FINAL HONOURS. POLITICAL SCIENCE. FIFTH PAPER. 1. What are the necessary elements in a legal act? 2. What are the essentials of fraudulent representation? Give illustrations. 3. Discuss the current tendency of court decisions in boycott and picketing cases. To what extent is the use of the injunction in labor disputes justifiable? 4. What recourse has the workingman in Germany, Great Britain, and Canada, respectively, in the contingencies of accident, sickness, and old age disability? 5. Compare and account for the different forms of representative institutions developed in medieval Europe. 6. (a) To what extent is International law to be considered law at all? Discuss the bearing of the recent Hague Conferences on this point. (b) What is the present ruling of International law on (i) contraband, (ii) continuous voyages, (iii) combatant persons, (iv) diplomatic immunities. PRELIMINARY HONOURS. MODERN GEOMETRY. 1. (a) For a pencil of four prove sin AOB sin COD+ etc.-0. (b) A and B, C and D, E and F are the opposite vertices of a quadrilateral. Prove sin A sin B+sin C sin D+sin E sin F=0. 2. The mean centre of the vertices of a triangle with multiples proportional to the opposite sides is the incen tre. 3. (a) Invert (i) two given circles, (ii) three given circles, into equal circles; and point out a case where your method fails. (b) Give a construction for finding a circle to touch three given circles. 4. Find an expression for the polar radius of a triangle in terms of the angles and circumdiameter; and from it show when the radius is real, zero, or imaginary. 5. Two circles touch three others. Show how the centres of similitude and radical axis of the two are related to the radical centre and axes of similitude of the three. 6. A quadrilateral circumscribes a circle, touching it at the vertices of an inscribed one. Show that the four internal diagonals meet in a point, that the two external diagonals are on the same line, and that the point and line are pole and polar. 7. Two tangents and a secant from the same point determine a harmonic system of points on a circle. 8. (a) Prove Pascal's theorem for a concyclic hexagram; and state it for a concyclic (i) pentagon, (ii) quadrilateral, (iii) triangle. (b) State the polar reciprocal of the theorem of (a) (iii). 9. The sides and diagonals of a normal quadrilateral cut any line in a six-point involution. PRELIMINARY HONORS. ALGEBRA I. 1. If f(x) is divisible by (x-a) but not by (x−b) show that the remainder when f(x) is divided by 2. How many different sums of money can be formed with a dollar bill, a 50-cent piece, a quarter, a dime, a nickel and a cent? Find the total value of all the sums. 3. Employ undetermined coefficients (a) to expand x/(x-1) to four terms. (b) to separate (2-x-4x3)/(1—x)2(1+2x2) into its partial fractions; and find the coefficient of x 2+1 in its expansion. nth 4. (a) Prove by induction the rule for finding the convergent to a continued fraction. (b) Find the fraction with smallest numerator and denominator which differs from 1/6 by less than two three-thousandths. 5. (a) Assuming the expansion for et deduce the relation log,x=log,(x−1) (b) If ex3=173′47, find x to 4 decimal places, |