1. Subtract (1) 38792 from 42635; (2) 2.8453

from 45.5.

Explain in either case the principle upon which you "borrow" and " carry."

A

2. If 12 men can perform a piece of work in 8 days, in how many days will one-third the number of men do double the amount of work?

Find a fourth proportional to,, and 9.

3. Find the greatest common measure of 36585 and 109484.

Can the greatest common measure of two numbers exceed the difference between them?

4. From 13 subtract 8. Why do you reduce the fractions to a common denominator?

By what fraction will ++ 2 differ from unity? What fraction of £4 6s. 8d. is of 13s. 4d.?

2

5. Reduce to a decimal, and state why the decimal terminates.

6. Find (2·01)3. Extract the square root of— (1) 465-2649

(2) 12.1.

7. What is the amount of £2674 15s. for 4 years, at 5 per cent, simple interest?

At what rate per cent., simple interest, will £450 amount to £576 in four years?

8. If the civil year, when not a leap year, contains 365 days, and if there are 97 leap years in 400 years, what is the average length of the year?

2. Reduce to their smallest forms

a-2b+3c a+2b-c

(1)

(2)

and

3. Multiply

x3-3x2+3x-1 by x3+3x2+3x+1,

4. Divide

(x3+xły3+y3) by (x3 — y1)

x1 — 9x2y2 – 12xy3 – 4y1 by x2+3xy+2y2

and

5. Prove

(x+y) by (x+y3)

(1) (1+2x+3x2)2 − (1 − 2x − 3x2)2=4x (2+3x)

7. Divide the number 91 into two such parts, that if the greater be divided by their difference, the quotient shall be 7.

8. A luggage train going at the rate of 10 miles an hour is some distance in advance of an express engine, which starts to overtake it, and just comes up with it in 5 hours; had the express engine travelled 10 miles an hour less, it would have taken

7 hours to have come up with the luggage train. Determine the distance between the train and the express engine at first, and the rate at which the latter travelled.

Euclid.

1. Upon the same base and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those terminated in the other extremity.

Prove this for the case in which the vertex of one triangle falls without the other.

If the words printed above in italics be omitted, would the proposition as then stated be true?

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

Divide a given triangle into four equal parts.

3. What is a rectangle? If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, and to twice the rectangle contained by the parts.

If the line be bisected, what do the rectangles contained by the parts become?

4. Divide a given straight line into two parts, so that the rectangle contained by the whole line and one of the parts shall be equal to the square of the other part.

In Euclid's figure can you point out any other straight line, besides the given line, that is divided according to the proposition.

NATURAL SCIENCES.

By MAJOR-GEN. PORTLOCK, R.E., LL.D., F.R.S.,

IN THE ABSENCE OF

W. WARINGTON SMYTH, ESQ., A.M., F.R.S.

Maximum Number of Marks

600

[The correct answering of twelve of the Mineralogical Questions, or of twelve of the Geological, or of a corresponding number in both subjects, will entitle the Candidate to the full number of Marks.]

Mineralogy.

1. In the description of mineral substances, what is meant by the terms amorphous, botryoidal, crystalline ?

2. Define the terms crystal and crystallization. 3. What conditions are necessary towards the formation of crystals?

4. How many systems of crystallization are now admitted by mineralogists? Name and define

them.

5. What is meant by the cleavage of minerals, and what are the most important laws relating to it?