EXERCISE LXXIII. 1. Simplify the following expressions: (i) (æTM‚‚—(x1+m)m+1 + “√μ2m, (ii) x2-[x2+ y2+(x − y) (x + y) − {x2 - (y2 — x2 + y2)}], 3. Find two numbers in the proportion of 9 to 7 such that the square of their sum shall be equal to the cube of their difference. (ii) (√2+√√3) (√3+ √5) (√5+√√2) 5. If a b c : d, shew that Ja−b: √c-d:: Ja- √b: Nc- √↓d, N and if d be the least term in the proportion, that 6. The sum of n terms of an Arithmetic series is - n (b2 + x2) — n (n − 3) bx, find the 7th term and determine the series. 7. How many permutations can be made of the letters in the word "Alellein" taken all together? How many of those in the word "Ilicifolia"? (x+a)-2(x-1)+n-2 2. Reduce to its lowest terms 7x3-2x2y — 63xy2+18y3 4. In how many different ways is it possible to pay £100 in Napoleons (worth 16s. each), and five-franc pieces (worth 4s. each) ? 5. If ax2+ by x cx2-dy, shew that xx √ÿ; a, b, c, d being constants. The first two terms of a series in harmonical progression are 4 and 3, continue the series to 3 more terms. 7. Find the coefficient of x in the expansion of 一番 15 and expand (2+3x) to four terms. 8. A certain crew, who row 40 strokes per minute, start at a distance equivalent to five of their own strokes behind another crew, who row 45 strokes per minute. Assuming that 5 strokes of the hindmost crew are equivalent to 6 of the foremost, in how long will the former overtake the latter? = (ii) { (x+5) (y +7) = (x + 1) (y−9)+112, 2x+10 = = 3y+1, 4. If x is to y in the duplicate ratio of a to b, and a to b in the subduplicate ratio of a +x to a-y, shew that x+y: x-y :: x+a: x. 5. If in an infinite Geometric series any term be equal to the sum of all that succeed it, shew that the common 6. A boat club consists of 12 members, how many different crews of 8 can be formed, and from how many of these must an individual member be excluded? 7. Expand (-21); write down the 8th term of (5x-4y), and the nth term of (1-x)-3. 8. Transform 5279 from the denary scale to the senary. The number 136 in the denary scale is expressed by 253 in another scale, what is the radix of this scale? 1. Multiply EXERCISE LXXVI. x2+2x1y2+3y2 by x3⁄4 — 2xy” + 3y3r ; and prove the result by division. 2. Extract the square root of (xy) (~* + 45') + 4 +1 +252 (1+2μ3). * 4y 2y (iv) 3x+5y=17 in positive integers. 4. Two men, A and B, sell a quantity of wheat for £28. 88. B sells 4 quarters more than A, and if he had sold the quantity that A sold would have received £10 for it, while A would have got 16 guineas for what B sold. Find the quantity of wheat sold by each, and the rate at which they sold it. Interpret all the solutions. d, shew that 5. If a b c and find x if 4+x: 2x+8 :: 2x−1 : 3x + 2. |