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7. Show generally that when the two quadratic

3. At the beginning of the year a set of books equations, ax2+2h,x+b2=0 and a,x2+2h,x is re-opened, and there are balances at the follow+b=0, have a common root, then always (a,b,ing accounts, viz. :+ba,-2hh,)=4(a,b,—h3)(a,b,—h23).

8. Clear of radicals, each separately by itself, the equations, x+y*+z1=0 and xz+y*+z3=0; and arrange symmetrically in xyz the result in each case.

9. Divide £8 11s. 8d. between four men, five women, six boys, and seven girls; giving each woman a fifth, each boy a fourth, and each girl a third less than the share of each man.

10. A schoolboy, having purchased m dozen of apples at p pence per dozen, and n dozen more at q pence per dozen, disposes of the entire purchase to his companions at (p+q) pence per dozen; show clearly the condition that he should gain by the transaction, and give the amount in pence his profit when he does.

Cash.

Bills receivable.
Bills payable.
Goods.
Furniture.

On which side of the ledger will each be put ?

4. In the Cash-book form supplied for the purpose, enter all the cash and bank transactions which appear in the following general record, balancing on 31st January :

1882.

January 1, Cash on hands

of

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Do., in Bank...
5, Bought of Smith and
Sons, goods...
,, Do., James Murray do.
9, Handed Smith and Son
my acceptance at three
months for £95 4s. 3d.
and cheque for £195,
which, with £5 dis-
count allowed, settle
their account

12, Sold Francis Burke,
goods

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325 13 0

248 15 0

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18, Received from Francis
Burke, cash

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650

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5. Journalise all the foregoing transactions. Cash paid for petty charges, and discounts, may go to Profit and Loss direct.

6. Post all the transactions from the Journal into the Ledger, opening the necessary accounts. 7. Balance and close the Ledger-taking the value of goods on hands at £65 6s. 8d.

8. Under your last entry in the Journal make out a statement of the assets and liabilities on the 31st January.

EASTERN CADETSHIPS.-A competitive examination will be held in London for Eastern Cadetships (age 21 to 24) on 22nd August. Preliminary examinations will be held in London, Dublin, and Edinburgh, on the 11th August.

EXAMINATIONS FOR MEN CLERKSHIPS, OR LOWER DIVISION CLERKSHIPS.

ADVERTISEMENTS

AGE, 17 to 20:

to time by the Civil Service Commissioners, and published in many of the daily papers, intimating that a Test examination will be held on a day named at which eligible candidates may attend. The same advertisement also gives the date of the competitive examination, which is usually a couple of weeks after the test.

"No candidate can be admitted to the competition who has not previously satisfied the Civil Service Commissioners that he possesses the requisite amount of proficiency in handwriting, orthography, and arithmetic (including vulgar and decimal fractions)." The number of persons to be selected at each examination will be published as part of the notice of every such examination.

The result of each competitive examination is published a few weeks later, the candidates appearing on the list in the order of merit.

From this list the Civil Service Commissioners supply clerks to the departments for the existing vacancies, or those that may afterwards arise. The subjects of examination are as follows:Subjects. Maximum Marks,

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400

400

400

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The following are the maximum marks scored by the four successful candidates placed first on the list in the latest return issued:

The first scored 2,117; the second got 2,083; the third, 2,069; and the fourth, 2,025. About 300 competed on this occasion. We annex some of the Examination Papers.

DICTATION (1).

The influence of William was no longer what it had been. The general opinion of Europe was, that the riches and the credit of England were completely exhausted; and both her confederates and her enemies imagined that they might safely treat her with indignity. Spain, true to her invariable maxim that everything ought to be done for her and nothing by her, had the effrontery to reproach the Prince to whom she owed the retention of the

ships to defend her possessions in Italy. The Imperial ministers formed and executed resolutions gravely affecting the interests of the coalition without consulting him who had been the author and the soul of the coalition. Lewis had, after the failure of the Assassination Plot, made up his mind to the disagreeable necessity of recognising William, and had authorised his minister to make a declaration to that effect. But the defection of Savoy, the neutrality of Italy, the disunion among the allies, and, above all, the distresses of England, exaggerated as they were in all the letters which the Jacobites of Paris received from the Jacobites of London, produced a change. The tone of the French minister became high and arrogant; he refused to give any pledge that his master would acknowledge the Prince of Orange as King of Great Britain. The joy was great among the Jacobites. They had always, they said, been certain that the Great Monarch would not be so unmindful of his own glory and of the common interest of Sovereigns as to abandon the cause of his unfortunate guests, and to call a usurper his brother. They knew, from the best authority, that His Most Christian Majesty had lately given satisfactory assurances to King James. Indeed, there is reason to believe that the project of an invasion of our island was again seriously discussed at the French Court.

DICTATION (2).

