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If 0, & be the greatest and least angles of a triangle, the sides of which are in Arithmetical Progression, show that

4(1-cos 0) (1 − cos 4) = cos + cos φ.

7. In order to determine the height of a mountain a base was measured of 2750 feet. At either extremity of the base were taken the angles formed by the summit and the other extremity. These angles were 58° 28′ and 111° 53′. Also at the extremity from which the latter angle was taken the angular height of the mountain was II° 19'.

Find the height of the mountain.

8. Show how to construct a regular pentagon. Choose the most convenient axes and write down the equations of its sides and of its inscribed circle.

9. Find the angle between the straight lines represented by

the axes being rectangular.

x2+ y2 = 2xy cosec a,

Find the area of the triangle formed by these lines and the line

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and show that they intersect at right angles and also at an angle tan-11.

II. From P, a point on an ellipse whose centre is C, an ordinate PM is drawn parallel to the minor axis. On PM a point Q is taken such that QC = PM.

Find the locus of Q.

12. Establish an equation to a hyperbola. Prove that the eccentricity is the secant of half the angle between the asymptotes.

13. In an ellipse S, Fare the foci and P, Q any two points on the If tangents at P, Q intersect at 7, show that the angles QTS, PTF are equal.

curve.

14. Prove that the rectangle contained by the intercepts made by any tangent to a hyperbola on its asymptotes is constant.

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[Full marks may be obtained for about two-thirds of this paper. Great importance is attached to accuracy. N.B.-g may be taken = 32 feet per second per second.]

I. Three forces proportional to the sides of a triangle act at the middle points of these sides in directions perpendicular to them. Prove that the forces will be in equilibrium if they act all inwards or all outwards. Forces k. AB, k. BC, k. CD, and k. AD act perpendicularly to the sides AB, BC, CD, DA of a quadrilateral at their middle points, the first three acting inwards and the last outwards. Find their resultant.

2. Explain how to find the resultant of a number of parallel forces applied at given points in a straight line.

A rectangular portmanteau 3 feet in length and 2 feet in height and weighing 56 lbs. is carried up a staircase by two men supporting it along the front and back edges of its bottom face. If this face be held at an inclination of 30° to the horizon, find, to two decimal places of a pound, what portions of the weight are supported by the two men, supposing the centre of mass of the portmanteau to be at its centre of figure.

3. A body is made up of a number of different portions whose centres of mass are all in one plane. If the positions of these centres of mass are given and also the weights of the several portions, find the position of the centre of mass of the whole body.

Prove that the centre of mass of any quadrilateral lamina may be found by the following construction:-Let E, F be the middle points of the diagonals of the quadrilateral, O their point of intersection. On OE, OF cut off OM = & OE and ON = OF and complete the parallelogram OMGN having OM, ON as adjacent sides. Then G will be the centre of gravity of the lamina.

(This question may be solved either geometrically or analytically.)

4. Find the mechanical advantage in the common screw-press working without friction.

The coupling between two railway carriages consists of a right-handed and left-handed screw at opposite ends of a long bolt, working in nuts attached to the two carriages, each screw having five threads to the inch. Find the force which when applied to the end of an arm 18 inches long will cause the carriages to be drawn together with a force of I ton, taking the ratio of the circumference to the diameter of a circle as 37.

5. Explain what is meant by the angle of friction.

A body is placed on a rough plane whose inclination to the horizon is 45° the angle of friction being 30°. Prove that the force which must be applied to the body in a horizontal direction in order to just prevent it from slipping down is the same as if the plane were made smooth and its inclination to the horizon were decreased to 15°.

(This question may be solved either geometrically or analytically.)

6. A slip-carriage is detached from a train and brought to rest with uniform retardation in two minutes during which time it travels two-thirds of a mile. With what velocity was the train travelling when the carriage was detached?

The weight of the whole train was 130 tons and that of the slip-carriage 10 tons. Before the carriage was detached the train was just kept going with uniform velocity, by the pull of the engine, the resistance due to friction amounting to 35 lbs. weight per ton. Supposing the engine to pull the train with the same force after the carriage is detached, find the acceleration.

7. In Atwood's machine, two weights of 5 lbs. and 7 lbs. are attached to opposite ends of a string which passes over a light smooth pulley, and a rider weighing 4 lbs. is placed over the smaller weight. Find the acceleration.

When the system has moved through one foot from rest, the rider is detached by coming in contact with a fixed ring. How far will the 5 lb. weight descend below the ring before coming to rest?

8. A mass of m lbs. is making n complete revolutions per second on a smooth horizontal table in a circle of radius r to whose centre it is connected by a string. Find the tension of the string expressed in pounds' weight.

In the Watt's governor of a steam-engine, two equal light rods OA, OB, I foot long, are freely hinged at one end O to a vertical axis, and to their other ends A, B are attached equal weights of 7 lbs. The whole system is revolving about the vertical axis so that the rods OA, OB make equal angles with the vertical on opposite sides of it. Taking n = 22, find the number of revolutions per minute when the rods include an angle of 120°.

9. A particle is projected in vacuo from a point O with a velocity whose horizontal and vertical components are 32 feet per second and 80 per second (upwards) respectively. Draw a careful figure exhibiting the positions of the particle after 1, 2, 3, 4, and 5 seconds respectively, and state the coordinates of the particle at these instants, referred to horizontal and vertical axes through O.

Find also the coordinates of the focus of the parabola described.

10. A sphere of mass M moving with velocity U overtakes and impinges directly on a sphere of mass m, moving with velocity u in the same straight line. If the elasticity of the spheres be perfect, find by how much the kinetic energy of the first sphere is decreased and that of the second increased.

Three equal spheres whose coefficient of restitution (i.e. elasticity) is are ranged at rest in a straight line. The first sphere is then projected in the same straight line with velocity U, so as to impinge directly on the second. Find the final velocities of the three spheres after the last collision has taken place between them.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JUNE, 1897.

OBLIGATORY EXAMINATION.

I. EUCLID.

[Ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. In the absence of special directions to Candidates, any of the propositions within the limits prescribed for examination may be used in the solution of problems and riders. Great importance will be attached to accuracy.]

I.

Define carefully the following geometrical terms:-point, plane, plane angle, circle, parallelogram.

2.

Prove that if two angles of a triangle are equal to one another the sides which are opposite to the equal angles are also equal to one another.

3. Prove that if the square described on one of the sides of a triangle is equal to the sum of the squares described on the other two sides of it, the angle contained by these two sides is a right angle.

4. Prove that if a straight line be divided into any two parts, the squares on the whole line, and on one of the parts, are together equal to twice the rectangle contained by the whole and that part, together with the square on the other part.

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