). It strikes a horizontal bar fixed at height \(2a/5\), rotates about the bar until it touches the ground again, and then rolls along the ground with velocity \(V\). If the hoop does not slip on or rebound from the bar or the ground, show that \(10ga < 16U^2 < 16ga\) and \(25V = 16U.\)