11. If 1 qr. 7 lbs. of iron cost 3s. 9d., what must be paid for 87 tons 17 cwt. at the same rate? Ans. £1054. 4s. 12. Find the simple interest on £5640 for 3 years at 2 per cent. per annum. Ans. £423. 13. Add together 5, 10, 11, and 11. 14. Subtract from 41. Ans. 3ğ. Ans. 261. 15. Multiply together 1, 2, 7, and 223. 16. Divide 7 by 418. Ans. 18583. Ans. 3888. 17. Add ⚫0056 acres to 84.75 square yards, and give the result in square feet and the decimal of a square foot. Ans. 1006.686 square feet. 18. Subtract 5'017 of a gallon from 1.358 of a quarter, and give the answer in pints and the deeimal fraction of a pint. Ans. 655.16 pints. 19. Multiply 709-52 by 10506. Ans. 74.5421712. 20. Multiply together 4.08, 001, and ·17. Ans. 0006936. 21. Divide 3010-9712 by 36200. Ans. 083176. 22. Divide 064665 by 0000135. Ans. 4790. 23. Express 3 qrs. 22 lbs. 6.4 oz. as the decimal of 1 cwt. Ans. 95. 24. If eggs be bought at the rate of three a-penny, how must they be sold to gain 40 per cent.? Ans. 15 for 7 pence. 25. The simple interest of £956 for 2 years is £119. 10s. Required the rate per cent. per annum. Ans. 5 per cent. GEOMETRY. THURSDAY, 30TH NOVEMBER, 1876. 3.45 P.M. TO 5.30 P.M. 1. If two triangles have two sides of the one equal to two sides of the other, each to each; and have likewise the angles contained by those sides equal to each other; they shall likewise have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz. those to which the equal sides are opposite. 2. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. 3. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel. 4. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle. 5. The diameter CA of the parallelogram ABCD is produced through A to E so that AE is equal to AC, and parallelogram EADF is completed having EA and AD for adjacent sides. Prove that AF is equal to and in the same straight line with AB. Since FD is equal and parallel to ACE, therefore it is equal and parallel to AC; and, therefore, the figure AFDC is a parallelogram. Hence, the side AF is equal and parallel to the side CD, but CD is equal and parallel to AB. Therefore AF is equal to AB and in the same straight line with it. MATHEMATICS (1). SATURDAY, 2ND DECEMBER, 1876. x" mn 1. Prove that x - 1 is divisible by "-1 without remainder; and divide a (b-c) + b3 (c − a) + c3 (a - b) by a+b+c. To prove that x-1 is divisible by "-1 without remainder. n-1 1 n-1 z" z"-1 — 1 be divisible by z-1, 2"-1 is also; or if 2"-- 1 be divisible by z-1 for any one value of the index, it is proved to be so for that index increased by unity; but z-1 and z2-1 are known to be divisible by z-1; therefore by induction 2"-1 is divisible by z-1, whatever positive integer n may be, and similarly "-1 is divisible by x-1 whatever positive integers m and n are. To divide a (b-c) + b3 (c− a) + c3 (a - b) by a+b+c, arrange in descending powers of a. Result a2 (bc) — a (b2 — c2) + bc (b − c). 2. Reduce to their simplest forms: 3. If m and n are two whole numbers, each of which leaves the remainder 1 when divided by 4, shew that m+n leaves the remainder 1 when divided by 4. Let m=4p+1, and n=4q + 1; therefore mn = 16pq+4(p + q) + 1, which leaves the remainder 1 when divided by 4. Prove that every uneven square number leaves either the remainder 1 or the remainder 9 when divided by 16. Every uneven square number may be written in the form (4n+1)=16n" +8n + 1. If n is even, this may be written 16 (n2± √n) + 1. 16 {n2 + 1⁄2 (n − 1)} + 9 or 16 {n2 — † (n + 1)} + 9. We see that these expressions will always give the remainder 1 or 9 when the expression is divided by 16. 4. Find the highest common divisor of 3x10x9x-2x and 2x-7x+2x2+8x. Result x(x-2). Find the least common multiple of x2 - 1, x2-x-2, x2+x−2. Result (x2-4) (x2 — 1). Shew that if d is the greatest common divisor of two ab numbers a and b, their least common multiple is. 5. Extract the square root of (1) 9a"-30ab3ab2 + 25b2 + 5b3 + 16*. (2) 87 - 12 √(42). Results 3a-56+163. 2 √(6) − 3 √√/(7). Prove that a quadratic surd cannot be equal to the sum or difference of two dissimilar quadratic surds. (2) √{(x − 3) (2x − 3)} + √/ {(x − 1) (2x − 5)} = √√/(2). therefore X2+ Y2 - 2XY=8 or X- Y=±2 √(2), and X+Y+2XY=4; |