SET I. EUCLID. 1. Any two sides of a triangle are together greater than the third side. If D be a point on the base BC of the triangle ABC, prove that AD is less than the greater of the two sides AB and AC, and if E be a point on either of these two sides, prove that DE is less than the greatest side of the triangle. 2. The complements of the parallelograms which are about the diameter of any parallelogram, are equal to one another. 3. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. 4. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. 5. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc. A triangle ABC is inscribed in a circle of which O is the centre and the arc BC is bisected at D. Prove that the angle ADO is half the difference of the angles ABC and ACB. 6. If from any point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on the line which meets the circle, the line which meets the circle shall touch it. 7. Describe a circle about a given triangle. Given the radius of the circumscribing circle, the length of one of the sides, and the area of the triangle, construct the triangle. 8. Inscribe an equilateral and equiangular quindecagon in a given circle. (3) 9. The sides about the equal angles of triangles which are equiangular to one another are proportionals; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or the consequents of the ratios. The chords AB and CD in the circle ABCD are produced towards B and D respectively to meet in the point E, and through E, the line EF is drawn parallel to AD to meet CB produced in F. Prove that EF is a mean proportional between FB and FC. 10. If four straight lines be proportionals the similar rectilinear figures similarly described upon them shall be proportionals. ALGEBRA. 1. When (m) and (n) are whole numbers, express the result of the division of am by a"; (1) when (m) is greater than (n); (2) when (m) is less than (n), and trace the steps in the theory of indices from which it is inferred that a-" 1 that an represents a. 1 represents and Express by an arithmetical fraction (83 + 4*) × 16. 2. Multiply (x + 2y* + 3z*) by (x* − 2y+ — 3z3). 3. Divide (x + y)2 + (x + z)2 + (y + z)2 + 2 (x + y) (x + z) +2(x + y) (y + z) + 2 (x + z) (y + z) by (x + y + z). 4. Express a2 (c — b) + b2 (a — c) + c2 (b simple binomial factors. 5. Reduce to its lowest terms 6. Prove a) as the product of three (a2 - b2) (x2 - y2) - 4abxy 7. Show how to find a factor which will rationalise the surd quantity ар +b2, where (p) and (q) are any whole numbers. 9. What are the roots of a quadratic equation? If (a) and (6) are the roots of the equation 2 − px + q = 0, and (a1) (B) the roots of x2- Px+Q = 0; express P and Q in terms of (p) and (g). 10. Two workmen (A) and (B) of unequal efficiency, working together are capable of making a drain, which they undertake to do, in 60 days. After working for 20 days A falls ill, and his place is supplied by another workman (C), whose efficiency is the same as that of B, and it is found that the whole time consumed in making the drain is 80 days. In how many days would A or B working alone make the drain? 11. A cubical tank contains 512 cubic feet of water. It was required to enlarge the tank, the depth remaining the same, so that it should contain seven times as much water as before, subject to the condition that the length added to one side of the base should be four times the length added to the other side. Find the sides of the new rectangular base. 12. Define the three progressions, Arithmetical, Geometrical, Harmonical; and show that the three equations following severally determine for the three quantities (a) (b) (c) the conditions of each progression : A watch, which is set right at noon, gains two minutes the first hour afterwards, three the second, four the third, and so on; after what interval will the watch be an hour and a half fast, and what time will it then indicate? 13. Assuming the expression for the number of permutations of (n) things taken (r) together, find the expression for the number of combinations of (n) things taken (r) together. A dealer has for sale 8 bay, 7 grey, and 5 black horses. A purchaser requests that 12 horses, four of each colour, may be sent him; in how many different ways can the dealer execute the order? 14. Assuming the form of the expansion of a binomial when the index is a positive integer, and that f(m) denotes the series 1 + mx + m (m − 1) 1.2 x2+ &c. for any value of (m); show how to obtain a proof of the theorem when the index is a negative integer. = Since (1 1, it would follow that if the two expanded series for (1-x)" and (1 - x)" be multiplied together all the coefficients of the different powers of (x) should vanish separately; show that this is the case for the whole coefficient of x1. PLANE TRIGONOMETRY. 1. What in plane geometry is the limit of the magnitude of an angle? In what sense in analytical Trigonometry is the magnitude of an angle considered unlimited, and either positive or negative? Which of the ordinary trigonometrical functions are limited in magnitude and which unlimited? If sin A = a, express by a general formula all the angles which have (a) for their sine; if cos A = b, prove a2 + b2 = 1. 2. Find sin A and cos A in terms respectively of the sines and cosines of their supplements. Through what portions of the first four quadrants is the sine greater than the cosine in magnitude, and the tangent greater than the cotangent; when are they respectively equal and of the same sign? Find sin 120°, tan 60°, cos 135°. 3. Prove cos (A+B) = cos A cos B - sin A sin B, and deduce from the formula the corresponding expressions for sin (A+B), sin (A — B), and cos 2A. 4. Prove (3.) Sin A. sin (A + 2C) + sin B. sin (B+ 2A) + sin C. sin (C + 2B) 0, if A+B+C = = 180°. 4 = 90°. 5 +tan-1 (8 — a) (8 — c) +tan 8 (8-b) may be properly taken for the measure of an radius angle; of what angle is 1.5708 the circular measure. 1 Prove sine less than 0 — 03, and verify this when 0 = π 6 |