II. Prove that parallelograms which are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides. 12. Prove that, in any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. 3. Find the Least Common Multiple of 34, 42, 119, and 255; and obtain the sum of the fractions 4, 4, 11, and 2, by reducing them to a common denominator. 4. Find the value of Tf of £1. 19s. 81d. 5. Find the cost of 25 quarters 3 bushels of oats at 19s. 3d. a quarter. 6. One side of a rectangular field is '054 of a mile, and the adjacent side is 13 of a furlong. Find the length of each side in yards and feet, and express the area of the field as a fraction of an acre. 7. A sum of £377. 13s. 4d. lent at simple interest amounts in a year and a half to £396. 11s. Od. What is the rate of interest ? 8. A garden whose length is 67 ft. 9 in. has a path 4 ft. wide on the two sides and at one end: if it costs £4. 10s. 3d. to turf the remainder at 6d. a square yard, what is the width of the garden ? 9. A, B, and C who are engaged on piece-work do amounts in the same time which bear to one another the proportion of 10, 9, and 14 respectively. What ought each to receive if the amount paid for the whole is £24. 15s.? 10. What is meant by an odd number and an even number? Show that it is only necessary to look at the digit in the unit's place to ascertain if a number is odd or even. Several numbers have to be added together. What is the condition that their sum should be odd ? II. A cistern is filled in 3 hours by a pipe 3 sq. in. in cross section through which water flows at the rate of 6.4 miles an hour. What is the volume of the cistern? 12. If coffee and chicory cost £8. 10s. and £2. Ios. 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 24 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 221 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.) [N.B.-Great importance will be attached to accuracy.] (1+6a+6a2)+(3+12a + 12a2)3 is divisible by (2+6a+6a2)2; and find the value of the quotient when a = - . 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. 71d., 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- 1. 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 1 × 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 r 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. 10. Write down the expansions for a*, ex and loge(1+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 + z + z + etc. |