12. If coffee and chicory cost £8. 10s. and £2. 10s. per cwt. respectively, what is the proportion of coffee and chicory in a mixture of which 7 lbs. are worth 7s. 6d. ? 13. A man buys goods and finds that the cost of carriage is 4 per cent. on the cost of the goods. He is compelled to sell at a loss of 5 per cent. on his total outlay; if however he had received £3. 5s. more than he did he would have gained 2 per cent. What was the original cost of the goods? 14. A and B set out from the same place in the same direction and travel uniformly; after 9 days' travelling A finds he is 72 miles ahead of B: he then turns and travels back the distance B would travel in 9 days, he then turns again and overtakes B in 22 days from the start. What is the rate of travelling of each ? III. ALGEBRA. (Up to and including the Binomial Theorem, and the theory and use of Logarithms.) I. [N.B.-Great importance will be attached to accuracy.] What are the factors of x6+3? Hence, or otherwise, show that (1+6a+6a2)6+(3+12a+12a2)3 is divisible by (2+6a+6a2)2; and find the value of the quotient when a = -1. 5. I sent cash to a grocer for a certain number of lbs. of sugar, at the rate of 7 lbs. for Is. Id. But before the order reached him the price of sugar had risen, and the money was sufficient only to buy a quantity less by 10 lbs. than that which I had intended; so I sent an additional 5s. 7 d., and received one-fifth as much again as I had at first ordered. Find the number of lbs. ordered at first, the rise in the price being less than a halfpenny a pound. 6. Show that, in a scale of notation of which the radix is r, when the sum of the digits of any whole number is divided by r-1, there will be the same remainder as when the whole number is divided by r- I. If A and B be numbers in the scale of 10, and if 5 and 7 be the remainders when the sums of the digits of A and B are respectively divided by 9; find the remainder when the sum of the digits of the product of A and B is divided by 9. 7. Find expressions for the sum and general term of an Arithmetical Progression. In a series of right-angled triangles, one side has, in succession, the lengths I × 4 in., 2×6 in., 3×8 in., 4 × 10 in., etc., and the hypotenuse differs in length from this side by one inch; show that the lengths of the other side form an Arithmetical Progression. 8. Find the number of permutations of n dissimilar things taken together. Find the number of arrangements of three different letters which can be formed of the ten letters Q to Z; Q, when it occurs, being always followed by U. 9. Prove the Binomial Theorem when the index is a positive fraction, assuming its truth for a positive integer. Calculate by logarithms the numerical value of the 6th term of the expansion, by the Binomial Theorem, of (1-x)120, when x = *220793; and show that it is nearly 105. IO. Write down the expansions for a*, ** and log.(I+x), each in a series of ascending powers of x. If z = *9999999999, and e = 2*71828; find, to four places of decimals, the value of z+22+23+ etc. I. IV. PLANE TRIGONOMETRY AND MENSURATION. [N.B.-Great importance will be attached to accuracy.] Show that the angle subtended by any arc of a circle at the centre, may fairly be measured by the ratio of the arc to the radius of the circle. Why may it not be measured by the ratio of the chord of the arc to the radius of the circle? 2. Define the tangent of a positive angle less than 360°. If the angle A be known to be positive and less than 360°, what possible values can it have when (i.) tan A = √√3. (ii.) tan A= = I. (iii.) tan 24 tan A ? 3. Prove the expression for sin (A+B) in terms of sin A, cos A, sin B, cos B, showing that the result is true for all sizes of the angles A, B. Hence find all the trigonometrical ratios of 105°. 4. Prove that in any triangle the sides are proportional to the sines of the angles opposite to them; and that the cosine of any angle of the triangle is expressible, in terms of the sides, by the formula 2bc cos A = b2+c2 − a2. If the sides of a triangle be 4, 5, 6, find the cosines of the angles; and hence, with the table of logarithms, determine the smallest angle to the nearest minute. If b = 11, c = 3, A = 57°, find B and C to the nearest second, using the table of logarithms. 6. Prove the relations (i.) tan A+tan B+tan C = tan A tan B tan C where A+B+C = 180°; (ii.) sin2B sin2(C – A)+sin2C sin2(A – B) +2 sin B sin C sin ( C – A) sin ( A – B) = sin2A sin2(B – C). 7. What is the meaning of tan-1x? Show that (i.) cos-1[xy-√ 1 − x2 − y2 + x2y2] = cos 1x+cos-1y, 8. If D be the orthocentre of the triangle ABC, that is, the point of intersection of the perpendiculars drawn from A, B, C to the opposite sides, prove that the radius of the circle drawn through A, B, C is such that Prove also that, if O be the centre of the circle drawn through A, B, C, OD2 = R2(1-8 cos A cos B cos C). 9. Prove that the area of any quadrilateral ABCD is given by √(s − a)( s − b) (s − c ) (s — d) — abcd cos2}(A+C), where a = AB, b = BC, c = CD, d = DA, 2s = a+b+c+d, and A, C are the angles DAB, BCD respectively. 10. Prove that the area of a triangle is half the product of the base and height, and that the volume of a right circular cone is one third the product of the area of the base, and the height. What is the height of a right circular cone when its volume is equal to that of a sphere of which the radius is equal to the radius of the base of the cone? |