Problems

Give an account of some method of finding the rectilinear asymptotes of a curve whose \(x,y\) equation is given and show how to determine in what manner the curve approaches an asymptote. Consider the cases

1. [(i)] \(y(x^2-1)=x^3+x^2+1\),
2. [(ii)] \(x(y-x)^2-3y(y-x)+2x=0\).

Give a general account of the motion of a projectile, neglecting air resistance. Consider the possible paths through a given point \(P\) when the velocity at the point of projection \(O\) is given in magnitude, and the envelope of the paths when the direction is varied for a given magnitude. A fort is on the edge of a cliff of height \(h\). Show that there is an annular region in which the fort is out of range of the ship, but the ship is not out of range of the fort, of area \(8\pi kh\), where \(\sqrt{2gk}\) is the velocity of the shells used by both.

A particle on a smooth table is attached to a string passing through a small hole in the table and carries an equal particle hanging vertically. The former particle is projected along the table at right angles to the string with velocity \(\sqrt{2gh}\) when at a distance \(a\) from the hole. If \(r\) is the distance from the hole at time \(t\), prove the results

1. [(i)] \(2\left(\frac{dr}{dt}\right)^2 = 2gh\left(1-\frac{a^2}{r^2}\right)+2g(a-r)\),
2. [(ii)] the lower particle will be pulled up to the hole if the total length of string is less than \(a+h+\sqrt{ah+\frac{1}{2}h^2}\),
3. [(iii)] the tension of the string is \(\frac{1}{2}mg\left(1+\frac{2a^2h}{r^3}\right)\), \(m\) being the mass of each particle.

Give a short account, without proofs, of the principal properties of the three transformations: (1) conical projection, (2) inversion, (3) reciprocation. Discuss the chief characteristics in which these transformations are alike or different, e.g. that in (1), (2), but not in (3), a point corresponds to a point, and that in (1) and (3), but not generally in (2), a conic corresponds to a conic.

\(ABCDE\) is a structure consisting of 7 equal light rods lying in a plane and freely jointed at their ends. It is supported at \(A\) and \(D\) so that \(AD\) is horizontal and it supports a load \(W\) at \(E\). Draw a force diagram and determine the stresses in the various members. Find the changes in the stresses in the various members when extra rods \(AC, BD\) are inserted in the positions shown by the dotted lines in the diagram and are so adjusted that they each take a thrust equal to \(W\). % Diagram of a 7-rod structure is present in the original document. % It shows a symmetrical roof-truss like structure. % Vertices are A, D on the bottom, horizontal. % From A, rods go to B and E. From D, rods go to C and E. % Rods connect B to C, B to E, and C to E. % The shape consists of three triangles: ABE, BCE, CDE. % ABE and CDE are congruent isosceles triangles. BCE is isosceles. % The load W is at E. Dotted lines show extra rods AC and BD.

Each edge of a tetrahedron \(OPQR\) is equal to the opposite edge, and \(A, B, C\) are inverse to \(P, Q, R\) with respect to \(O\). Prove that the foot of the perpendicular from \(O\) on the plane \(ABC\) is the incentre \(I\) of the triangle \(ABC\).

\(P\) is a point inside a quadrilateral \(ABCD\) such that the sum of the areas \(PAB, PCD\) is constant. Prove that the locus of \(P\) is a straight line.

Find how many conics (not necessarily real) can be drawn to pass through \(m\) given points and touch \(5-m\) given straight lines, in the six cases when \(m = 5, 4, 3, 2, 1, 0\). It is assumed that no three given points are collinear, no three given straight lines concurrent, and that no given point lies on any of the given straight lines. Give geometrical constructions for the points of contact in the two cases \(m=4, 3\).

A uniform rectangular door of depth \(a\) weighing \(W\) lbs. slides in vertical grooves and is supported by a vertical chain which is attached at a distance \(c\) from the centre line of the door. The distance apart of the grooves is slightly greater than the width of the door, the coefficient of friction between the door and the grooves is \(\mu\) and \(\mu\) is less than \(a/2c\). Show that the difference between the tensions of the chain when the door is being raised and lowered slowly is \[ \frac{4a\mu c W}{a^2 - 4\mu^2 c^2}. \]

A variable straight line through the centre \(O\) of a regular hexagon \(ABCDEF\) meets \(AC\) in \(G\) and \(AE\) in \(H\). Prove that \(BG, FH\) meet on the circle circumscribed to the hexagon.