(b) Reduce the following expressions to simpler forms: (1) √45-√80c3+√5a2c; 2m p a (2) [(1)]; m 2n 2n 3 (3) 4× 2√. II. 6. Given 3x2+x=7, to find x. √p+x+√p-x 7. Given √x 8. Given x + y = a ; x2 + y2 = b2; to find x and y. 9. Expand (a) log (amb"); ; 1+2x terminate coefficient. 10. Write equivalent expressions adapted to computation for the following: (b) log √a2x2. b = α September, 1882. [State what text-book you have studied, and to what extent.] x4 1. Multiply b4 x2 + bx by into a series by the method of inde =c and y to find x. so as to obtain the d, to find x and y. 3. Given √4a+x=2√b+x−√x, to find x. 4. Write the 6th power of (a — 2b) by the binomial formula. 5. Simplify the following expressions: 5 32m3 n1 10 (c) √36 a2b2; p15 920 (d) √8+√50-V18; (e) 3a b x 5b V 2 c. (a) (x2)"; (b) 6. Solve the equation 6x2 −13x+6=0, and resolve the first member into its factors. 7. Given x+y=p and xy = q2, to find x and y. 8. A traveller has a journey of 132 miles to perform. He goes 27 m. the first day, 24 m. the second, and so on, travelling 3 m. less each day than the day before. In how many days will he complete the journey? 9. What is the present value of a dollars due n years hence, at c per cent compound interest? 10. How many distinct straight lines can be drawn in joining, two and two, five distinct points, no three of which are in the same straight line? June, 1883. NOTE 1. Candidates for examination in this subject, as a whole, should take the whole of this paper; those for the first year's partial examination, the first part of it; those for the second year's partial examination, the second part. NOTE 2. State at the head of your paper what text-book you have studied, and to what extent. 2. Given 1. Reduce to their simplest forms the fractions, (a) ac+bd+ad + bc af +2 bx +2 ax + bf ace d I. (a+b)2x a ; (b) axm bxm+1 b3x3 - bx = ae — 3 bx, to find x. 3. A sum of money, at simple interest, amounted in m years to a dollars, and in n years to b dollars. Find the sum and the rate of interest. 7. Given <: 5. (a) Simplify (a2b3)}+ (a2c®)}. (b) Extract the square root of 6 hm2n + h2 + 9 m1n. √x + a + √x (c) Reduce √x + α-√x-α with a rational denominator. II. 6. Given 15a2 - 20x 35, to find x. = = x+√x2-9 8. Given x2 - xy = 48, and xy — y2 = 12, to find x and y. 9. The number of permutations of n things taken r together is equal to 10 times the number when taken r 1 together; and the number of combinations of n things taken r together is to the number when taken r— 1 together as 5 to 3; required the value of n and r. a to an equivalent fraction a (x-2)2, to find x. into a series of ascending powers of x 3+2x 10. Expand 5+7x by the method of indeterminate coefficients. (Four terms of the series will be sufficient.) September, 1883. [State what text-book you have studied, and to what extent.] 1. Given x + y + 1 x-y+1 2. Simplify (a) √27+2√48+3√108. (b) (√a2b)3 (√a3b12)*. x2p (q−1) — y2q (p−1) (c) 2o (4-1)+y? (p-1) =α, and 3. Form an equation whose roots shall be 2 and -3. Resolve 2-3x+4 into two factors. 3 x + √ 4 x x2 3x-4x-x2 1 1 5, and + = 13, to find x and y. 9. Express log = b, to find x and y. 2, to find x. 6. To deduce a formula for the sum of a geometric progression in terms of the first term, the ratio, and the number of terms. 7. Having 10 different letters, how many sets of two each can you form of them, differing by at least one letter? 1 8. Expand 1-2x+x2 of x by the method of indeterminate coefficients. terms of the series will suffice.) into a series of ascending powers (Four 3 abc in a form adapted to computation. d5 10. To deduce a formula for the amount of a given sum of money for a given time at a given rate of compound interest. TRIGONOMETRY. July, 1880. [State what text-book you have studied, and to what extent.] 1. Given tan A=√3; find all the other functions of A when A is an angle of the third quadrant. 2. Given sin 30° = ; find the sine and cosine of ±60°, 120°, 150°, 210°, 240°, 300°, 330°. 3. Deduce the formula, sinsiny 2 sin(x + y) cos(xy). = 5. Write the formulæ for solving the several cases of right triangles. 6. In a plane triangle the side b is 304, the side c is 280.3, and the included angle A is 100°; find the remaining parts. September, 1880. [State what text-book you have studied, and to what extent.] 1. The length of an arc is 1.5 that of its radius; what is the number of degrees in the angle it subtends? 2. Find all of the functions of the following angles: ±45°, 135°, 225°, 315°. 3. Given sin A = m sin B and tan A = n tan B; find sin A and cos B. 4. Deduce the formulæ for the sine of the sum of two angles in terms of the sines and cosines of the angles. 5. Given A = sin13, B = sin-14, to show that A+B= 90°. |