2. Find the value of 2.72324801 of £2475. 15s. 6d., correct to the

nearest sixpence; and that of
places of decimals.

•782634 x 42.5682

13.5647

correct to four (10)

3. Sixteen hundred dollars were divided as follows amongst A, B, C, and D. A, B, and C continued to go up in that order to D, beginning with A, and D gave them 4, 7, and 12 dollars respectively on each occasion, until there were not enough dollars left to pay the one of them whose turn it was to be paid, and that residue was D's share. How many dollars did each one of the four get? (8) 4. A merchant bought a ship's cargo and sold it at such a price that he obtained a profit at the rate of 10 per cent. If he had paid 5 per cent. less for the cargo than he did pay, and sold it for 5 per cent. more than he did sell it for, his profit would have been increased by £315. What sum did the merchant pay for the cargo? (10)

5. For what value of x is 2x1— 3x3 — 27x2+53x-4 exactly divisible by x2+2x-7?

Find the numerical value of the quotient when x has that value.

6. Simplify

(6)

ab (2a+b)(a+2b) — (a2 — b2)2 + (a2 —ab+b2)2; and find the factors of

(i) a2+2bc+b2-2a (b+c), (ii) 15x2+34x-72.

106

(10)

and A spent on an average What was B's average daily

8. A spent 102 dollars in a certain number of days, and B spent 3 more dollars in 3 more days, 1 dollars more per day than B. expenditure?

(12)

9. Draw the graph of y = 44+8x-x2 for positive values of x not exceeding 10, and from it write down as accurately as you can, to one decimal place, the values of x when y = 55.

(10)

10. Obtain the expression for the sum of n terms of an Arithmetical Progression whose first term is a and common difference d. What is the greatest number of terms of the series 13, 17, 21, ... which may be taken without their sum exceeding 5500, and what will be the last term taken ?

(12)

11. Find the difference between the sum of the first 5 terms of the series

1. Construct a triangle ABC, with its sides AB, BC, and CA 23, 29, and 37 units long respectively. Bisect the angle BAC. In the line AF, which bisects the angle BAC, take F so that AF is 17 units long. Draw from F a perpendicular to AC, and measure its length. (10) 2. Make a square ABCD whose sides shall be 14 units long. In the diagonal AC produced through C, find a point X so that the area of the triangle ABX shall be 133 square units. Measure the length of BX. (10)

3. Describe an equilateral triangle about a circle whose radius is 19 units long. Measure the length of a side of the triangle. (10)

B.-Theoretical.

Figures must be drawn neatly.

Recognized abbreviations may be used.

4. If two angles of a triangle are equal, prove that the sides which are opposite to them are equal.

ABCD is a quadrilateral of which the side AB is equal to the side AD, and the angle DBC is equal to the angle BDC. Show that AC bisects the angle BAD.

(8)

5. Prove that the sum of the angles of a triangle is equal to two right angles.

If the bisectors of the angles ADC, BCD of a quadrilateral ABCD meet at E, prove that the angle DEC is half the sum of the angles DAB and ABC.

(8)

6. If the square on one side of a triangle is equal to the sum of the squares on the other two sides, prove that the angle contained by those two sides is a right angle.

Show, without proof, how to divide a given straight line into two parts, so that the sum of the squares on the two parts may be to the square on the whole line as 9 is to 16.

(12)

7. Prove that the rectangle contained by the sum and the difference of two straight lines is equal to the difference of the squares on the lines.

ABC is an isosceles triangle whose vertex is A, and a straight line XY parallel to BC meets AB in X and AC in Y. Show that the square on BY is equal to the square on CY together with the rectangle contained by BC and XY. (12) 8. Prove that chords of a circle which are equal are equidistant from the centre.

If two equal chords intersect within a circle, prove that the segments of one chord are respectively equal to the segments of the other chord.

(10)

9. Prove that angles in the same segment of a circle are equal. ABCD is a parallelogram, and the angle BAD is an obtuse angle. The circle passing through the three points B, A, and D intersects BC and DC produced in E and F respectively. Prove that E and F are equidistant from A. (10)

10. Prove that the angle in a semicircle is a right angle.

AB is the diameter of a semicircle, and C any point in the arc. In BC, or BC produced, a point D is taken so that BD is equal to AC. What is the locus of D?

(10)

MECHANICS (Associateship).

Saturday, June 22nd, 1912.-Afternoon, 2.30 to 5.30.

Work neatly.

Graphic methods of solution are allowed, unless otherwise stated.

1. State the Parallelogram of Forces.

If the resultant of forces P and Q is at right angles to the resultant of Q reversed and P, prove that P= Q, and that P and Q may act at any angle.

(25) 2. Prove that the centre of gravity of a triangle is the intersection of its medians.

A quadrilateral ABCD is made up of two isosceles triangles ABC, BCD, common base BC, equal sides AB, AC and CD, BD. The heights of ABC, BCD are 9 inches and 6 inches. Prove that the centre of gravity of ABCD is one inch from BC. (20)

3. Describe a Wheel and Axle. If the radius of the wheel be a and the radius of the axle b, find the relation between effort and resistance, and show that what is gained in power is lost in speed. (20) 4. An engine draws a train of 250 tons at 30 miles an hour against a resistance of 12 lb. to the ton. Find the work done per second and the horse-power of the engine.

(20) 5. How are work and kinetic energy measured? and state the kinetic energy in foot-pounds of mass m lb. moving with v foot-seconds.

Masses 12 lb. and 3 lb. balance on a system of pulleys of the first kind having two weightless movable pulleys. One pound is added to the 3 lb., and the 4 lb. so formed descends x ft. from rest and gains velocity v foot-seconds. State the following four things in terms of v and x:— (a) the work done by the 4 lb., (b) the distance risen by the 12 lb., (c) the work absorbed by the 12 lb., (d) the kinetic energies of the masses 4 lb. and 12 lb. (30) 6. Define uniform acceleration, and, using ordinary notation, prove that v2 = u2+2as.

A body A falls from the top of a tower 144 ft. high, and a body B is projected from the base a little later; so as to meet 64 ft. from the top with equal velocities. Find (a) their velocities when they meet, (b) the velocity of projection of B.

(30)