Donnez! pour Donnez! afin qu'un jour, à votre heure dernière, ARITHMETIC. 1. Find the cost of 11 tons 13 cwt. 3 qrs. 5 lbs. at £7 5s. 10d. per ton. 4. If the compound interest on £1000 for 4 years amounts to £169 17s. 2.0544d., find the rate per cent. per annum. 5. In what proportion must tea at 1s. 4d. per lb. be mixed with tea at 10d. per lb., so that a profit of 36 per cent. will be realised, if the mixture is sold at 1s. 5d. per lb.? 6. The average price of wool this season is 50 per cent. greater than it was last season, but the quantity has fallen off 15 per cent. The total value of this season's clip is £2,000,000 more than last season's. Find the value of this season's clip. 7. The capital of a certain mine consists of 20,000 £1 shares. The mine contains stone yielding 1 oz. 5 dwt. of gold to the ton, and its expenses amount to £1000 per annum, and £2 9s. 5d. per ton of stone treated. If the gold is worth £3 15s. per oz., how many tons of stone must be treated per annum in order that dividends of 3d. per share per fortnight may be paid out of profits? 8. By how much will a man's income be affected if he sells out £1375 of 3 per cent. funded stock at 102 and invests the proceeds in shares costing £25 12s. 6d. per £12 10s. share, on which dividends at the rate of 9 per cent. per annum are paid? 9. Find, correct to a square inch, the area of a rectangular field whose sides measure 9 chains 71 links and 7 chains 13 links respectively. Find 10. A tunnel is 100 yards long and 24 feet wide. Its cross section consists of a rectangle 24 feet wide and 13 feet high, surmounted by a semicircle of 12 feet radius. the cost of excavating it at 4s. 6d. per cubic yard. 11. A and B ascend a mountain and return by the same track. A walks at the rate of 3 miles an hour up the mountain, and at the rate of 4 miles an hour down; B walks at the rate of 2 miles an hour up and at the rate of 5 miles an hour down. B starts 20 minutes before A and reaches the summit 10 minutes after him. Which reaches home first and by how long? ALGEBRA. 1. If w=3(−1+√−3) prove that w2=−(1+w), and find the continued product of x+y+z, x+wy+w2z, xw+3y+wz. Express a+b3—2 √/2.c3 +3 √/2.abc as the product of two real factors. 3. Solve the equations (i.) qзt—b a- ·b -x [Find two of the roots by inspection.] (ii.) ax2+by+cxy=ax+cy+b=bx2+ay2+cxy. 4. A basket of oranges is emptied by one person taking half of them and m more, a second person taking half the remainder and n more, and a third person taking half the remainder and p more. How many oranges did the basket contain? 5. If a, ẞ are the roots of the equation x2+pa+q=0, find the equation whose roots are a+ß2 and 1 1 +. a2 B B2 7. S is the sum of a geometrical progression and R is the sum of the reciprocals of the terms of the progression. Prove S that the product of the first and last terms of the R given progression. 8. How many words can be formed from the 21 consonants and 5 vowels of the alphabet, each of which contains 4 consonants and 2 vowels? 9. If a, b, c be three consecutive coefficients in the expansion of a power of (1+a), prove that the index is 10. Prove that log blog,c=logac. 2ac+b(a+c). b'-ac Given log,9=a, log,5=b, log,7=c, find the logarithms to base 10 of 5, 6, 7, 8, 9. GEOMETRY. 1. Describe a parallelogram that shall be equal in area to a given triangle, and have one of its angles equal to a given angle. 2. The side AB of a triangle ABC is greater than the side AC; shew that the median AX makes a smaller angle with AB than it does with AC. 3. Enunciate and prove a proposition of the second book equivalent to the algebraical equation a2+b2=2ab+(a—b)2. 4. A, B and C are three fixed points in a straight line, and P is any other point in this line; prove that PA.BC+PB.CA+PC.AB=0, due regard being had to the representation of opposite directions by the signs + and 5. A straight line is drawn from one angular point of a triangle to bisect the opposite side; prove that the sum of the squares on the sides containing this angle is equal to twice the sum of the squares on this bisector and on half the side bisected. Prove also that in a triangle the shortest median bisects the longest side. 6. The diameter is the greatest chord in a circle, and of others that which is nearer the centre is greater than one more remote. 7. If two chords of a circle cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. 8. The bisector of the vertical angle A of a triangle ABC meets the circumscribing circle in D, and DK is drawn perpendicular to AB; shew that AK is equal to half the sum of the two sides containing the angle A. 9. AB, CD are two parallel straight lines cut by two transversals OAD, OBC; shew that the circles circumscribing OAB, OCD will touch one another at O. 10. Circumscribe a circle about a given triangle. TRIGONOMETRY. 1. What is meant by a radian? A circular cylinder is found by measurement to be 1.94 inches in diameter and 6.09 inches in circumference. Assuming these estimates to be correct, find in degrees, correct to one decimal place, the value of a radian. 2. Prove the following formulae, cos(A+B)=cos A cos B-sin A sin B, cos A+cos B=2 cos(A+B) cos (A-B). 3. Assuming the values of the sine and cosine of 30° and 45°, find those of 15°. Also find the value of cos 973°, explaining the reason for the sign. 4. Prove that a2b2+c2-2bc cos A. ADEB and AFC are a square and equilateral triangle c+ a = (c—a) (c+a—b) a+b × tan(3B+A)=(a−b)(a+b—c) tan (C+B). 6. Discuss the ambiguous case in the solution of triangles. If b, c, B are given and a, and a are the two values of a, prove that a3a3=2c(c2 cos 3B+362 cos B). 7. If m=cosec 0―sin 0, n=sec 0-cos show that m 3+n3=(mn) ̄3 ̧. where P1, P2, P3 are the perpendiculars from the angular points on the opposite sides. 9. A person standing on the bank of a river observes the elevation of the top of a tree on the opposite bank to be 51°, and when he retires 30 feet from the edge he finds the elevation to be 46°. Find the breadth of the river, having given log 3 log 1.55823.192631, L sin 46°=9.856934, |