POLITICS. Extra-mural. I. Discuss Aristotle's verdict on democracy. How far are his conclusions applicable to present-day conditions? 2. What are the essential features of Aristotle's theory of education? What is the connection between his theory of education and his general political theory? 3. Compare the political teachings of Stoicism and of primitive Christianity. 4. State briefly the contributions to political theory made by Bodin, Machiavelli, Hobbes, Montesquieu, Burke, and Bentham. 5. Outline Locke's theory of the social contract and estimate its value from the standpoints of (a) historical validity, (b) interpretation of the relation of the state and the individual, and (c) serviceability for democratic agitation. 6. Can any general rule be laid down for the limitation of the functions of the state? JUNIOR MATHEMATICS. (At least 20% must be made on each section). 1. Factor (a) xa—9x2+4x+12, (b) Σ„a3 (b2—c2). Ο 2. If is a cube root of unity, show that (1) (1+2)=-9. 3. (a) How many numbers greater than 100 can be formed with the digits 3, 4, 5, 6, 7? (b) In how many different ways can an elector vote if there are 5 candidates, and he may vote for 1, 2, or 3 of them? 4. (a) Find the sum of 56 terms of the series 1+1.3+1.6+1.9+2.2+. (b) Find in simplest form the first four terms in the expansion (1-2x)-4. 5. Divide a number into two parts such that the sum of their squares may be a minimum. B. 6. Construct a triangle, having given its three medians. 7. Prove that the sum of the perpendiculars from any point within an equilateral triangle to the sides is constant. Explain the truth of the theorem when the point is without the triangle. 8. Show from continuity the relation between the following theorems : (a) the angle in a segment of a circle is constant, (b) the angle between a tangent and a chord is equal to the angle in the opposite segment, (c) an exterior angle of a concyclic quadrangle is equal to the opposite interior angle. 9. (a) The square on a side of a right-angled triangle is equal to the rectangle contained by the hypothenuse and the projection of the side on it. (b) From (a) deduce common expressions for (i) the square on the hypothenuse, (ii) the square on the altitude to the hypothenuse. 10. P is any point in the base BC of an isosceles triangle ABC. Prove AB2-AP2=BP PC. C. 11. A ball of 2 feet radius subtends an angle of 43′. Find its distance from the eye. a b c 12. Prove (a) R=4A · (b) cos (A-B)=cos A⋅ cos B+sin A sin B, and deduce values for cos(A+B), sin(A+B), and sin (A-B). 13. In ▲ ABC, a=7, b=6, c=8. D is the mid point of AB, E is on AC twice as far from C as from A. Find the sides, angles, and circumdiameter of A ADE. |