4. If x2-yz=a2, y2—zx=b2, z2-xy=c2, shew that a2x+b2y+c2z - (a2 +b2 + c2) (x+y+z)=0. 5. Find the product of the first 9 terms of the series The first four terms of a geometrical series are together equal to 45, and the first six terms to 189; find the series. 6. Find the coefficient of x20 in the expansion of 7. Express 23 125 and 957 in the septenary scale, and the latter also in the scale whose radix is -7. 8. Out of a sphere of clay whose diameter is 8 feet, two spheres are formed with diameters 5 and 6 feet respectively; having given that the volume of a sphere varies as the cube of its radius, find the diameter of the sphere which can be formed of the clay which is left. EXERCISE LXXXVIII. 1. Extract the square root of (i) ax1-2ax +3-2α-x+a-1x, 4. (iii) (-12xy=432, There is a number composed of two figures, of which the figure in the unit's place is triple of that in the ten's; if 36 be added to the number the sum is expressed by the same digits reversed. What is the number? 5. In a Geometrical Progression, if S1, S2, S3 are the sums to n terms, 2n terms, 3n terms respectively, find the relation between S1, S and Sq. 6. In what numbers should 20 men be combined, so as to form the greatest possible numbers of different companies? In how many of these will the same pair of men be found together? 7. Shew that the difference between any number in the denary scale and that formed by reversing the digits is divisible by 9. Extend this theorem to any scale. 3. If x be real, shew that the expression 2-8x+20 can never be less than 4; and that x2-x+1 lies between 3 and 4. If x+y: 3a−b :: y + z: 3b−c :: z +x : 3c-a, prove that x+y+z: ax+by+cz :: a+b+c: a2+b2+c2. 5. There are two cricket-clubs of 14 and 16 members respectively; how many different matches can they make, and in how many will two particular members, one out of each club, both play? x\ & 6. Expand (+2) in ascending powers of x, and find the coefficient of a" in the expansion of 2n 7. A man saves £p a year from his income, which he invests annually at r per cent. compound interest: at the end of how many years will he have accumulated a capital, the interest of which is equal to his yearly saving? 8. There are two towns A and B; a traveller sets out from A towards B, and another from B towards A on the same day, and the travellers meet half way between A and B; one travels 5 miles the first day, 10 the second, 15 the third, and so on; the other travels 20 miles a day for two days, and 35 miles a day afterwards until they meet: find the number of days they travel, and the distance between A and B. EXERCISE XC. 1. Shew that 10404 is a perfect square in every system of notation; and if x+y be a multiple of p, and also x-y, then x2 y2 is a multiple of p2. (ii) (4+5x − x2)3=2ax3 + (x2+3x−4)§, (iii) 3x+4y+7z=25 (in positive integers). 4. A and B run a race; B has 50 yards' start, but A runs 20 yards while B runs 19; what must be the length of the course that A may come in a yard a-head of B ? |