5. Describe a square on a given straight line as a diagonal.

6. Show that the sides of any four-sided rectilineal figure are together greater than the two diagonals.


1. Add together 2, 4, 8, and .

2. Subtract 21 from 201.

3. Multiply by 14.

4. Divide 2 by 1.

5. Add together 16·41215, 9.376, 00403, and 270·3.

6. Subtract 17.2398 from 27.06.

7. Multiply 46.2375 by '0074.

8. Divide 92.3784 by 623 to three places of decimals. 9. Find the value of 1.25 of £1. 13s. 4d.

10. Reduce 4 tons 3 cwt. 1 qr. 9 lbs. to ounces.

11. Find the income-tax on £356. 10s. at 5d. in the £.

12. Find the simple interest on £576 for 6 years at 4 per cent. per annum.

13. Add together 17, 3, and 41.

14. Subtract 5 from 10.

15. Multiply together 194, 27, 94, and 24.

16. Divide 214 by 44.

17. Add together 2.6 of a day and '85 of an hour,

and give the answer in minutes.

18. Subtract 4:42 of a cwt. from 3.64 of a ton.

19. Multiply 62.5 by 579.

20. Divide 473928 by 24.18.

21. Express 0425 of a mile in yards.

22. In 156704 square inches how many square yards and square feet are there?

23. What is the income of a man whose income-tax at 4d. in the pound amounts to 23 guineas?

24. At what rate per cent. will £230. 15s. amount to £305. 14s. 10d. in 13 years at simple interest?

25. Find the dividend on £274. 10s. at 8s. 3d. in the £.

26. If 2 horses can plough 7 acres of ground in a day, how many horses will be required to plough 161 acres in 11 days?



1. Find the numerical value of (x2 - 2ax + a2) when x = 1 and a = 1; find also the value of x, if (10000);= 10. 2. Perform the operations indicated in the following examples:

(A) {2 (a + b) — 3 (c − d)} − {2 (a − b) + 3 (c + d)}.


(B) (a3 — a1b+aba − b3) × (a+b).

(c) (16x1 – 72x2a2 + 81a1) ÷ (4x2 + 12ax+9a2).

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3. Find the least common multiple of (1-x), (1-x),

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3x3 — 27ax2 + 78a2x − 72a3

2x+10ax2 - 4a2x - 48a3

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6. A boat's crew row 9 miles with the tide in & of an hour, and when the tide is flowing at half its former rate, the same crew row 9 miles against the tide in an hour and a half. Required the rate of the strongest tide and the rate at which the crew will row in still water.

7. Solve the following quadratic equations: (A) (x+3)2 = 6x + 58.

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8. If the lengths of two straight lines be each expressed in feet and inches, show how the product of their lengths may represent superficial measurement. Apply the rule of duodecimals to find the area of a room 32 feet 8 inches long by 18 feet 4 inches wide, and explain the different terms in the product.

9. Define the logarithm of a number. If the base were 3, of what numbers are 1, 2, 3, 1, -2, -3 respectively the logarithms? Why is it found most convenient to calculate logarithms to a base 10? Of what number is 5 the logarithm to the base 10?

Find by the aid of the tables


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10 A right-angled triangle has a base of 240 feet, and the hypothenuse is 400 feet; find the area of the triangle.

11. A pyramid has a regular hexagon for its base, each side being 20 feet; find the cubical contents of the pyramid, if its altitude is 12 feet.

12. The three edges of a rectangular parallelepiped that meet in an angle are respectively 25, 54, 160. Find the side and diagonal of a cube which has the same volume as the parallelepiped.

13. A field has 5 sides; show by what measurements the field may be divided into triangles and its area found.



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 of the line between the points of section, is equal to the square of half the line.

Illustrate the truth of this proposition by taking the whole straight line AB to be 12 feet, and the two parts AD and DB to be 9 feet and 3 feet respectively.

2. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles.

A point is taken within a circle and from that point two equal straight lines are drawn to the circumference. Prove that the straight line which bisects the angle between these two straight lines passes through the centre of the circle.

3. If one circle touch another internally in any point, the straight line which joins their centres being produced shall pass through that point of contact.

4. Inscribe a circle in a given triangle.

If the points of contact be joined prove that each angle of the triangle so formed together with half the angle opposite to it in the original triangle is equal to a right angle.

5. Describe a circle about a given square.

Prove that the area of the square inscribed in any circle is half that of the square described about the same circle.

6. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or these produced, proportionally.

7. Equal parallelograms, which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional.

8. Prove that the ratio of the diameter of the circle described about a triangle to any one of the sides is the same as the ratio of half the rectangle of the remaining two sides of the triangle to the area of the triangle.

9. Define the grade and the degree, and express the interior angle of a regular pentagon both in grades and degrees.

10. Define the circular measure of an angle, and find two angles such that their difference is one degree, and their sum is the unit of circular measure.

11. Define the principal trigonometrical ratios, and find the sine of 30° and secant of 45°.

12. Prove that sin (A-B) = sin A cos B-cos A sin B, stating what limitations your proof supposes in the values of A and B respectively.

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