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ARITHMETIC AND ALGEBRA.
1. Find the value of is of a guinea ; and reduce ; of 2s. 41 d. to the fraction of a half-crown.
2. Divide 27 by of 43, and reduce the result to a decimal ; compare also the values of *, 33, and of 9.
3. A bankrupt owes 3 creditors £.400, £.300, and £.420 ; but his assets are only £.224. What will each creditor receive, and how many shillings in the pound will he pay?
4. If 4 men can remove 50 cubic yards of earth in 18 days, how long will 8 men be in removing 800 cubic feet?
5. Which stock offers the best interest—the Indian 5 per cents. at 109-or the Russian 41 per cents. at 911 ? And if I invest £.5,000 sterling in each of these two stocks, what will be the difference of income from the two funds ?
7. Divide a + a® + af 64 + a2 bo + 18 by a* + ab + a' b3 + ab + b*; and multiply a + b +-abi-atc-blc by a + b3 + c.
8. Find the greatest common measure and least multiple of
Br + x2 + + 1 and r - 2? + x-).
9. Divide 8–512 by 3 — 2/2; and extract the square root of 14 +8/3.
11. Find a 4th proportional to 100, 120.5 and 41.76 ; also, show that if
a : 6::c:d; then, pa + qb : ra-sb :: pc+qd : rc—sd. 12. Determine the value of the sum of an Arithmetical Progression of n terms; and sum the following series, 1+9+17+ &c., to 50 terms
13. Sum the geometric series, 3–1 + }- ... to 9 terms; and insert 4 arithmetic means between – 2 and — 18; and also 3 geometric means between } and 128.
14. Expand (2a-3b); and find the 5th term in the expansion of (x−y)".
15. A merchant made a mixture of wine at 28s. a gallon, with brandy at 42s. a gallon; and he found that by selling the mixture at 35 s. a gallon, his gain amounted to 15 per cent. on the price of the wine, or 20 per cent. on the price of the brandy: in what ratio were the wine and brandy mixed together?
16. Find the time which 3 persons would take jointly to do a piece of work, which they can do separately in a, b, c days respectively.
3 June 1862.
1. DEFINE obtuse angled triangle, acute angled triangle, rhombus, segment of a circle, and sector of a circle.
2. If from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.
If a trapezium and a triangle stand upon the same base, and the trapezium falls within the triangle, the perimeter of the trapezium shall be less than that of the triangle.
3. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
How many right angles are the interior angles of a pentagon and a decagon equal to ?
4. If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.
To construct a square which shall be equal to the difference of two squares. 5. The two diagonals of a parallelogram mutually bisect each other.
Also, the sum of the squares of the two diagonals is equal to the sum of the squares of the four sides.
6. If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Hence prove that the difference of the squares of two unequal lines, is equal to the rectangle contained by their sum and difference.
7. The diameter is the greatest straight line in a circle ; and, of all others, that which is nearer to the centre is always greater than one more remote.
8. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
A circle cannot be described about a rhombus.
9. Jf from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets shall touch the circle.
Hence, describe a circle which shall touch a given straight line, and pass through two given points on the same side of the given line.
10. If the angle of a triangie be divided into two equal angles, by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another.
il. If a solid angle be contained by three plane angles, these three angles are together less than four right angles.
12. To make a triangle equal to a given pentagon ; and to make a square equal to this triangle.
5 June 1862.
MECHANICS AND HYDROSTATICS.
1. EXPLAIN the principle of the parallelogram of forces ; and prove that if three forces acting on a point, keep it at rest, each of these forces is propor.. tional to the sine of the angle made by the other two.
2. Four forces represented by 15, 20, 25, 30, act on a point A. The two first are inclined to the left of the line AB, at the angles 60° and 135°; and the two others to the right of this line at the angles 30° and 90°; it is required to find, both by construction and calculation, a force which shall keep these four forces at rest.
