5. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.

How must a straight line be divided into two parts, so that the rectangle contained by them may be the greatest possible?

6. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.

Draw the common tangents to two circles which cut one another.

7. Define the segment of a circle.

A segment of a circle being given, describe the circle of which it is the segment.

8. Inscribe a circle in a given triangle.

Inscribe also a second circle in the space intercepted at one of the angles, so as to touch the circumference of the circle and each of the sides containing the angle.

9. Describe an isosceles triangle BAC, having each of the angles at the base double of the third angle BAC.

If a point Ɗ be taken in AB so that AD is equal to BC, and DE be drawn parallel to AC to meet BC in E, shew that AB touches the circumscribing circle of the triangle CDE.

10. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

D is any point in AC, the base of an isosceles triangle ABC. DE and DF are straight lines making equal angles with AC, and meeting the equal sides BC and AB in E and F respectively. Prove that the triangles AED, CDF are equal in area.

II. Describe a rectilineal figure which shall be similar to one given rectilineal figure and equal to another given rectilineal figure.

[N.B.-Great importance will be attached to accuracy in numerical results.]

4. Divide 3 of 12 by 51, and the result by 14.

5. Add the difference between 035 of a ton and 064 of a cwt. to the difference between ·27 of a qr. and 78 of a lb., and give the answer in lbs. and the decimal of a lb.

9. Reduce 03257 of an acre to square yards and the decimal of a square yard.

10. Express 4 ozs. 17 dwts. 12 grs. as the decimal of a lb. troy.

II.

12.

Divide 12 miles 2 furlongs 20 poles 4 yards 2 feet 6 inches by 47.
Find the dividend of £3,407. 15s. at 135. 9d. in the £.

13. At what rate per cent. simple interest will £245 amount to £324. 18s. 7d. in 7 years?

14. A man leaves £32,818 to be divided among his four sons in the proportion of the fractions, , and . Find the share of each.

3

15. By selling goods for a certain sum a man gains 5 per cent. If he had sold them for 3 shillings more he would have gained 6 per cent. Find their cost price.

16. A buys a pipe of wine and sells it to B at a profit of 5 per cent., B sells it to C at a profit of 5 per cent., and C sells it to D for

£49. 125. 3d., making a profit of 12 per cent. What did the wine cost A?

17. Find the square root of 968 and the cube root of

182284*263.

18. A commences business with a capital of £4,000, and after

4 months takes B into partnership with a capital of £300. Two months later they take C into partnership with a capital of £5,000. At the end of the year their net profits amount to 16 per cent. on the whole capital invested. What should each receive of the profits?

19.

I cwt. 2 qrs. 12 lbs. of lead are rolled into a sheet 18 feet long and 6 feet wide. Find its thickness. (A cubic foot of lead weighs 720 lbs.)

20. By buying 3 per cent. consols at a certain price I find I obtain 3 per cent. for my money and derive a net income therefrom, after paying an income-tax of 6d. in the £, of £421. 45. Find the amount of stock and the price at which I bought it.

21. A train leaves London for Brighton at 9 A.M., travelling at a uniform rate of 15 miles an hour. An express train leaves Brighton for London at 10 A.M. and travels at a rate of 40 miles an hour. At what time will they pass each other and at what distance from London, the distance from London to Brighton being 50 miles?

III. ALGEBRA.

(Including Equations, Progressions, Permutations and Combinations, and the Binomial Theorem.)

[Great importance will be attached to accuracy in results.]

1. State the rule for affixing the correct sign to the product of two algebraical quantities which are affected with the same or different signs. Verify by reference to arithmetic the correctness of this rule.

Find the value of x2-6x+7 if x = 3−√3.

2. Multiply (a2 + b2+c2 - ab - ac - bc) by (a+b+c).

(2) (x-4)+(x-5)3=31 {(x-4)2- (x-5)2}.

7. Form a quadratic equation whose roots are - 3+ √2 and − 3 - √2.

8. If (p) and (q) are whole numbers, find generally the factor which

Hence rationalise 3a + 2a, and obtain the numerical value of the result.

9. In any scale of notation whose radix is (r), prove that if the sum of the digits of any whole number be divisible by (r− 1) the number itself will be divisible by (r − 1).

Given 222.22, in the scale whose radix is 5, reduce it to the denary scale.

IO.

If (a) and (r) are whole numbers, find the sum to (n) terms of the

a a a

series a + + + +&c., and explain how the sum of an infinite number

r

of terms of a series may have a finite value.

Express the circulating decimal 9 as a geometrical series, and find its value.

II. Find the total number of permutations of (n) things, taken all together, when there are (p) things of one sort and (9) of another, and the rest are unlike.

12. Two casks, each containing 20 gallons, are filled, one with water, the other with spirit, (x) gallons are drawn from each cask, mixed, and the casks are again filled up with the mixture; when this is done a second time it is found that the quantity of spirit in one cask is to the quantity in the other as 5 to 3. Find (x).

13. Assuming the form of the expansion by the binomial theorem of (1+x)-1 when (n) is any positive whole number, shew that the coefficient of xa in (1+x). (1+x)"−1 is the number of combinations of (n) things taken (r) together. State briefly how this property is applicable to the proof of the binomial theorem. Express the sum of the coefficients of a binomial whose index is a positive whole number in terms of a power of 2, and verify the property in the expansion of (1+x)6.

The coefficient of the 3rd term in the expansion of (1 − x)-" is 2; find (n) and the coefficient of the fifth term.