3. If a, ẞ are the roots of the equation ax2 + bx+c=0, prove that ax2+bx+c=a(x−a)(x —B). 4. Find the relation between p and q when the roots of the equation x2-pr+q=0 are equal. 5. What is the relation between a and c when the product of the roots of the equation ax2+ bx+c=0 is equal to unity? 6. Shew that the roots of a quadratic equation are either real and equal, real and unequal, or both impossible. (14) =13 4 — = x2 + xy + y2 = 1} } (15) x2 - 3 x y + 212 = 3 } 4 xy+ y2=6 XVIII. PROBLEMS PRODUCING QUADRATIC EQUATIONS. 1. Find two numbers such that their difference is 18, and the square of the one is 9 times the other. 2. The sides of a rectangle are 16 and 24 feet respectively: what breadth of border must be taken off all round, that the remaining area may be 240 square feet? 3. A person bought 2 pieces of cloth, one 6 yards longer than the other, for 20l. 10s.; and each cost as many shillings per yard as there were yards in its length: find the length of each. 4. A man walking of a mile above his ordinary rate gains of an hour in 39 miles: at what rate does he ordinarily walk? 5. A certain number of persons, going for an excursion, took two friends with them; the whole expenses were 321. 6s., and each share was 4s. more than it would have been without the friends: of how many persons did the party consist? 6. A person bought a certain quantity of tea for 81. 5s.; and if he had given sixpence a pound less he might have had 3lbs. more for the same money. How much did he buy? and what did he give for it per lb. ? 7. A person out walking has 18 miles to go, and finds that at the rate at which he is going he will be half an hour late, but if he quickens his pace by a mile an hour he will arrive just at the proper time: at what pace is he going? 8. A certain number, of two digits, is equal to three times the product of its digits; and if 18 be added to the number its digits will be transposed: what is the number? 9. The area of a field is 7 acres, and the distance between the opposite corners is 275 yards: find the lengths of the sides. 10. Find two numbers whose sum is 7, and the sum of their squares exceeds their product by 19. XIX. RATIO AND PROPORTION. 1. Which is the greater ratio, 4 : √17 or 5: √26? 2. Find a number which, when increased by 4, has the same ratio to 61 which, if diminished by 4, it would have to 59. 3. Find the fourth proportional to 6, 9, 12; the third proportional to 8, 12; and the mean proportional between 12, 27. 4. Find three numbers in the proportion of 1, 3, and 2, such that the sum of their squares is 724. 5. If a b c d, prove that (1) ma+nc: mb+nd :: ma-nc: mb―nd. (4) (a+b): bt :: (c+d): dr. (2) (ace)} : (bdf)3 :: (ma2+nc2+pe2)} : (mb2+nd2+pƒ2)3. 7. If a bb : c, prove that (1) a+b: b + c :: (ab)* : 8. If a bbc::c: d, prove that (1) a: b :: aś : d3 :: (ab+ac+bc)1⁄2 : (bc+bd+cd)*. (2) ac: bd:: a2 + b2 + c2: b2 + c2+d2. 9. If a+bb::b+c:c:c+a: a, shew that a=b=c. I I 10. If a + c : b + c :: a2 : b2, then either ++ -=0, or a=b. 11. Find the ratios of xyz from the equations ax+by+cz=0, lx+my+nz=0. XX. DEFINITIONS OF TERMS, &c. 1. Explain the terms coefficient, power, root, index, factor, term; and prove that the product of two negative quantities is positive. 2. Explain the terms multiple, two dimensions, reciprocal of a quantity. 3. What is an impossible quantity, and what is an irrational quantity? Define Involution, Evolution, Exponent. 4. Explain what you mean by simple factors, binomial expressions, homogeneous quantities, similar surds. 107 EXAMINATION PAPERS.1 1. Simplify ALGEBRAICAL. I. 2-3x-(4-6x) — {7—(9—2x)}. 2. Multiply 1+2x+3x2 by 1+4x+5x2+6x3, and find the continued product of x+y, x-y, and x2+2xy + y2. 3. Divide x3 + (a+b+c)x2+(ab+bc+ca)x+abc by x+a. 4. Reduce to its simplest form a2 + ab + b2 a3 +63 a2-ab+b2x a3— b3° 63 5. Find the G.C.M. of x3+6x2+11x+6 and x3+92 +27x+27. 8. Extract the cube root of x6+6x3 +21x1+44x3+63x2 + 54x+27. Examination Papers I.-XXVIII. extend only to Simple Equations. |