SCHOLARSHIP QUESTIONS, 1876. Candidates are not permitted to answer more than one question in each section. The solution must in every instance be given at full length. A correct answer, if unaccompanied by the solution, or if not obtained by an intelligible method, will be considered of no value. ARITHMETIC. SECTION I. 1. Multiply three millions seven thousand and five by four hundred and seven. Prove the sum by casting out the nines. 2. Divide eight hundred and sixteen millions eight hundred and eighty-seven thousand six hundred and sixty-five by four hundred and seven. Why do you begin at the left-hand side ? 3. Divide £404 8s. Odd. among 17 women and 19 men, giving each woman twice as much as a man. Section II. 1. How many turns does a hoop, 2 yds. 16 in. in circumference, make in a quarter of a mile ? 2. How many 9-oz. packets of tea are there in 40 chests, each containing 24 lbs. P 3. Find the value of the silver at 4s. 5d. per oz. in a bar containing 19 lb. 7 oz. SECTION III. 1. Make out the following bill : 273 lbs. of beef at 11 d. per lb. 16 lbs. of pork at 7d. per lb. 2. Divide £2944 9s. 7d. into an equal number of shillings, sixpences, fourpences, and threepences. 3. A table, five feet square, is covered with halfpence, placed in rows; find the value of the halfpence, each a . per lb. 34 halfpenny being one inch in diameter, and no one touch- 1. 50984 articles at £15 14s. 9zd. each. [The sum should be worked so as to be intelligible to a class learning Practice.] SECTION V. 1. What is meant by ratio and proportion? Find the fourth term in the proportion 118, 130, 177, and a mean proportional to 1764, 2304. 2. If 9 cwt. 3 qrs. 5 lbs. of cheese can be bought for £31 19s. 11d., what will 8 cwt. 2 qrs. 8 lbs. cost ? 3. If 575 men perform a piece of work in 180 days of 10 hours each, how long will it take 115 men working 9 hours a day? SECTION VI. 1. Add together 6:1, 11, 331, subtract from this sum 531, and divide the remainder by 57 of 3. 2. A job can be finished in 25 days by 30 men; at the end of each week (consisting of 6 days) 5 men are withdrawn. How many weeks must the last five men work by themselves to finish the job ? 3. Three fields are bought for £240, £270, and £470 respectively; they contain 4 ac. 3 r., r., 9 ac. 3 r. respectively: find the average cost per acre, and the highest priced field. SECTION VII. 1. Divide :15 by 30, 1500 by .005, .015 by :003. Prove the second result by vulgar fractions. 2. A's share is .09 of B's, B's is 3.08 of C's, C's 1:1 of D's; O's share is £2750: find A's, B's, and D's. 3. 1,802,830 gallons of wine, value £658,405, were imported in 1874; 1,902,415 gallons in 1875, value £671,374: find the increase per cent. in quantity and value to two places of decimals. SECTION VIII. 1. Define interest, principal. Find the simple interest of £1575 10s. for 4 years at 82 per cent. 2. Define discount, present worth. Find the present worth of £1296, due 9 months hence, at £10 138. 4d. per cent. 3. A man invests £1585 10s. in 3 rer cents. at 941, 5 ac. 2 Euclid, Algebra, and Mensuration. 99 þrokerage is charged at the rate of ith per cent. : what income does he derive? SECTION IX. 1. labourer's wages some years age were 15s. 2d. per week, and he could save ls. weekly ; his wages are now 188. 6d., but the cost of living has increased 17} per cent. : what can be save now ? 2. The rates of a parish amount to 38. 6d. in the £; frd is poor rate, ths highway rate, the rest is schoolboard rate. What will a man pay for school purposes who is rated at £170 per annum? SECTION X. 1. A cubical space containing 941,192 cubic inches is exactly filled by 64 cubical boxes : find the length of the side of each box. 2. How many deals 4 feet long and 8 in. wide are required for the floor of a room 16 feet long and 12feet wide ? 