If there be one fixed, and two moveable, pulleys, find how far the weight can be practically raised, if it be initially 24 feet below the lower moveable pulley.

6. Explain how velocity is measured, and if u be the measure of a velocity when s feet and t seconds are the units of space and time, find its measure when the units are s' feet and t' seconds.

Compare the velocities of two particles, one of which describes 9 miles in two hours and the other II feet in 4 seconds.

7. Express the space passed over by a particle moving subject to uniform acceleration in terms of its initial and final velocities and the time occupied.

An engine-driver reduces the speed of a train (at a uniform rate) from 40 to 30 miles per hour in a quarter of a minute. Find the distance passed over in this time, and also the velocity of the train when half this distance has been described.

8. A ball of mass m impinges directly upon a ball at rest of the same size but of mass m'. Show that after impact the balls will move in the same direction or in opposite directions according as mis > or < em', e being the coefficient of elasticity.

9. A particle is projected from O with velocity u in a direction inclined to the horizon at an angle a. Prove that the equation of its path is

the axes of x and y being the horizontal and vertical lines drawn in the plane of the motion through the point of projection.

Find the velocity and angle of projection of a particle which being thrown from the level of the ground just clears a wall 18 feet high at a distance of 36 feet from the point of projection, and strikes a wall parallel to the former and 60 feet beyond it, at a point 8 feet above the ground. The plane of projection is perpendicular to the walls.

10. A particle of weight W, attached by a string of length L to the vertex of a smooth cone whose axis is vertical and semi-vertical angle a describes a horizontal circle on the surface of the cone with uniform velocity v; find the tension of the string and the pressure on the surface.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1895.

OBLIGATORY EXAMINATION.

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

[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. Great importance will be attached to accuracy.]

I. If two triangles ABC, DEF, have the sides AB, AC of the one respectively equal to the sides DE, DF of the other, and the angle BAC equal to the angle EDF, the triangles are equal in all respects.

2.

Prove that triangles on equal bases and between the same parallels are equal to one another.

If two triangles have equal bases, but the height of one be double the height of the other, prove, by Euclid's methods, that one of the triangles is double the other.

3. If the square on one side of a triangle be equal to the squares on the other two sides together, prove that these sides include a right angle.

4. If a straight line AB be divided internally at any point C, prove that the square on AB is greater than the squares on AC, CB together, by twice the rectangle contained by AC and CB.

If the base BC of a triangle ABC be bisected at D, prove that the squares on AB, AC are together equal to twice the squares on BD and DA together. Show that this includes some of Euclid's propositions, in Book II., as particular cases.

5. Show how to divide a given straight line into two parts, so that the square on one part may be equal to the rectangle contained by the whole line and the other part.

6. Define a tangent to a circle, and prove that it is at right angles to the diameter of the circle through its point of contact.

7. Show that any two opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles.

Show that two opposite sides of a convex quadrilateral (i.e., a quadrilateral without re-entrant angles) described about a circle are together equal to the other two sides together.

State a sufficient condition that it may be possible to inscribe a circle in a given convex quadrilateral, proving your result.

8. Through a point O interior to a circle two chords AOB, COD, are drawn; prove that the rectangle AO, OB is equal to the rectangle CO, OD.

9. Describe a circle interior to a given triangle to touch the sides of the triangle.

Show that four circles can be drawn to touch the sides of a triangle, three of them being exterior to the triangle.

10.

Describe an isosceles triangle having each of the angles at the base double of the third angle.

II. If the vertical angle BAC of a triangle be bisected internally by a line cutting the base in D, prove that the ratio BD: DC is equal to the ratio BA: AC.

If the perpendicular from C, upon the bisector AD, meet AD in N, and O be the middle point of the base BC, prove that ON is half the difference of the sides AB, AC.

12. If in the triangles ABC, DEF, the angles BAC, EDF be equal, and the ratio BA: ED be equal to the ratio FD: CA, prove that the triangles are equal.

II. ARITHMETIC.

[N.B. The working as well as the answers must be shown.]

1. Simplify

5 of 3 of 24-64 of 3 of 1
7 of 3

2. Divide 3.425 by '002192.

3. What principal, if invested for 3 years at 24 per cent. per annum simple interest, will amount to £575. 10s. 7d.?

4.

Find the value of

2 of 3 of 4 lbs. 8 oz. 10 dwt. 12 grs. Troy.

5. Reduce £4. 18s. 101d. to the decimal of 5 guineas.

6. Find the cost of 276 tons 16 cwt. at £3. 18s. 111d. per ton.

7. Find the least common multiple of 385, 231, 165, 105.

8. A room, 21 ft. 4 in. long, 18 ft. 8 in. wide, and 15 ft. 6 in. high, is papered with paper 32 inches wide at one shilling a yard. What is the total cost?

9. If by selling a certain horse for £66 I should lose 28 per cent. of the cost at which I bought the animal, what is my loss?

Prove that the product of any two numbers which consist of three figures and four figures respectively must be a number consisting of not less than six nor more than seven figures.

II. A cubical box of external dimensions 17 inches each way would contain crushed ore of the value of £421. 17s. 6d. if it were made of material I inch thick; but by mistake it has been made of thicker material, and the difference in the value of the ore which it will hold is consequently £78. 17s. 6d.; what is the real thickness of the material?