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7. Any number of lines c,b,, c2b2, &c. are drawn parallel to the

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8. If C be the centre of an ellipse and P, Q be any two points in the ellipse, such that C P and C Q are at right angles to one another, then

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9. A P is the arc of a conic section of which the vertex is A: PG the normal, and P K a perpendicular to the chord A P meeting the axis in G and K. Shew that G K is equal to half the latus rectum.

MECHANICS.

Morning Paper.

1. Two pressures acting perpendicularly on a straight lever and on the same side of the fulcrum will balance one another if they be reciprocally proportional to their distances from the fulcrum. Ex. Weights of 18 and 21 lbs. are placed at distances x and 3 x respectively from the fulcrum, what pressure must act in a contrary direction at a distance 9 x from the fulcrum to balance them, and what will be the pressure on the fulcrum?

2. Give illustrations of the different kinds of levers. State what are the requisites for a good balance and shew by what means the sensibility of the common steel-yard may be increased or diminished.

3. If a point be kept at rest by three pressures acting upon it at the same instant, any three lines forming a triangle in the direction of the pressures will represent them in magnitude. What do you understand by a line representing a force in magnitude and direction? Could any line, 17 inches in length, represent a pressure equivalent to 15 lbs. in weight?

4. Find the centre of gravity of any number of heavy bodies in the same plane.

5. Find the relation between P. and W. in a system of Pulleys where each string is attached to the weight, the Pulleys being all equal in weight. 6. State the principle of virtual velocities and prove in the inclined plane that PX P's velocity =WXW's velocity.

7. What is the difference between the momentum of a body at any instant and the moving force acting upon the body at that instant. Explain the meaning of action and reaction being equal and opposite.

8. State the third law of motion and mention experiments which illustrate the truth of it.

9. In uniform motion when one body impunges directly upon another, the centre of gravity of the bodies moves in the same line, and with the same velocity before or after impact.

10. Shew that in uniformly accelerated motion s= f.t2.

11. Prove that the velocity of a projectile at any point of its path is equal to that which it would have acquired in falling down a distance equal to its distance from the focus acted on by gravity. Verify this when the point is at the vertex of the parabola.

Afternoon Paper.

1. In one-half of a sphere a cone is described having a great circle of the sphere for its base and the radius for its altitude; in the remaining half of the sphere another cone is placed of the same altitude and of one-fourth the volume of the former. Find the centre of gravity of the volume between the cones and the sphere.

2. The sides of any quadrilateral figure ABCD are bisected in the

D

d

B

points a, b, c & d. Two forces P&Q act at a & d in contrary directions, what forces must act at c & b to prevent the figure from turning round, and what relation must exist between P & Q if the four forces are in equilibrium.

C

3. A triangular plane ABC is kept in equilibrium by three systems of pulleys (in which the same string passes round all the pulleys) each having one block fastened to a fixed external point and another attached to an angular point of the triangle by a string whose direction bisects the angle. The same string passes round all the pulleys and is drawn tight by a certain weight. Shew that the numbers of the strings between the pul

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4. One corner of a triangle equal to part of its area is cut off

n

by a line parallel to its base, find the centre of gravity of the remainder.

5. In a vertical circular groove a rod AB of given uniform weight is placed subtending a right angle at the centre. If μ be the coefficient of friction and the greatest angle which the rod can make with the horizon

when equilibrium exists prove that μ = tan

04

2

6. Three equal bodies are moving in the same direction with velocities proportional to 3, 2 & 1, and the distances between them were at a given time the same; shew after impact the velocities will continue to be in Arithmetical progression.

7. A body weighing 15 lbs. is observed to move with a velocity which steadily increases at the rate of two feet per second. What weight would the force support which is acting on the body?

8. A weight of 6 lbs. resting on a smooth plane inclined to the horizon at an angle of 30° is fastened by a string passing over the top of the plane to a weight of 4 lbs. hanging freely. After gravity has acted on this machine for 3 seconds and the two weights are then in the same horizontal line the string breaks; find the positions of the bodies after 5 seconds more.

9. A body falling in vacuo under the action of gravity, is observed to fall through 144.9 feet and 177.1 feet in two consecutive seconds, determine the accelerating force of gravity and the time from the beginning of the motion.

10. A body is projected with a given velocity in a given direction and at the highest point of its path, its velocity is suddenly doubled. Determine the range and time of flight.

f

FOURTH CLASS.

EUCLID AND ALGEBRA.

Morning Paper.

1. Find a mean proportional between two given straight lines. In this case shew how similar triangles are to one another in the duplicate ratio of their homologous sides.

2. The parallelograms about the diameter of any parallelogram are similar to the whole and to one another.

3. If a solid angle be contained by three plane angles, any two of them are greater than the third.

4. Find the cube root of 207

measure of x

x} + 6 z} x}.

945 and the greatest common

− 2 x} z} + 4 (xyz)} + 8 z} ył and y} x} — 2 y} zł — 3 zł

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5. Find an expression for the sum of a Geometric series, first when the number of terms is limited, and secondly when unlimited.

6. In the expansion of (a + x)" where n is a whole number, find at what term the coefficients begin to converge. Investigate the general formula for the expansion of a multinomial and employ it to determine the coefficient of x27 in (a 3x2 + 4x5)12.

7. Explain clearly the principle of indeterminate coefficients and employ them to resolve into three other fractions and to sum

x4

3

2x2+8

the series 3.22 + 6.42 +9.62 + for n terms.

....

8. Investigate a formula for converting any continued fraction into a series of converging fractions by means of the partial quotients, and shew that every converging fraction is in its lowest terms.

9. Find the number of solutions in positive integers of the equation

ax + by = C.

10. Find an expression for the equated time of payment of two sums due at different times, and shew that the common rule is in favor of

the payer.

11. Find how many different divisors a composite number may have, and determine into how many factors prime to one another it may be resolved. Ex. 176400.

12. Prove that the chance of an event taking place which is contingent upon other events is the continued product of the chances of the separate events. With two dice what is the chance of throwing ten the first time, seven the second, and four the third.

Afternoon Paper.

1. Given an equilateral and equiangular pentagon, it is required to divide its area into three equal parts by describing within it two similar pentagons.

2. In a plane inclined to the horizon at a certain angle, there are three points, find a point in the horizontal plane equally distant from each of them.

3. Given the base, the ratio of the sides containing the vertical angle, and the distance of the vertex from a given point in the base; to construct the triangle.

4. Expand (1) in a series ascending by powers of sin. 0,

when 2 cos

α

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5. If be a fraction and if a and the successive remainders be multiЪ

plied by a series of quantities q, q', q′′, &c., and the successive products p', be divided by b giving quotients p, p', p", &c., shew that ++ զզր p", + &c. qq'q"

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6. Find a whole number which is greater than three times the integral part of its square root by unity. Shew that there are two solutions of the problem and no more.

7. A rupee is thrown up into the air twelve times, what is the probability that it will fall for an even number of times with the face uppermost. 8. Prove that N = a. 2 N + a3

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nearly, if a be the integer next 2a3+ N

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find the square root of 1410123404 in a scale whose radix is 5.

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11. If a, b, c, be in harmonic progression shew that +

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