5. A uniform ladder rests with one end against a vertical wall and the other on the ground, inclined to the vertical at 45°. Compare the least horizontal forces which, applied to the foot of the ladder, will move it towards or from the foot of the wall, the coefficient of friction being the same for both ends of the ladder. 6. A particle projected with a velocity u moves subject to an acceleration a in the direction of motion. Find the space described in the nth unit of time. A particle slides from rest down a smooth plane inclined at 30° to the horizon. Find the position of that length of 92 feet which is passed over by the particle in one second. 7. Two scale pans, each of 4 oz. mass, are connected by a light inextensible string, which passes over a smooth fixed pulley. If a mass of 2 oz. be placed in one pan, and a mass of 3 oz. in the other, find the tension of the string and the pressures of the masses on the scale pans. 8. Two spheres of elasticity e and masses m, m', moving with velocities u, u', impinge directly. Find their velocities after impact. The centres of two equal billiard balls of radius a and elasticity § move along the straight lines, whose equations are in rectangular coordinates 4y=3(x − a), 5y = − 12(x+a), in such a manner that the line joining them is always parallel to the axis of x, and impinge at the origin. Find the equations of their lines of motion after impact. 9. Prove that the path of a projectile in vacuo is a parabola, and that the velocity at any point is that due to falling from the directrix. IO. A particle of mass m attached to a fixed point by a light string of length 7, makes complete revolutions in a vertical plane under the action of gravity. If u be its velocity at the highest point, find its velocity in any other position, and also the tension of the string. If the ratio of its maximum to its minimum velocity be a: b, show that the maximum tension of the string will be to the minimum in the ratio of 5a2-62 to 562 – a2. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, NOVEMBER, 1896. 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. Give accurate definitions of the following geometrical terms:superficies, rectilineal angle, circle, rhombus, postulate, axiom. 2. Draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it. Construct a square which shall have an extremity of one of its diagonals at a given point, and the extremities of the other diagonal on a given straight line. 3. Prove that if a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and that the three interior angles of every triangle are together equal to two right angles. From a vertex of an equilateral triangle a perpendicular is drawn to the opposite side, and upon this perpendicular another equilateral triangle is constructed; show that its sides are perpendicular to those of the original triangle. 4. Prove that in obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. Prove that a triangle, the sides of which are three, four, and six inches in length, is obtuse-angled. 5. Construct a square which shall be equal to a given rectilineal figure. 6. Prove that the diameter is the greatest straight line in a circle, and that, of all others, that which is nearer to the centre is always greater than one more remote. 7. Prove that the opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. 8. On a given straight line describe a segment of a circle, containing an angle equal to a given rectilineal angle. Through a given point draw a straight line which shall cut off from a given circle a segment containing an angle equal to a given rectilineal angle. 9. Construct an isosceles triangle, having each of the angles at the base double of the third angle. Prove that the perpendicular drawn to one side of such a triangle, at its middle point, cuts the other side in extreme and mean ratio. IO. Prove that the sides about the equal angles of triangles which are equiangular to one another are proportionals. The altitude of a certain triangle is equal to its base; show that if a rectangle be inscribed in it so as to have one side along the base and the extremities of the opposite side upon the sides of the triangle, then the three triangles by which the original triangle exceeds the rectangle are together equal to half the square on the diagonal of the rectangle. II. Prove that parallelograms which are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides. 12. Prove that, in any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. 3. Find the Least Common Multiple of 34, 42, 119, and 255; and obtain the sum of the fractions 34, 72, 11, and, by reducing them to a common denominator. 5. Find the cost of 25 quarters 3 bushels of oats at 19s. 3d. a quarter. 6. One side of a rectangular field is ‘oj4 of a mile, and the adjacent side is 13 of a furlong. Find the length of each side in yards and feet, and express the area of the field as a fraction of an acre. 7. A sum of £377. 13s. 4d. lent at simple interest amounts in a year and a half to £396. 11s. od. What is the rate of interest? 8. A garden whose length is 67 ft. 9 in. has a path 4 ft. wide on the two sides and at one end: if it costs £4. 10s. 3d. to turf the remainder at 6d. a square yard, what is the width of the garden? 9. A, B, and C who are engaged on piece-work do amounts in the same time which bear to one another the proportion of 10, 9, and 14 respectively. What ought each to receive if the amount paid for the whole is £24. 15s.? IO. What is meant by an odd number and an even number? Show that it is only necessary to look at the digit in the unit's place to ascertain if a number is odd or even. Several numbers have to be added together. What is the condition that their sum should be odd? II. A cistern is filled in 3 hours by a pipe 3 sq. in. in cross section through which water flows at the rate of 6.4 miles an hour. What is the volume of the cistern? |