3. Find the centre of gravity of a triangle ABC; and prove that three forces which are proportional to the lines GĂ, GB, GC, will keep the point G at rest, G being the centre of gravity of the triangle.
4. Find the distance of the centre of gravity of any system of bodies considered as points from a given plane.
5. If any number of forces P, Q, &c.; p, q, &c., acting upon the arms of a lever without weight, to turn it in opposite ways, be such that
P.CM + Q.CN + &c. = p.Cm + q.Cn + &c., C being the fulcrum, and CM, CN, &c. the perpendiculars upon the directions of these forces respectively, there will be an equilibrium.
6. In a vertical screw when there is an equilibrium, prove that the power is to the weight as the distance between two contiguous threads measured in a vertical direction to the circumference of the circle which the power describes. Also, show that the principle of virtual velocities is true in the case of the
7. In a simple isosceles truss roof ABC, calculate the tension of the tiebeam AC. Determine also the strains on the braces drawn from the middle of AC, to the middle of BA and BC.
8. In the construction of arches, prove that the horizontal pressure is the same at every point of the arch ; and that the perpendicular pressure upon each of the voussoirs is proportional to the secant of the angle which the joint makes with the vertical.
9. Explain Atwood's machine, and show how it has been applied to prove the laws of motion.
10. If an imperfectly elastic body impinges obliquely upon an immovable plane, to determine its motion after impact.
11. How long would a body be in falling down the lower half of an inclined plane, whose length is 150 feet and height 60 feet?
12. A pendulum is found to make 640 vibrations at the equator in the same time as it makes 641 at Greenwich ; if a string, hanging vertically, can just sustain 160 lbs. at Greenwich without breaking, how many pounds can the same string sustain at the equator ?
13. From what height must a centrifugal carriage descend down an inclined plane, so that with the velocity acquired it may enter the interior of the circumference of a circle placed in a vertical position at its base, and just pass the top of the circumference without falling ; the diameter of the circle being 12 feet?
14. Explain the fundamental property of fluids. By what experiments is this property established? Is water compressible or not?
15. If two fluids communicate in a bent tube, they will be in equilibrium, when their perpendicular altitudes above the horizontal plane where they meet are inversely as their densities.
Explain the hydrostatical paradox or hydrostatic bellows.
16. Describe Nicholson's hydrometer; and show in what respect it is superior to that of Sykes'.
What is the specific gravity of alcohol, supposing that the weight of Nicholson's hydrometer is 250 grains; and 72 grains are required to sink it to the proper depth in water, and 9 grains in alcohol ?
17. Explain how the elastic force of air varies. 1st, when the density varies and the temperature remains the same; and 2dly, when the temperature varies and the density remains the same.
18. Required the pressure of sea water on the cork in an empty bottle, supposing that its diameter is fths of an inch, and that it is sunk to the depth of 600 feet.
N.B.—The specific gravity of sea water is 1.026.
19. Describe the construction and mode of working of the air-pump. Find the density of the air in the receiver after four strokes of the piston, the volume of the receiver being ten times that of the barrel.
20. Describe the mode of filling and graduating the thermometer. Explain also the different scales of Fahrenheit, Centigrade, and Reaumur.
What degrees on the Centigrade and Reaumur's scales correspond to 100° of Fahrenheit?
INDIAN ENGINEER ESTABLISHMENT.
Competitive Examination, June 1862.
LAND SURVEYING. 1. In measuring between two stations A and B, a very wide river intervenes, crossing the line at right angles. There is no instrument available but a 100 feet chain, which evidently cannot span the stream; show how the exact breadth of the river may be determined and the measurement continued.
2. I have a French plan in which the scale has been omitted; one of the lines on the plan however is numbered 804:584 metres, and measures exactly 25 English inches.
Construct an equivalent English scale to read to 10 yards.
3. From the following extract taken from a fieldbook, plot the survey, on a scale of 5 chains to 1 inch.
N.B.--Fill in (with ink) the spaces left blank, and draw the scale to read to i link.