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II. ARITHMETIC.

(Including the use of Common Logarithms.)

[N.B.-Great importance is attached to accuracy. Candidates are expected to use Arithmetical methods of solution.]

I. Divide 10101255 by 2185, and explain the process.

2. Give a definition of Multiplication which shall include the multipli. cation of one fraction by another.

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3. Find the Greatest Common Measure, and the Least Common Multiple of the numbers 15496 and 12665, and show, by general reasoning, that these four numbers form a proportion.

4. Multiply 4*327615 by '003248, and divide 292262016 by 327648.

5. Find the rent of a piece of ground covering 14 acres 16 poles 22 sq. yards 6 sq. feet at £3. os. 6d. per acre.

6. Explain what is meant by a Recurring Decimal, and find the vulgar fractions equivalent to the decimals, ∙123, 11*1234, and '0034568.

7. Reduce of 4 guineas to the fraction of £7, and £3. 175. 2§d., to the decimal of £4 to eight places of decimals.

8. Find the Square Root of 199.6569.

When a regiment of 962 men is drawn up in a solid square, one man is left out; find the number of men in the face of the square.

9. A can do a piece of work in 11 days, B in 20 days, and C in 55 days; how soon can the work be done if A is assisted by B and C on alternate days?

IO. Find the cost of carpet 2 feet 3 inches wide for a room 20 ft. 3 in. long by 13 ft. 4 in. wide, at 5s. a yard.

II.

If the simple interest on £4373.6s. 8d. for 14 years be £246, what is the rate per cent.?

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12. Find the discount on £51. 15s. 10d., due 4 years hence, at 3 per cent., simple interest.

13. A gravel walk 6 feet wide runs round a grass-plot 60 feet long and 40 feet wide. If gravel is 3s. per cubic yard, find the cost of a coat of gravel on the path 3 inches deep.

14. A person transfers £4865 from the 3 per cents. at 93 to the 3 per cents. at 87. Find the change in his income.

15. Explain what is meant by the characteristic of a logarithm, and state the rule for finding the characteristic of the logarithms to base 10 of the numbers less than unity.

16. Find the logarithms to the base 2 of—

64, 14, and 3/32.

17. Find from the tables the logarithms of 35726 and 357°26437.

18. Find from the tables the number whose logarithm is 4'9220534.

19. Employ the tables to find the value of the product of 52'4574 by 3'78472.

20. Employ the tables to find the mean proportional between 33°549 and 44 642.

21. Employ the tables to find the value of the expression—

(5'7432)1°246

22. A tricycle, going at the rate of 5 miles an hour, passes a milestone, and 14 minutes afterwards, a bicycle, going in the same direction at the rate of 12 miles an hour, passes the same milestone; find when and where the bicycle will overtake the tricycle.

III. ALGEBRA.

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

[N.B.-Great importance is attached to accuracy.]

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14x2 − 23xy+312 ̄ 35x2+47xy+612 *

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ab (b-c) (c-a) ac (a - b) (b −c) bc (a - b) (c − a) *

5. Apply the process for extracting the square root to find m and n when x+ax3+mx2+cx+n is a complete square.

6. If the roots of the equation

(1−9+) x2+p(1+g)x+9(9-1) +mo

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9. The metal of a solid sphere, radius v, is made into a hollow sphere whose internal radius is ; required its thickness. [The volume of a sphere, radius ", is Tr3.]

IO. Two rectangular lawns have the same area (a2), but the perimeter of the one is one-fourth longer than that of the other, which is a square; required its dimensions.

II.

q terms=

If the sum of the first p terms in an A.P. =o the sum of the next −a(p+q)¶ ̧

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12. Is the coefficient of x in the expansion of (1-x)" equal to the number of combinations of (n+r) things taken r together?

If not, amend the proposition, and prove it.

IV. PLANE TRIGONOMETRY.

(Including the Solution of Triangles.)

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

I. Define the circular measure of an angle.

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calculate the number of degrees, minutes, and seconds

in the unit of circular measure.

2.

Prove geometrically that cos 2a=cos2 a - sin2 a.

Find the value of cos 15°, and prove that

(sec 15° + cosec 15°)2=24.

3. Write down a single expression for all the angles which have the same cotangent as the angle a; and give, in a single expression for 0, the complete solution of the equation 3 tan2 20 = 1.

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5. Determine x from the equation sin cot-1=tan cos-1√x; and prove that if a tan1, ẞ=tan ̄1, then cos 2a=sin 4ß.

76. ABC is an isosceles triangle, right angled at C. D is the middle point of AC. Prove that DB divides the angle B into two parts whose cotangents are as 2 : 3.

7. Prove that in any triangle ca cos B+b cos A.

From this and the two corresponding formulæ

a=b cos C+c cos B, b=c cos A+ a cos C

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and if the sines of the angles are as 13: 14: 15 find the ratio of the cosines.

9. Obtain the expression for the area of any triangle in terms of the sides.

If the sides of a triangle are in Arithmetical Progression and its area ths of the area of an equilateral triangle of the same perimeter, prove that its sides are in the ratio 3:57.

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10. Given two sides of a triangle and the included angle, obtain a formula, adapted to logarithmic computation, which will enable us to determine the other two angles.

Given a=2*7402, b=*7401, C=59° 27′ 5′′, solve the triangle completely, with the aid of the logarithm tables supplied.

II. The extremity of the shadow of a flagstaff 6 feet high, standing on the top of a regular pyramid on a square base, just reaches a side of the base and is distant 56 feet and 8 feet from the extremities of that side. If the height of the pyramid be 34 feet, find the sun's altitude.

[Logarithm Tables to seven figures were supplied.]

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