(a) Prove that the three lines joining the mid-points of opposite edges of a tetrahedron meet in a point. (b) \(ABCD\) is a tetrahedron in which the edge \(AD\) is at right angles to the edge \(BC\), and the edge \(BD\) to the edge \(CA\). Prove that the edge \(CD\) is at right angles to the edge \(AB\). Prove also that in this case the perpendiculars from the vertices on to the opposite faces meet in a point.
Shew that the locus of a point \(P\) whose rectangular Cartesian coordinates are given by \[ x:y:1 = at^2+2bt+c : a't^2+2b't+c' : a''t^2+2b''t+c'', \] where \(t\) is a variable parameter, is in general a conic. Examine the particular cases in which (i) the determinant \[ \Delta = \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} \] vanishes, (ii) all first minors of \(\Delta\) vanish. Find the equation of the tangent at the point \(t\), and prove that the coordinates \((\xi, \eta)\) of the pole of the line joining the points \(t_1\) and \(t_2\) are given by \[ \xi:\eta:1 = at_1t_2+b(t_1+t_2)+c : a't_1t_2+b'(t_1+t_2)+c' : a''t_1t_2+b''(t_1+t_2)+c''. \]
\(g(x), h(x)\) are given polynomials, of degrees \(m, n\) respectively (\(m \ge n\)). Prove that the degree of a polynomial (not vanishing identically) which can be written in the form \[ \text{(1)} \qquad G(x)g(x) + H(x)h(x), \] where \(G(x)\) and \(H(x)\) are polynomials, can be as small as, but not smaller than, a definite integer \(\nu (\ge 0)\). Prove also that the polynomial \(\chi(x)\), which is of the form (1) and of degree \(\nu\), and in which the coefficient of \(x^\nu\) is unity, is unique. Prove that \(\chi(x)\) is a common factor of \(g(x)\) and \(h(x)\). Prove also that in the expression of \(\chi(x)\) in the form (1), \(G(x)\) and \(H(x)\) can be found of degrees less than \(n\) and \(m\) respectively.
A function \(\psi_n(x)\) is defined by the equation \[ \psi_n(x) = \frac{d^n}{dx^n}\frac{\sqrt{x}}{1+x} = \frac{\sqrt{x}}{(x+1)^{n+1}}\Psi_n(x). \] Shew
The portion of the curve \(y=f(x)\) included between the ordinates \(x=a\) and \(x=b\) (\(a < b\)) is rotated about the axis of \(x\). Prove that the volume of the surface of revolution so obtained is \[ V = \pi \int_a^b \{f(x)\}^2 dx, \] and that the area of the curved surface is \[ S = 2\pi \int_a^b f(x) \{1+(f'(x))^2\}^{\frac{1}{2}} dx, \] where \(f'(x) = \dfrac{d}{dx}f(x)\). Find the volume and area of the surface of the solid obtained by rotating the portion of the cycloid \[ x=a(\theta+\sin\theta), \quad y=a(1+\cos\theta) \] between two consecutive cusps about the axis of \(x\).
A coplanar system of forces acts on a rigid body. Shew that the system is equivalent either to a single force, or to a single couple, stating clearly the assumptions which are needed in the course of the proof. Shew further that the work done by the system in an arbitrary infinitesimal displacement of the body is equal to the work done by the equivalent force or couple.
A uniform lamina in the shape of an equilateral triangle \(ABC\) of side \(a\) is free to move in a vertical plane, the edges \(AB, BC\) resting on two smooth pegs \(P, Q\) at the same level and at distance \(b\) apart. If \(G\) is the centroid of the lamina, shew that, whatever the inclination of the lamina, \(BG\) passes through a fixed point. Shew also that the height of \(G\) above this fixed point when \(BG\) makes an angle \(\theta\) with the vertical is \[ (a\cos\theta-b-b\cos 2\theta)/\sqrt{3}. \] Hence shew that the position of equilibrium with \(BG\) vertical is stable if \(a < 4b\), and unstable if \(a > 4b\). Is the equilibrium stable or unstable if \(a=4b\)?
A particle moves in a straight line under the action of a given (variable) force. What physical quantity is represented by the area lying under the curve and bounded by two ordinates in the several cases when the abscissae and ordinates represent graphically (1) time and velocity, (2) time and force, (3) distance and force? A cable is used for raising loads, the greatest tension that the cable will bear being \(W\) tons weight. Shew, by consideration of the time-velocity graph, that the least time in which a load of \(W'\) tons can be raised through \(h\) feet from rest to rest by means of the cable is \(\sqrt{\left\{\dfrac{2h W}{g(W-W')}\right\}}\) seconds. The weight of the cable itself is negligible.
Establish the formulae \(dv/dt, v^2/\rho\), for the tangential and normal components of acceleration of a point moving on a given curve, where \(v\) denotes the velocity of the moving point, and \(\rho\) the radius of curvature of the curve. A light inextensible string \(AB\) of length \(l\) has the end \(A\) attached to a point of the surface of a fixed cylinder, whose cross-section is a simple closed oval curve whose intrinsic equation is \(s=f(\psi)\). Both \(\psi\) and \(s\) vanish at \(A\), and \(s\) increases always with \(\psi\). A particle of mass \(m\) is attached to \(B\), and is acted on by a constant force \(mc\) at right angles to the string, so that the string wraps itself round the cylinder, the whole motion being in a plane at right angles to the generators. Find the relation connecting the time with the inclination \(\psi\) of the straight part of the string to the tangent at \(A\), and shew that the tension of the string is \[ m \frac{u^2+2c\{l\psi - F(\psi)\}}{l-f(\psi)}, \] where \(u\) is the velocity of the particle when \(\psi=0\), and \(F(\psi)=\int_0^\psi f(x)dx\).
A particle of mass \(m\) moves in a plane, and is attracted towards a fixed origin \(O\) in the plane with a force \(mn^2r\), where \(r\) denotes distance from \(O\). It is projected from the point \((c,0)\), the axes being rectangular, with velocity \(nb\) and in a direction inclined at an angle \(\theta\) to the axis \(Ox\). Shew that the path of the particle is the ellipse \[ b^2(x\sin\theta-y\cos\theta)^2 + c^2y^2 = b^2c^2\sin^2\theta. \] Shew further that the points of the plane which are accessible by projection from the given point with the given velocity lie within the ellipse \[ \frac{x^2}{b^2+c^2} + \frac{y^2}{b^2} = 1. \]