The colony of a civilised nation, which takes possession either of a waste country, or of one so thinly inhabited that the natives easily give place to the new settlers, advances more rapidly to wealth and greatness than any other human society. The colonists carry out with them a superior to what can grow up of its own accord in knowledge of agriculture and of other useful arts, the course of many centuries among savage and barbarous nations. They carry out with them, too, the habit of subordination, some notion of the regular government which takes place in their own country, of the system of laws which supports it, and of a regular administration of justice; and they naturally establish something of the same kind in the new settlement. But among savage and barbarous nations, the natural progress of law and government is still slower than the natural progress of arts, after law and government have been so far established, as is necessary for their protection. Every colonist gets more land than he can possibly cultivate. He has no rent, and

scarcely any taxes to pay. No landlord shares with him in its produce, and the share of the sovereign is commonly but a trifle. He has every motive to increase as much as possible a produce, which is thus to be almost entirely his own. But his land is commonly so extensive, that, with all his own industry, and with all the industry of other people whom he can employ, he can seldom make it produce the tenth part of what it is capable of producing. He is eager, therefore, to collect labourers from all quarters, and to reward them with the most liberal wages. In other countries the superior orders of people oppress the inferior one. But, in new colonies, the interest of the superior orders obliges them to treat the inferior one with generosity and humanity, at least where the inferiors are not in a state of slavery.

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7. Find the simple interest on £251 5s. from June 15, 1881, to August 27, 1882, at 31 per cent. per annum.

8. Extract the square root of 105404-440241 in the scale of 6, and express the result in the duodenary scale.

9. How much 3 per cent. stock must be sold out at 89, in order that the owner's income may be increased £12 by investing the proceeds in 4 per cent. stock at 917 ? The brokerage on each sale is per cent. 10. The interest on a certain sum in 4 years is

£70 6s. 3d., and the true discount for the same time, at the same rate per cent., is £56 5s. Find the sum, and the rate of interest.

11. The first of 5 clerks can copy 3 lines as soon as the second can copy 4, the second can copy 5 as soon as the third can copy 3, the third can copy 7 as soon as the fourth can copy 8, and the fourth can copy 14 as soon as the 5th can copy 17. lines will the fifth copy while the first is copying 490 ?

How many

12. A grocer buys one kind of tea at £7 14s. a cwt., and another kind at 2s. a lb. In what proportion must he mix the two to gain 50 per cent. by selling the mixture at 2s. 6d. a lb. ?

13. Two towns, A and B, are 30 miles apart. The road goes over a hill, the summit of which is 6 miles from A. Two men set out at the same time from A and B ; the former walks 4 miles an hour up hill and 5 down. The latter 3 miles an hour up hill and 4 down. How far from A will they meet?

14. A cylindrical reservoir whose diameter is 10 feet and height 10 feet is supplied by two pipes. One of them alone could fill it in 20 minutes, and water flows in through the other at the rate of 200 gallons a minute. If both pipes are opened simultaneously, how long will they take to fill the reservoir, supposing the volume of a gallon of water to be 277.25 cubic inches? (N.B.—The area of a circle is 3.1416 times the square of its radius.)

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As one of the chief functions of THE COMPETITOR is to assist in teaching the various subjects treated of in its pages, the following solutions are made more explanatory than would be necessary, or even advisable, at an examination. Some questions have been worked in two ways where there appears to be a decided advantage in departing from the ordinary method.

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Or thus. The quantity of lower dimension, viz. 2x3+3x3-2x-3, contains the factor -1, which will be at once seen if this quantity is written:

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The factor 2x+3 cannot be a measure of the other given quantity, since 2x is not contained in its first term 35, nor 8 in its last term 1: therefore, if the given expressions have a common measure it must be x3-1, or a factor of it.

Then divide a2-1 into 3x+5x1+4x3 −6x2-7x+1: if there is no remainder, -1 is the G.C.M.; if the remainder is +1 or 2-1 (the two factors of x2-1), then that remainder is the G.C.M. Subtracting and adding 3a and arranging, this becomes :

3x3-3x2+5x1 — 5x2+7x2 −7x−x2+1,

i.e., 3x23(x2−1)+5x2(x2−1)+7x(x2 −1)−1(x2—1)
=(x2−1) (3x2+5x2+7x−1)

3. Let B's present age be a years, then A's present age is 3x years, and their respective ages 8 years ago would be denoted by x-8 and 3x-8 years.

.. 5 (x-8)=3x-8;

or 5x-3x=40-8,
i.e. 2x=32:

x=16 and 3x=48.

A's present age is therefore 48, and B's 16 years.

4. The difference of two squares is equal to the product of the sum and difference of their

square roots.

.. 16b2c-(a-b2-4c22

= [4bc+(a2—b2-4c2)] [4bc-(a2- b2 — 4c2)]

=(a2-b2+4bc-4c2) (4bc-a2+b2+4c2)

={a-(b-2c)} {(b+2c)3—a2}

=(a+b-2c) (a+2c-b) (a+b+2c) (b+2c-a). Ans.

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