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MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,
JUNE, 1882.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. AND VI.).

[Great importance will be attached to accuracy.]

I. What is Euclid's definition of a straight line? Is this definition available for demonstration? What axiom concerning straight lines furnishes the test required? Enunciate and prove the proposition in Euclid in the proof of which this axiom is first referred to.

2. Equal triangles upon equal bases in the same straight line and towards the same parts are between the same parallels.

ABC is a triangle; join D and E, the middle points of AB and AC: prove, by the use of propositions of the first book only, that DE is parallel to BC.

3. Describe a parallelogram equal to a given triangle and having one of its angles equal to a given rectilineal angle.

Describe a parallelogram, the area and the perimeter of which shall be each equal to the area and perimeter of a given triangle.

4.

Enunciate and prove the proposition from which the corollary is inferred that "the difference of the squares of two unequal straight lines is equal to the rectangle contained by their sum and difference."

Find the straight line the square of which shall be equal to the rectangle contained by the sum and difference of two given straight lines.

5. Divide a 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.

W. P.

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If AB be divided in C so that the rectangle AB, BC is equal to the square on AC, and CD be taken equal to BC, shew that AC is divided in D so that the rectangle AC, AD is equal to the square on CD.

6. The diameter is the greatest straight line in a circle, and of all others that which is nearer to the centre is greater than one more remote.

Two circles cut one another, through a point of intersection draw a straight line terminated by the circumferences, so that the chords so intercepted in each circle may be equal.

7. The angle in a semicircle is a right angle, the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle.

When are segments of circles said to be similar? If two circles touch each other externally, any straight line drawn through the point of contact will cut off similar segments; when will all the four segments cut off be similar?

8. Describe a circle about a given triangle.

Hence shew that the perpendiculars drawn from the middle points of the sides of a triangle meet in the same point.

9. Describe a circle about an equilateral and equiangular pentagon; assuming that the straight lines bisecting each of the angles of the pentagon intersect in the same point.

If A, B, C, D, E be the angles of the pentagon taken in order, prove that the line CE is parallel to AB.

IO.

Define similar rectilineal figures; if the figures be triangles, is there anything superfluous in the definition?

Similar triangles are to each other in the duplicate ratio of their homologous sides.

ABC is a triangle, AE and BF intersecting in G are drawn to bisect the sides BC, AC in E and F; compare the areas of the triangles AGB, FGE.

II. If an angle of a triangle be bisected by a straight line which cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line which bisects the angle.

If ABC be a right-angled triangle, whose right angle B is bisected by BF, cutting the base in F and meeting the circumference described about ABC in D, prove that the rectangle contained by BD and BF is equal to twice the area of ABC.

II. ARITHMETIC.

(Including the use of Common Logarithms.)

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

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4.

Divide 4 of 17 by 53, and the result by 21.

5. Add together 0045 of a mile, 052 of a furlong, and 153 of a pole; subtract the result from 35 yards, and express the answer in inches and the decimal of an inch.

6. Multiply 17.63745 by 004905.

7. Divide 364 353 by 00671.

8. Divide 3714285 by 5°0571428, and express the answer as a decimal correct to four places.

9. Reduce of 25. 7d. to the decimal of 15s. 10d.

10. Express 15 cwt. 2 qrs. 21 lbs. as the decimal of 5 tons.

II.

Divide 8 acres 1 rood 31 poles 27 sq. yards 2 sq. feet and 116 sq. inches by 53.

12. What is the rateable value of a parish if a rate of 3s. 5d. in the £ produces a sum of £14,352. IS. 8d.?

13. What principal will amount to £10,672. 95. 3d. in 7 years and months at 5 per cent. per annum simple interest?

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14. A cistern which can be filled by one tap in 28 hours is emptied by four others. The first of these alone would empty it in 10 hours, the second in 12 hours, the third in 15 hours, and the fourth in 21 hours. Supposing the cistern to be full, in what time would it be emptied if all the five taps were set running together?

15. A man buys a parcel of coffee and re-sells it, losing per cent. on the transaction. If he had obtained £14 more, he would have gained 4 per cent. What was the original sum paid for the coffee?

16. A cubic foot of Canadian elm weighs 725 as much as a cubic foot of water; and a cubic foot of water weighs 1000 ozs.: what will be the weight of a beam of Canadian elm 12 feet 6 ins. long, 1 ft. 6 ins. deep, and I ft. 3 ins. thick?

17. It is desired to put a cubical case, whose content is 4019'679 cubic feet, through a square hatchway whose area is 37791*36 square inches. Shew whether this can be done.

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18. In a rectangular piece of level ground 160 feet long and 80 feet broad it is required to make a rectangular bath 145 feet long and 65 feet broad. The earth excavated is to be placed evenly on the surrounding ground. What must be the depth of the bath in inches in order that the surrounding ground may be 8 feet above its former level?

19. A person gave £75 for 20 casks of oil, each containing 30 gallons. He sold 5 casks at three shillings per gallon, one cask was stove in and the whole of its contents lost, and 15 gallons were also lost by ordinary leakage. He then sold the remainder at a price per gallon which made his gain amount to 20 per cent. on the whole transaction. What was his selling price per gallon at the second sale?

20.

A and B start at the same time from London to Blisworth, A walking 4 miles an hour, B riding 9 miles an hour. B reaches Blisworth in 4 hours, and immediately rides back to London. After 3 hours' rest he starts again for Blisworth at the same rate. How far from London will he overtake A, who has in the meantime rested for 6 hours?

21. £5000 was invested in 3 per cent. stock at par, subject to an income-tax of 6d. in the £. Another £5000 was invested in 3 per cent. stock at 104, free from income-tax. What investment was the more advantageous, and by how much?

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III. ALGEBRA.

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

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

I. Prove the rule of signs in Algebraical Subtraction.

From

{m (2m – 3p) – 2n (4n − 3p)} x+ {m ( p − m) − p (2n+p)}y,

3 { 1 (2n-32) - 1⁄2 (2m - 3p)} x

2m − 3p)} x − {p (p − m)+2n (2n+p)} y,

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4a (a+b+c)+10bc − 3 (b2+c2) by 2a+3b-c,

and find the continued product of

2aa – zax1, za ̄1+2x ̄1, 4a3x3+9a ̄3x§.

3. Find the Greatest Common Measure of

7a3 – 6a2b – 18ab2+4b3, 14a3 – 19a2b − 32ab2+28b3,

and the Least Common Multiple of

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5.

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Write down the fifth power of a+ √x, and shew that it will be both real and rational if x=(5±2 √5) a2.

6. Extract the square root of

81 (x2+1)+36x (x2 – 1) – 158x2.

The first and second of the three digits by which a perfect square is expressed are 1 and 2n respectively; find the third digit, and shew that, whatever be the scale of notation, a perfect square would be obtained either by reversing the order of these three digits, or by inserting the same number of cyphers between the first and second and between the second and third of the digits.

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