The coordinates \((x,y)\) of a point on a curve are given in terms of a parameter \(t\) by the equations \[ x=at^2, \quad y=bt^3. \] Prove that, if \(t_0 \ne 0\), the tangent at the point \(t=t_0\) cuts the curve again at the point \(t = -\frac{1}{2}t_0\). Prove that there are two lines, each of which is both a tangent and a normal to the curve, and obtain their equations.
Prove that in general three normals (real or imaginary) can be drawn to a parabola from an arbitrary point in its plane. If \(M\) is a fixed point on the parabola and \(P\) a variable point on the normal at \(M\), show that the line joining the feet of the other two normals from \(P\) is parallel to a fixed direction.
A tractor of mass \(M\) is moving against a constant frictional resistance \(R\) up a hillside inclined at an angle \(\alpha\) to the horizontal. If the power \(P\) developed is constant, show that the speed changes from \(V_1\) to \(V_2\) in a time \[ \frac{P}{MB^2} \log\left(\frac{P-MBV_1}{P-MBV_2}\right) - \frac{V_2-V_1}{B}, \] where \(B=g \sin\alpha + R/M\). Find an expression for the distance travelled during this change of speed.
Trace the curve \[ x^5 + y^5 = 5ax^2y^2 \quad (a>0). \] By writing \(y=tx\), or otherwise, prove that the area of the loop is \(\frac{5}{2}a^2\).
Four variables \(u, t, p, v\) are such that any one of them can be expressed as a function of any two of the others. Prove that \[ \left(\frac{\partial u}{\partial t}\right)_p = \left(\frac{\partial u}{\partial t}\right)_v + \left(\frac{\partial u}{\partial v}\right)_t \left(\frac{\partial v}{\partial t}\right)_p, \] where \(\left(\frac{\partial u}{\partial t}\right)_v\), for example, means that \(u\) is expressed as a function of \(t\) and \(v\), and \(v\) is kept constant in the differentiation.
If \(A\) and \(B\) are two fixed points, and \(P\) is a variable point, lying in a fixed plane through \(AB\) and on one side of \(AB\), such that \(\angle PAB - \angle PBA\) has a constant positive value \(\alpha\), prove that the locus of \(P\) is a part of a rectangular hyperbola. Specify which part of the hyperbola is involved. Show that, as \(\alpha\) varies, the centre of curvature at \(A\) of the locus lies on a fixed straight line perpendicular to \(AB\).
Obtain the expressions \[ \frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2, \quad \frac{1}{r}\frac{d}{dt}\left(r^2 \frac{d\theta}{dt}\right) \] for the components of the acceleration of point with the polar coordinates \((r, \theta)\). A bead of mass \(m\) can slide smoothly along a straight rod that is made to rotate in a horizontal plane with constant angular velocity \(\omega\) about one end \(O\). Initially \(r=a\) and \(\frac{dr}{dt}=0\), where \(r\) is the distance of the bead from \(O\). Express in terms of \(t\) the reaction of the rod on the bead. Verify that the work done in rotating the rod is equal to the increase of the kinetic energy of the bead.
Taking as your starting point the triangle of forces, develop the theory of the composition of parallel coplanar forces (proving all the theorems you need) so as to lead up to the following theorem, the proof of which should be given: \(P, Q, \dots\) are fixed points in a plane. Parallel forces of given magnitudes \(p, q, \dots\) act in the plane through \(P, Q, \dots\) respectively. Then in general the forces have a resultant of magnitude \(p+q+\dots\) which passes through a point \(G\) which is independent of the direction of the forces.
When \(x=a\), the functions \(f(x)\), \(g(x)\), \(f'(x)\) and \(g'(x)\) have the values \(0, 0, b\) and \(c\) respectively. Prove that, if \(c \ne 0\), \(f(x)/g(x)\) tends to \(b/c\) as \(x\) tends to \(a\). Evaluate the limits as \(x\) tends to \(0\) of \[ \text{(i) } \frac{3^x - 3^{-x}}{2^x - 2^{-x}}, \quad \text{(ii) } \frac{1}{x} \int_0^x \sqrt{(3t^2+4)} \, dt. \]
Prove that, in general, two conics of a given confocal system pass through an arbitrary point \(P\) of the plane, and that they cut orthogonally at \(P\). If the tangents at \(P\) to the two confocals through \(P\) meet one of the axes of the confocal system in \(X, X'\), and the other axis in \(Y, Y'\), prove that, for all positions of \(P\), the circle \(PXX'\) belongs to a certain coaxal system and the circle \(PYY'\) to the orthogonal coaxal system.