Arithmetic.

Junior, Senior, and Higher Local.

Junior Work, Nos. 1-10 inclusive.

Senior Work, Nos. 4-14 inclusive.

Higher Local Work, Nos. 6-17 inclusive.

1. Find the product of nine millions eight hundred and sixty four thousand three hundred and two, and three hundred thousand and seventy-one. Express your answer in words.

2. In a division sum the quotient is 596, the dividend 20573290, and the remainder 11290. What was the divisor?

3. Reduce to prime factors 1155 and 2717.

4. What must be the circumference of a wheel which makes 129280 revolutions in 202 miles?

5. Find by Practice the value of 3 acres, 2 roods, 20 poles, 7 yards, at £10 10s. per rood.

6. If £805 48. 44d. be the price of 63 cwt. 3 qrs., what is the price of 1 cwt.? Continue this reasoning, and find the price of 96 cwt.

7. Express 19s. 11 d. as the decimal of £1, and also as the Vulgar Fraction of £1 11s. 6d.

(ii) 2.3975÷0000005; and ·0000013932 ÷ 32.

(iii) 3 × 2.97 x 32·6×124.

49

9. Find the value of x in 35 x 125 0145.

10. What principal would amount to £591 12s. 4d. in 4 years at 21% simple interest?

11. In what time would £345 17s. 6d. amount to £387 7s. 7 d. at 4% simple interest?

12. What sum must be invested in the 3 per cents at 81 to produce an income of £180 a year?

sq. yds., what length What would be the

13. If the area of a square court be 420 of wall would be required to enclose it? thickness of this wall supposing its height to be 6 feet, and its cubic contents 566,784 cubic inches?

14. If 800 soldiers consume 560 lb. of flour in 24 days, how many days will 1,680 lb. last 2,400 soldiers?

15. If A advance £1,000 for 10 months, and B advance £1,500 for 6 months, and C advance £2,000 for 4 months, what share of a gain of £4,500 belongs to each ?

16. A and B can do a piece of work in 15 days, A and C in 20 days, and B and C in 25 days, how long would it take A, B, and C to do the work together? and how long would it take each of them to do it alone?

17. Simplify 68.971442301.

Geometry.

Junior, Senior, and Higher Local.

Junior work, Nos. 1-8 inclusive.

Senior work, Nos. 5-12 inclusive.

Higher Local work, Nos. 9-16 inclusive.

1. If two straight lines cut one another, the vertically opposite angles must be equal.

If the vertical angles made by four straight lines at the same point be respectively equal to each other, each pair of opposite lines shall be in the same straight line.

2. In Euclid's definition of an angle, what generally recognized angle is excluded by the words, "not in the same straight line"? Suggest a definition which shall not make the same omission.

Explain the terms a supplement and a complement of an angle.

3. By two different methods describe a rectilineal angle equal to a given rectilineal angle.

4. If a straight line fall upon two parallel straight lines it makes the two interior angles upon the same side together equal to two right angles, and also the alternate angles equal each to the other, and the exterior angle equal to the interior and opposite angle upon the same side.

Enunciate the converse of this proposition.

5. If any straight line joining two parallel straight lines be bisected, any other straight line drawn through the point of bisection to meet the two lines will be bisected in that point.

6. The opposite sides and angles of a parallelogram are equal to one another and the diagonal bisects it.

If both diagonals be drawn they bisect each other.

7. By synthesis and analysis draw from a given point two straight lines which shall make equal angles with two given straight lines which intersect each other.

8. If a straight line be divided into two equal 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.

9. If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square on the other part, is equal to the square on the straight line which is made up of the whole and that part.

10. Find the side of a square equal to a given equilateral triangle.

11. The square on the base of an isosceles triangle whose vertical angle is a right angle, is equal to four times the area of the triangle.

12. In two different ways find the centre of a circle.

=

13. What are the conditions essential to the equality of triangles? Enunciate each case which Euclid proves.

If two triangles have their three angles equal, each to each, are they equal in every respect?

14. Which propositions in Euclid, Book I., do you consider embody the most important truths? Give reasons for your

answer.

15. A quadrilateral of which the diagonals bisect each the other is a parallelogram.

16. The perimeter of an isosceles triangle is less than that of any other equal triangle upon the same base.

Algebra.

Junior, Senior, and Higher Local.

Junior Work, Nos. 1-10 inclusive.

Senior Work, Nos. 2-11 inclusive.

Higher Local Work, Nos. 4-15 inclusive.

1. If ́ x3 + y3 + z3 — m.xyz be divisible by x + y + z, what is the value of m?

5. What fraction is that whose numerator being doubled, and denominator increased by 7, becomes equal to ; but the denominator being doubled, and the numerator increased by 2, the value becomes ??

6. Define a surd. If x+√a=y+√b, √σ and √ being surds, show that x=y, a=b. If ab is rational, show that is rational.

b