A piece of uniform wire is bent into the shape of an isosceles triangle \(ABC\) in which \(AB=AC\). The triangle hangs in a vertical plane with \(BC\) in contact with a rough peg. Shew that the triangle will rest in equilibrium whatever point of \(BC\) is in contact with the peg provided that the coefficient of friction \(> 2\tan\frac{A}{2}(1+\sin\frac{A}{2})\).
A motor bicycle with side car weighing 3 cwt. attains a speed of 20 miles per hour when running down an incline of 1 in 20 without the use of the engine. But up the same incline the greatest speed that can be attained is 40 miles per hour. Assuming that the resistance varies as the square of the velocity, determine the horse-power developed by the engine.
Find the direction in which a particle must be projected from a point with given velocity in order that the range on an inclined plane through the point may be a maximum. Prove that, if the difference in level between two points \(A\) and \(B\) is \(L\), the velocity of projection from \(A\) in order that \(B\) may be just within range of \(A\) is \(\sqrt{\{g(AB \pm L)\}}\) according as \(B\) is above or below \(A\).
A pencil of conics \(S\) passes through the four fixed points \(A_1, A_2, A_3, A_4\). Shew that the locus of the poles of a fixed line \(l\) with respect to the conics \(S\) is a conic \(C\) passing through the diagonal points of the quadrangle \(A_1A_2A_3A_4\). If \(A_1A_2\) meets \(l\) in \(A_{12}\), and \(A_{12}'\) is the harmonic conjugate of \(A_{12}\) with respect to \(A_1, A_2\), prove that \(C\) passes through the six points such as \(A_{12}'\). Shew also that \(C\) passes through the double points of the involution determined on \(l\) by the conics \(S\). What does \(C\) become when \(A_1, A_2, A_3, A_4\) are the vertices and orthocentre of a triangle and \(l\) is the line at infinity?
Shew that by suitable choice of homogeneous coordinates any conic can be represented by the parametric equations \[ x:y:z=t^2:t:1. \] Two conics \(S\) and \(S'\) have the two points \(A\) and \(B\) in common. \(P\) is a variable point on \(S'\) and the lines \(PA, PB\) meet \(S\) again in \(Q, R\). Prove that the line \(QR\) envelopes a conic.
Prove that the polar lines of a fixed line with respect to a system of confocal quadrics generate a paraboloid, which touches each of the three principal planes of the confocal system.
Give definitions of the tangent, principal normal, binormal, curvature (\(1/\rho\)), torsion (\(1/\sigma\)), and centre of curvature of a twisted curve, explaining carefully any conventions of sign involved in your definitions. Shew that, if from a fixed point \(O\) lines \(OT, ON, OB\) are drawn parallel to the positive directions of the tangent, principal normal, and binormal, respectively, at a point \(P\) of a curve \(C\), then if \(P\) moves along \(C\) with unit velocity the triad \(OTNB\) has at any instant an angular velocity whose components about \(OT, ON, OB\) are respectively \(1/\sigma, 0, 1/\rho\). A curve \(C\) drawn on the surface of a right circular cone of semi-vertical angle \(\alpha\) cuts the generators at a constant angle \(\beta\). Shew that the curvature and torsion of \(C\) at a point \(P\) at distance \(r\) from the axis of the cone are given by \[ \frac{r}{\rho} = |\sin\beta|\sqrt{1-\cos^2\alpha\cos^2\beta}, \quad \frac{r}{\sigma} = \pm \cos\alpha\sin\beta\cos\beta, \] explaining the ambiguity in the second formula. Shew that as \(P\) describes \(C\), the centre of curvature describes a curve lying on the surface of a right circular cone and cutting the generators at a constant angle.
Establish Newton's formulae for expressing the sums of powers of the roots of an equation \[ x^n+a_1x^{n-1}+\dots+a_n=0 \] in terms of the coefficients. Shew that, if \(1 \le m \le n\), then the sum of the \(m\)th powers may be expressed in the form \[ s_m = (-1)^m \begin{vmatrix} a_1 & 1 & 0 & \dots & 0 & 0 \\ 2a_2 & a_1 & 1 & \dots & 0 & 0 \\ 3a_3 & a_2 & a_1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ (m-1)a_{m-1} & a_{m-2} & a_{m-3} & \dots & a_1 & 1 \\ ma_m & a_{m-1} & a_{m-2} & \dots & a_2 & a_1 \end{vmatrix}. \]
A function \(f(x)\) satisfies, in the interval \((\alpha,\beta)\) (\(\alpha<\beta\)), the conditions \[ f(\alpha)<0, \quad f(\beta)>0, \quad f''(x)>0 \quad (\alpha \le x \le \beta). \] Shew that, if the equation \(f(x)=0\) has a single root \(\xi\) in \((\alpha,\beta)\), and that if \(\alpha', \beta'\) are defined by \[ \alpha' = \frac{\alpha f(\beta) - \beta f(\alpha)}{f(\beta)-f(\alpha)}, \quad \beta' = \beta - \frac{f(\beta)}{f'(\beta)}, \] then \(\alpha < \alpha' < \xi < \beta' < \beta\). Explain the geometrical significance of these results.
The function \(y=f(x)\) is continuous in the interval \(a \le x \le b\) (\(a < b\)), and increases (in the strict sense) from \(A\) to \(B\) as \(x\) increases from \(a\) to \(b\). Shew that there is a unique (inverse) function \(x=F(y)\), defined in \(A \le y \le B\), and satisfying the equation \[ f[F(y)]=y \] identically throughout this interval. Shew further that \(F(y)\) is a continuous strictly increasing function in \(A \le y \le B\). Shew that, for every \(a >1\), the equation \[ \tan x = ax \] has a unique root in the interval \(0 < x < \frac{1}{2}\pi\), and that this root is a continuous function of \(a\).