4th, and 5th squares in the third column, the 4th, 5th, and 6th squares in the fourth column, the 2nd, 3rd, 4th, 5th, and 6th squares in the fifth column, and the 3rd, 5th, and 6th squares in the last column. Show that the centre of gravity of the figure formed of these 20 squares divides the line AB in the ratio of II to 13. 5. A uniform rod rests on the rim of a plate with its middle point at the centre of the rim. The plate itself rests on a horizontal plane. A downward vertical force, just sufficient to disturb equilibrium, being applied to one end of the rod, show that the plate and rod will begin to move together, or the rod only, according as the weight of the rod is greater or less than rds of the weight of the plate; 4a, 5a being the diameters of the base and rim of the plate respectively, and 6a the length of the rod. 6. Define work, and show how it is measured. A right-angled triangle ABC turns stiffly in its own plane about the middle point of the hypotenuse AB. If forces P, Q, R, just sufficient to overcome the resistance, be applied at right angles to the sides BC, CA, AB at the angular points, all tending to turn the triangle in the same direction, show that the work done by them in turning the triangle through a right angle = {Rc+(P-Q)(a – b)}; the forces remaining throughout the motion parallel to their original directions and constant in magnitude. 7. A heavy particle slides down a rough inclined plane; find the space described from rest in a given time. Two particles are projected with a velocity of 40 feet per second from points 88 feet apart, the one up and the other down, a rough plane (μ = 1) inclined to the horizon at an angle tan 14. Find when and where they will meet, and account for the double solution [g = 32]. 8. Within a smooth circular tube fixed in a vertical plane are two particles of mass P, Q connected by a string whose length is equal to half that of the tube. Find the acceleration of each particle in the direction of motion, and the tension of the string supposed tight, when the line joining the particles makes an angle o with the horizon. 9. Two smooth imperfectly elastic balls, moving in one plane with given velocities in given directions, impinge obliquely on each other; determine the motion of each after impact. 10. A particle hangs from a fixed point in a wall by a string of length a, find the least velocity which must be given to it in order that it may make a complete revolution, without the string becoming slack. If the string come in contact with a nail in the wall situated in the horizontal line through the point of suspension and at a distance b from it, find the least initial velocity in order that the particle may make a complete revolution round the nail, without the string becoming slack. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1892. 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. The sum of any two sides of a triangle is greater than the third side, and their difference is less than the third side, 2. If two quadrilaterals ABCD, EFGH have the four angles A, B, C, D respectively equal to the four angles E, F, G, H; and likewise the sides AB, DC respectively equal to the sides EF, HG; show that if AD and BC meet when produced the quadrilaterals are equal in every respect. 3. Define parallel straight lines; and show that parallelograms on the same base and between the same parallels are equal to one another. 4. Enunciate the proposition which is represented in algebraical symbols by (a+b)2 + a2 = 2a(a+b)+b2; and give the construction by which the proposition is proved. If in a quadrilateral ABCD, the sides AD and BC are each perpendicular to the side AB, show that the square on DC is less than the sum of the squares on the three other sides by twice the rectangle contained by AD and BC. 5. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. 6. Define circle, tangent, angle in a segment. If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle. 7. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle. If the bisectors of the adjacent angles A and B of a quadrilateral ABCD meet in E, and the bisectors of the adjacent angles B and C, C and D, D and A meet respectively in F, G, H; show that a circle can be described round the quadrilateral EFGH. 8. If from any point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it, and 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 on the line which meets the circle, the line which meets the circle shall touch it. 9. About a given circle, describe a triangle equiangular to a given triangle. 10. Inscribe an equilateral and equiangular hexagon in a given circle. Show that if A1 A2 A3 A4... be an equilateral and equiangular figure of 24 sides, the straight lines A1A10 and A5A18 are at right angles. |