13. Given the axes of an ellipse, find expressions for (i.) the distance between the foci. (ii.) the distance between the directrices. (iii.) the latus rectum. (iv.) the product of the lengths of the perpendiculars from the foci on any tangent. 14. Explain carefully the nature of an asymptote to an hyperbola. Find the equation to a curve, of this description, such that the smaller angle between the asymptotes is 45°, and the distance between the foci 10 units of length. [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.] I. Show how to find the resultant of a number of coplanar forces acting at a point. Forces of magnitudes 3, 4, and 5, act at a point O in directions lying in one plane and making angles of 15°, 60°, and 135° respectively with a line OA in the same plane. Find to two places of decimals the magnitude of the resultant. 2. A small ring of weight W, which can move without friction on a circular wire fixed in a vertical plane, is in equilibrium at a point P on the lower half of the wire under the action of a force R in the direction of the tangent at P to the wire. If the pressure of the ring on the wire is equal to W, find the magnitude and direction of the force R. 3. Define the centre of gravity of a rigid body. What assumptions with regard to the action of gravity are made for the purpose of the definition? ABCDE is a lamina of uniform thickness and density, and of such a shape that BCDE is a square, and AB = AE. If the centre of gravity of the lamina is in BE, find the ratio of AB to BC. 4. Find the relation between the power and weight in the wheel and axle. Show how to arrange three wheels and axles, having radii R and r respectively, so that P/W = 3/R3. 5. A body of weight 16 lbs. rests on a rough inclined plane inclined at an angle of 30° to the horizon. If a force of 2 lbs. acting up and parallel to the plane is just sufficient to prevent the body from slipping down, find the least force in the same direction which will balance the maximum resistance of the body to motion up the plane. 6. State the proposition known as the "parallelogram of velocities." Prove that if a point possesses two independent velocities represented by λ. OA and μ. OB, where OA and OB are two straight lines meeting at O, the resultant velocity will be represented by (\+μ)OG, where G is a point on AB such that λ. AG = μ. GB. 7. A person rows with a velocity of 6 miles an hour across a river a quarter of a mile wide, which runs with a velocity of 4 miles an hour. The head of the boat makes a constant angle with the bank while he rows across, and he arrives at a point 36 yds. 2 ft. lower down the bank than the point opposite his starting point. Prove that tan 0 = §. 8. Prove that if a point moves from rest with a constant acceleration ƒ, the distance s passed over in a time t is given by s = =ft2. A body falls from the top of a tower, and after 2 seconds another body is projected downwards with a velocity of 192 feet per second. The two bodies reach the ground at the same time. Find the height of the tower. feet from the ground begins When the balloon is at a height 9. A balloon when at a height of 2021 to fall with a uniform acceleration of g of 500 feet from the ground, ballast to the amount of one-tenth the whole mass of the balloon is thrown downwards with a velocity, relative to the balloon, of 10 feet per second. Find the time the ballast will take to reach the ground. IO. A particle is projected from a point P with velocity v in a direction making an angle a with the horizon. Prove that the greatest height above P, to which the particle rises, v2sin2a A stone is thrown from a height of 4 feet so as just to pass horizontally over a wall which is 25 yards distant and 54 feet high. Find the velocity and direction of projection. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 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 ABC, DEF be two triangles which have the sides AB, AC equal to the sides DE, DF, each to each, and also the angle ABC equal to the angle DEF; then shall the angles ACB, DFE be either equal or supplementary. 2. Define parallel straight lines, extreme and mean ratio; and draw some simple figure of a superficies which is not plane. If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. 3. In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle. 4. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. 5. 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. ACB is a straight line, and on AC is described an equilateral triangle DAC; show that the square on DB is equal to the squares on AC and CB, together with the rectangle contained by AC and CB. 6. If two circles touch one another externally, the straight line which joins their centres shall pass through the point of contact. 7. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. BEAC is a semicircle whose diameter is BC; D is any point on BC; AD is perpendicular to BDC; EB is equal to AD; and F is on DA produced so that DF is equal to AB; show that CE is equal to CF. 8. If from any point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it. 9. Describe a circle about a given triangle. 10. What is Euclid's method for describing about a circle, a regular pentagon, hexagon, or quindecagon? Show that this method would not apply to the describing about a circle of a triangle equiangular to a given triangle. II. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another. |