12. When is an hyperbola said to be equilateral? Find the eccentricity of such an hyperbola. In a rectangular hyperbola, the distance of any point of the curve from the centre is a mean proportional between the distances of the same point from the foci. XCVI. 1. Add together 33, 4%, and 57. 2. Subtract 145 from 21. 3. Multiply together 213, 45, 18, and 187. 4. Divide 13 by 11. 5. Add together 15.7, 4·092, 0075, and 3.6505. 6. Subtract 72-0975 from 73.332. 7. Multiply 22.48 by 1.125. 8. Divide 917-3245 by 65.29. 9. Express 1 oz. 5 grs. as a decimal fraction of 3 oz. 4 dwts. 16 grs. 10. Add together 13 of 14, 114 of 51, 14 of 9%, and 12. Multiply together 5, 14 of 1, and 21 of 55. 13. Divide 27 by fr 29 14. Add together 1.1375 of a fathom, 875 of a yard, 2.965 of a foot, and 9.75 of an inch, expressing the answer in feet and a decimal fraction of a foot. 15. Multiply together 2.56, 1-75, and 000125. 16. Divide 273 by ⚫0028. 17. Reduce 15 cwts. 2 qrs. 21 lbs. to the decimal of 2 tons 10 cwts. 18. Divide 10 miles 1 furlong 6 poles by 22. 19. An estate of 625 acres 1 rood 10 poles is bought for £7,878 18s. 9d. What is the average price per acre? 20. Find by practice the cost of 3,257 tons of coals, at £1 3s. 74d. per ton. 21. In what time will the simple interest on £2,987 10s. 6d. amount to £248 19s. 21 d., at 3 per cent. per annum? 22. If 7 horses draw 5 tons along 56 yards on a road in 10 minutes, how many horses will be required to draw 11 tons along 50 yards in 5 minutes? 23. Find the difference between the simple and compound interest on £933 6s. 8d. in 2 years at 2 per cent. per annum. 24. Find the cube root of 10.218313. If the diagonal of a square is 3 ft. long, find the length of each side, correct to a thousandth of an inch. 25. Compute by means of the tables the value of 26. If a sum of £1,000 is lent on the condition that it shall bear compound interest at the rate of 5 per cent. per annum for the first 5 years, 10 per cent. per annum for the next 5 years, and afterwards 15 per cent. per annum; what will the debt amount to in 20 years? XCVII. 1. The product of two algebraical quantities having like signs is positive, and having unlike signs negative.' State briefly the steps by which this rule is obtained, and show its arithmetical correctness when in the product obtained for (a-b)x(c-d), a=12, b=9, c=10, d=4. If x and y differ in value, explain why Show how the rule of signs stated above leads to 2. Multiply (1+x+2x2)2-(1-x-2x2)2 Find the product of x2-(a+b) x+ab by x2+(a-b) x-ab, and examine what the product becomes if in it either a or b be substituted for x. 3. Divide x4-2bx3 — (a2 — b2) x2+2a2bx—a2b2 by x2-(a+b) x+ab. Reduce to its simplest form (x−y) (y−z)+(x−y)(z−x)+(y − z) (z−x) 4. Ifm be a whole number, prove that x+y is divisible by xm+ym, when m is odd. Write down the three last terms of the quotient of 1 x+y divided by x2+1+2+1, and examine how many terms the quotient will contain. 5. Find the fraction in its lowest terms which is the square root of 6. Solve the following equations: (1) (x-4)(x−8)(x−10)=(x−6)(x−7) (x−9). (3) 2cx2-abx+2abd-4cdx. (4) (x+y)2-z2=651 x2-(y+2)2=13 x+z-y=9 7. Form a quadratic equation whose roots are-5+6√ − 1. If x2-px+q=0 and x2-p1x+9=0 have a common root, determine it, and in that case show that— (9—91)2=(p—p1) (P1q—pq1). 8. A bill of £63 5s. was paid in sovereigns and halfcrowns, and the number of coins used in the payment was 100, how many sovereigns were paid, and how many halfcrowns ? 9. A cask A is filled with 50 gallons of water and a cask B with 40 gallons of brandy; C gallons are drawn from each cask, mixed, and replaced. The same operation is repeated. Find C when there are 87 gallons of brandy in A after the second replacement. 10. Find the sum of n terms of the series (a−b)+(a−3b)+(a−5b), &c., without assuming a formula for the sum of an arithmetic series, and apply the result to find the sum of 10 terms of 76+70+64 +, &c. 11. The sum of the two first terms of a geometric series is 12, and of the three first terms is 39, find the series, and determine if the condition is satisfied by more than one series. Find the sum of the infinite series 1+r+(1+b) r2+(1+b+b2) μ3 +, &c., r and b being proper fractions. 12. Show how to transfer a whole number from one scale of notation to another, and prove that if r be the radix of the scale, the number itself will be divisible by r+1 if the difference between the sum of the digits in the odd places and the sum of the digits in the even places be divisible by r+1. 13. Find the number of different permutations of n letters taken altogether when one letter occurs p times and another q times in each permutation. If the n letters contain only a and b, and n be even, show that the number of permutations will be greatest 14. Assuming the form of the binomial theorem, find the greatest term of (1+x)" when n is a positive integer and x is also positive. Apply the result to the determination of the greatest term of (1+5), and express the value of the greatest term. Show by actual multiplication that the whole coefficient of x in the product of the expansions of (1+x)" and (1-x)" is equal to the coefficient of x1 in the expansion of (1−x2)». XCVIII. 1. Define the degree and the unit of circular measure, and show how to pass from one of these units to the other. If the radius of a circle be 25 feet, find to four decimal places the length of the arc subtending an angle of 3o, taking to be 3-1416. 2. Define the principal trigonometrical ratios, and trace 3. Prove that sin (A+B)=sin A cos B+cos A sin B where A and B are positive angles each less than 90°. Assuming that the formula holds for all values of the angles, deduce that for cos (A-B). 4. Find cos 3A and tan 3A in terms of cos A and tan A respectively. 5. Prove that for all values of A (1) 2 cosec 4A +2 cot 4A=cot A-tan A; |