9. If 250 men, in 4 days of 11 hours each, can dig a trench 190 yards long, 3 wide, and 2 deep; in how many days of 9 hours each, can 194 men dig a trench of 350 yards long, 24 wide, and 14 deep? 10. Having given logarithm 2 = .30103, and logarithm 3 = 4771213, determine the number of years in which L.200 will amount to L.500, at compound interest at the rate of 8 per cent. ALGEBRA. 1835. 1. Find the square of a + x, and of a bers. x, and illustrate by num 2. State what geometrical truths these two operations establish. 3. Divide TM3—‹3 by x—y. 4. Reduce 81 to its simplest terms, and add √27 to √48. to find x. 6. Given √x+√a+≈= Na + 7. Given x2 + Ja3 6x, to find x. 8. Bought a certain number of sheep for L.24, lost 3 of them, and sold the remainder at an advance of 2s. a-head, by which I gained on the whole 6s. Required the number of sheep. 1836. 1. Suppose a 3, what are the values of 5a and as? 2. What is the difference between 3 and x? between 4 and 4? and between x and x} ? 3. Extract the square root of 94 + 12x3y — 20x2y2 — 16xy3 +16y+; 4. Expand (a + b)5, and find four terms of (a - b)m. 5. The logarithm of 2301030 and of 3=477121. Find from these the logarithms of 45 and 144. 6. How many values may ≈ have in the equation x + 3x1 - 2x3 7x=240. 8. Are the roots of these equations possible or not? 9. The sum of L.43 was paid in guineas and crowns, and the whole number of coins used was 60. How many of them were guineas, and how many crowns? 10. The first term of an arithmetical progression is 2, and the sum of 10 terms is 245. What is the common difference of the terms? 2. Find the Square Root of 2 + y2 by the Binomial Theorem. 3 8 5. Given 2+√ (2x2 — 5x + 1) − 5 (x + 1) = 0, to find the value of x. 7. xn+yn xn and xy yn xy 8y-9x 8y-9x 15 3x2 - 2xy' what kind of number must n be (odd or even) that these quantities may be integers? ma na 8. If m and n are any two unequal numbers, show that + n must be greater than 2a. m 9. Find the sum of 749 terms of the arithmetical series, 2, 34, 5, &c. 10. Find the sum of the series 1, 4, 3, &c. to infinity. 11. Given the Logarithm of 18 = 1.225273, and the Logarithm of 3 = 0·477121: find from these the Logarithm of 15. 12. Given + 30 = 31x ; find all the values of x. 13. A and B can do a piece of work in 16 days, A and C can do it in 18, and B and C can do it in 20: How many days will they take when they all work together? 14. The sum of two numbers is 8, and the sum of their fifth powers is 15868: What are the numbers? 2. Explain distinctly in words the meaning of a", a−m, a' m 5. Given 1 + √3x 1 3x 1 = 2 √3x + 1' x ++*= 85, and 2 x+y=85; find x, y, and z. 3xy 2y9 85, also + = and 3x 6x 7. Given 8. Find two numbers such that their product added to the sum of their squares shall be 304, and the square of their product added to their fourth powers be 34048. 9. The first term of a decreasing Arithmetical series is 10, the common difference, find the sum of twenty-one terms. 10. Find the sum of the series 2,,, g, &c. to infinity. 3. In attempting to arrange a number of counters in the form of a square, there was a deficiency of 4; and on diminishing the side of the square by 1, there was an excess of 11 counters: required the number of counters. 4. There is a certain number consisting of three places of figures, and the sum of its digits is 9; also, half the digit in the place of units, a third of that in the place of tens, and half of that in the place of hundreds, are together equal to 4; lastly, if to the number itself 198 be added, the sum is a number consisting of the digits inverted: required the number. 5. A sum of L.1000 was to have been divided among several per sons who claimed equal shares: but two new claimants having appeared, each of the former found that he must now receive L.25 less than he expected: required the number of claimants at the last. State also, if you can, the meaning of the negative solution. 6. Find a value of x in the cubic equation 23 6x11x-12= 0, by the formula for the solution of cubics; and then find the other two roots by reducing the given equation. 1840. 1. Reduce √20 + √12 to an equivalent fraction with a rational de nominator. 2. Divide a +3 by n-3; and (no—m2)} by n−m. 3. The Logarithm of 2 is 3010300; find the Logarithm of 25. Am 5. Extract the cube root of (a + b) 6m x3 +6 cao (a + b) 1m x2 + 6. (1.) Given x y + x y2 = 12; also x + xy3 = 18; find x and y. (2.) Given x + y: a :: x-y: b; and x2-y2 = c; find x and y. 7. Find the sum of the series} + √ } + { √ }+&c. in infinitum. Also of the series 3-1+- &c. to infinity. |