3. Find the cost of papering a room 18 feet long, 12 feet wide, 12 feet high, with paper 18 inches wide, at 1 d. per yard. The square EUCLID, ALGEBRA, AND MENSURATION. Candidates in Scotland may answer two questions out of Section IV. if they omit Section IX. With this exception Candidates EUCLID. on AB may be written “sq. on AB,” and the rectangle .contained by AB and CD, “rect. AB. CD." SECTION 1. 1. Define a “superficies,” a “circle,” a rhombus," and write out the three postulates of Euclid. 2. What is meant by saying that one proposition is the converse of another? Give examples from the first book of Euclid. 3. Into how many sections would you divide the first book of Euclid ? To what properties of figures do the last fourteen propositions refer? SECTION II. 1. To bisect a given rectilineal angle. In what particular case will the quadrilateral figure required by the construction be a rhombus ? 2. Given two points, find two other points that shall be at the same given distance from each of them. What is the least possible length of the given distance ? 3. Any two sides of a triangle are together greater than the third side. Construct the triangle when each side is equal to half the sum of the other two. SECTION III. 1. The three interior angles of every triangle are together equal to two right angles. What ratio does the angle of a regular hexagon bear to a right angle? 2. Triangles upon the same base and between the same parallels are equal. A line drawn through the middle points of the sides of a triangle is parallel to the base. 3. In any right-angled triangle the square described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle. SECTION IV. 1. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Construct a rectangle equal to the difference of two given squares. 2. If a straight line drawn through the centre of a circle cut a straight line in it which does not pass through the centre at right angles, it shall bisect it. Lines drawn at right angles to the sides of any figure inscribed in a circle from their middle points meet in one point. 3. The angles in the same segment of a circle are equal to one another. Given three points in the cir. cumference of a circle, required to find a fourth. ALGEBRA. The solution must in every instance be given at full length. A correct answer, if unaccompanied by the solution, or if not obtained by an intelligible method, will be considered of no value. SECTION V. 1. If a=2, b=3, c=0, find the value of 3a + 46, 3a — 4c, a2 + 2? + c?, abc (a + b + c) x (a + bc), (a? + 12 + c + 3). a y by až a a с a + b c+ d a Euclid, Algebra, and Mensuration. 101 2. Multiply 23 – 5x2 – 3x – 18 by x2 + 3x – 4. 3. Divide x SECTION VI. 1. Find the G.C.M. and L.C.M. of 202 + 5x + 4,203 + 4dc2 – 2x – 8. 2. Simplify the expressions,- a2 46 a? — 2ab and a + b b a? — 62 a2 + 4ab ab + 462 3. If show that b d b 0 - 0 SECTION VII. Solve the equations1. (a) 5x + 6 = 7x — 10. (6) 3 (2x - 1) + 4 (3x − 2) = 5 (3x + 2). Or 2. 2x – 3y + 2 = 7x — By 3 = 0. Or 3. 32.c2 – 3 = 20.c. SECTION VIII. 1. The length of a garden exceeds its breadth by 50 yards; the garden contains 9375 yards : find its length and breadth. 2. Two numbers are in the ratio of 5 : 6; show that four times their sum = 44 times their difference, whatever the numbers may be. 3. A labourer can save out of his weekly wages 10 per cent.; his wages rise one shilling a week, but, his expenses increasing also 10 per cent., he can now only save 5 per cent. of his increased wages. Find his weekly wages. MENSURATION. Section IX. 1. Find the number of turfs, 4 feet by 8 inches, required for a garden plot, 50 feet by 75 feet, allowing for four circular beds, diameter 6 feet. 2. A triangular piece of ground, whose sides are 800, 500, and 500 yards respectively, is let for £30 a year : find the letting price per acre. 3. A schoolroom is 60 feet long, 20 feet broad, 10 feet high to the wall plate, 16 feet high to the ridge of the roof. How many children would it contain, allowing 80 feet of cubic space for each child ? By how much does this exceed the workable number at 8 square feet per child? |