Shew by actual multiplication, that the whole coefficient of x in the product of the expansions of (1 + x)” and (1-x)" is equal to the coefficient of x in the expansion of (1 − x2)”.

therefore the coefficients of x in the product of these

2n3 (n−1)(n-2) 2n (n−1)(n − 2)(n−3)

+

1.2.3

6n2 - 6n

n (n-1)

1.2

1.2.3.4

which is the coefficient of x in the expansion of (1 - x2)”.

ARITHMETIC.

FRIDAY, 1ST DECEMBER, 1876. 2 P.M. TO 5 P.M.

1. Add together 33, 4%, and 53. Ans. 1411.

2. Subtract 143 from 211. Ans. 67g.

3. Multiply together 21, 45, 45, and 1187.

4. Divide 13 by 11. Ans. .

Ans. 9113.

5. Add together 157, 4.092, 0075, and 3.6505.

Ans. 23.45.

6. Subtract 72.0975 from 73.332. Ans. 1.2345.

7. Multiply 22.48 by 1.125. Ans. 25.29.

8. Divide 917.3245 by 65.29. Ans. 14.05.

9. Express 1 oz. 5 grns. as a decimal fraction of 3 oz. 4 dwts. 16 grns. Ans. 3125.

10. Add together 13 of 4, 114 of 51, 114 of 91%,

2

12. Multiply together 5, 14 of 1%, and 214 of 5. Ans. 221.

13. Divide 2 by Ans. 11.

14. Add together 1·1375 of a fathom, 875 of a yard, 2.965 of a foot, and 9.75 of an inch, expressing the answer in feet and a decimal fraction of a foot.

Ans. 13.2275 feet.

15. Multiply together 2.56, 1.75, and 000125.

16. Divide 273 by 0028. Ans. 97.5.

Ans. 00056.

17. Reduce 15 cwts. 2 qrs. 21 lbs. to the decimal of

2 tons 10 cwt. Ans. 31375.

18. Divide 10 miles 1 furlong 6 poles by 22.

Ans. 3 furlongs 27 poles 3 yards.

19. An estate of 625 acres 1 rood 10 poles is bought for £7,878. 18s. 9d. What is the average price per acre? Ans. 12 guineas.

20. Find by Practice the cost of 3257 tons of coals at £1. 3s. 7 d. per ton. Ans. £3807. 6s. 74d.

C

21. In what time will the simple interest on £2987. 10s. 6d. amount to £248. 19s. 21d. at 33 per cent. per annum? Ans. 2

22. If 7 horses draw 5

years.

tons along 56 yards on a road in 10 minutes, how many horses will be required to draw 11 tons along 50 yards in five minutes.

Ans. 25.

23. Find the difference between the simple and compound interest on £933. 6s. 8d. in 21⁄2 years at 2 per cent. per annum. Ans. £1. 3s. 53d.

24. Find the cube root of 10.218313. Ans. 2.17. If the diagonal of a square is 3 feet long, find the length of each side, correct to a thousandth of an inch. Ans. 25.4558 inches.

25. Compute by means of the tables the value of
(34·3) ÷ (144).
Ans. 7927926.

26. If a sum of £1000 is lent on the condition that it shall bear compound interest at the rate of 5 per cent. per annum for the first five years, 10 per cent. per annum for the next five years, and afterwards 15 per cent. per annum, what will the debt amount to in 20 years? Ans. £8315. 10s.

EUCLID.

ȘATURDAY, 2ND DECEMBER, 1876. 10A.M. TO 1P.M.

1. To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

How would the process fail if the last condition were not fulfilled?

If any two of the straight lines given were together less than the third, the two circles described would not intersect, and therefore the construction for the triangle required would fail.

2. Parallelograms upon the same base, and between the same parallels, are equal to one another.

Shew that if two triangles have two sides of the one equal to two sides of the other, each to each, and the sum of the two included angles equal to two right angles, the triangles are equal.

Let ABC and DEF (fig. 1) be two triangles having the two sides BA, AC equal to the two sides ED, DF respectively, and the angles BAC and DEF together equal to two right angles; then the triangle DEF shall be equal to the triangle ABC. Produce the side ED to E', making DE' equal to DE, and join FE'. Then, since the angle E'DF equals the angle BAC, and BA and AC are equal to E'D and DF respectively, the triangle FDE is equal to the triangle BAC. But the triangle FDE is equal to the triangle FDE', "since DE is equal to DE'; therefore the triangles BAC and FDE are equal.

3. In any right-angled triangle the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle.

Shew how to construct a straight line, the square on which shall be any given multiple of a given square.

Let it be required to find a straight line, the square

on which shall be any multiple of the square on a given straight line AB.

From B (fig. 2), one extremity of the straight line AB, draw BD at right angles to AB and equal to it. Join AD. Then the square on AD is equal to twice the square on AB. From D draw DE at right angles to AD and equal to AB. Join AE. Then the square on AE, being equal to the squares on DE and AD, is equal to three times the square on AB. In a similar manner the process may be continued, and a straight line, the square on which is any required multiple of the square on AB, may be found.

4. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

5. The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference.

6. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.

In a given straight line AB find a point O, such that the rectangle contained by the segments 40 and OB shall be equal to a given rectangle not greater than the square on half of AB.

Take a straight line C (fig. 3), such that the square on C is equal to the given rectangle. Let AB be the