13. A dealer has two sorts of tea, one of which he could sell at Is. 8d. per lb. and make 25 per cent. on his outlay, the other at 2s. 3d. and make 12 per cent. What profit per cent. will he make if he mixes them in equal quantities and sells the mixture at Is. 11d. per lb. ? 14. Goods are imported from abroad at an expense equal to 35 per cent. of the cost of production; and the importer makes 15 per cent. on his whole outlay by selling them to a tradesman at £7. 15s. 3d. per ton. Find the cost per ton of production. III. ALGEBRA. (Up to and including the Binomial Theorem, the theory and use of Logarithms.) I. 2. 3. 4. [N.B.-Great importance will be attached to accuracy.] Divide +x1+4x3 +21x2+23x-40 by x2+4x+5. Resolve into factors the expressions: x3-873, x2-24xy +128y2, and x1+x22+ya. Find the Highest Common Factor of the expressions: Prove that is divisible by and find the other factor. 5. Find the square roots of the expressions (i.) _n(n+1)(n+2)(n+3) + 1. (ii) 24+√√572. 7. (ii.) x2+3x+4√√x2+3x-3= 48. If a and ẞ are the roots of the equation x2+px+q = 0, prove that a+ẞ= -p and aß = q, and find, in terms of p and q, the equation of which the roots are a+28 and B+2a. 8. At present the ratio of B's age to A's age is the ratio 5:2, but in 30 years' time the ratio will be 35: 23; find their ages. IO. If the expression x(1 − 5x+6x2)−1 be expanded in powers of x, prove that the coefficient of x is 3′′ – 2′′. II. Write down an expression for the number of combinations of n things taken together, and, if "Cr represent this expression, prove, by general reasoning or otherwise, that 12. nCr=nCn-r, and n+1Cr=nCr + nCr−1. Having given log10 5 =*6989700, log107=8450980, and log10 II = 1'0413927, find the logarithms, to the base 10, of 385 and 1, and solve approximately, to two places of decimals, the equation 5*.7x-1= 11x+2. IV. PLANE TRIGONOMETRY AND MENSURATION. (Including the Solution of Triangles.) [N.B.-Great importance will be attached to accuracy.] I. Explain what is meant by the circular measure of an angle, and find the circular measure of an angle of x degrees. 2 A wire AB, i foot in length, is bent so as to form an arc of a circle whose diameter is four inches; find the angle subtended at the centre of the circle by the chord AB. 2. Express the Trigonometrical ratios of an angle in terms of the secant, the angle being less than a right angle. If in a triangle ABC, find the value of 3. Determine the values of the Trigonometrical ratios for an angle of 60°, and for an angle of 30°. A ladder rests against a vertical wall at an angle of 60° with the horizon, and when the foot is drawn back 18 feet further from the wall the inclination to the horizon is found to be 30°. Find the length of the ladder. 4. = Prove that cos(90° + A) = − sin A for the case when A is less than two right angles. Write down the values of sin 225°, cos 210°, tan 315°, and cosec 420°. 5 Show geometrically that cos (A+B) = cos A cos B- sin A sin B, where A, B are two positive angles whose sum is less than a right angle. Find the value of cos 75°. that 6. Express sin 3A and cos 3A in terms of sin A and cos A, and show 7. Prove the following identities: (i.) (x tana+y cot a)(x cot a + y tan a) = (x + y)2+4xy cot2 2a. (ii.) cos ẞ cos(2a - ẞ) = cos2a - sin2(a - ẞ). (iii.) cos a + cos 3a + cos 5a + cos 7a = sin 8a cosec a. 8. In any triangle ABC find tan in terms of the sides a, b, c. Find the angles of the triangle whose sides are proportional to 3, 5, 7. 9. Show how to solve a triangle when two sides and (i.) an angle opposite to one of the sides, and (ii.) the included angle, are given. From a boat at sea the angle subtended by the line joining two fixed objects A, B on land is observed to be a; after sailing x yards directly towards A the angle subtended by AB is found to be ẞ, and then after sailing y yards directly towards B, the angle is found to be y. Find an expression for the distance AB. IO. In a triangle ABC, a = 35, b = 43, and C = 75° 10' 40", find the angles A and B. II. Find the area enclosed by 200 hurdles placed so as to form a regular polygon of 200 sides, the length of each hurdle being 6 feet. 12. A leaden sphere one inch in diameter is beaten out into a circular sheet of uniform thickness = Tooth inch. Find the radius of the sheet. |