SENIOR FRESHMEN. Mathematics. A. MR. PANTON. 1. Find the coordinates of the two points of intersection of the circles and verify analytically that the middle point of the common chord lies on the right line joining the centres. 2. Find the value of a for which the following equation represents two right lines; and determine the tangent of the angle between the lines: 12x210xy + 2y2 + 11x − 5y+λ=0. 3. Determine x from the equation m sec2x tan (a - x) - n tan x sec2 (a − x) = 0. 4. If a be the arc drawn from the right angle C of a right-angled spherical triangle perpendicular to the base, and 8 the arc bisecting the base; determine expressions for cot a and cot ẞ in terms of the sides a and b. 5. If a point acted on by any number of forces be situated at the centroid of their extremities, prove that it will be in equilibrium. 6. Find the condition of equilibrium in a system of Pulleys containing n cords, all attached to the moveable weight (Burton of the third kind). MR. F. PURSER. 7. Form the equations of the two circles which pass through the origin, and touch the two lines 3x+4y+2=0, 7x+24y+8=0. and verify that the circle described on the line joining them as diameter is coaxal with the two circles. 9. Show the equivalence of the equations 10. Assuming, as is nearly the case, sin 23° 28', show that the comparative areas of the torrid, frigid, and temperate zones of the Earth may be represented q. p. by the following numbers : Torrid zone, 48. Temperate zones, each 31. II. The arms of an ordinary balance being suspected to be unequal in length, are tested by weighing the same object, first in one scale, then in the other; if W, W' be the two weights, find the true weight and ratio of arms. 12. Two bodies, A, B, placed at different elevations on a smooth inclined plane, A being the higher, commence to move down the plane at the same moment, continuing their motion after descent on the horizontal plane, supposed also smooth. Show that A will overtake B, and determine the time and place in which it does so. MR. W. R. ROBERTS. 13. A rigid rod without weight passes through two fixed rings, and is urged by a force P in the direction of its length against a plane, to which it is inclined at an angle a; find the pressure on the plane. 14. A sphere rests on two inclined planes; find the pressure on each. 15. Find the equation of the circle which touches the lines 3x2-4y2 + 3xy = 0, and passes through the point 1, 10. 16. Find the equations of the bisectors of the sides of the triangle formed by the lines whose equations are 2x + y + I = 0, 3y+ 2x + 2 = 0, x + y + I = 0. 17. In a spherical triangle prove tan E = √ tan § s tan § (s − a) tan §} (s — b) tan § (s — c). 18. Given base and area of a spherical triangle; find the locus of its vertex. B. MR. PANTON. 1. Express the diameter of a circle circumscribing the triangle formed by two tangents to an ellipse, and their chord of contact, in terms of the diameters parallel to the tangents and the perpendicular from the centre on the chord of contact. 2. Prove the equation by means of which the points of inflexion are determined when the equation of a curve is expressed in polar coordinates. 3. Determine the integrals x2-3x+3 dx, ex cos3x dx. 4. If from any point, O, perpendiculars Oa, OB, Oy, &c., be drawn to the sides of a polygon, and forces act along them, either all outwards or all inwards, each force being proportional to the side on which it falls, show that the system is in equilibrium. 5. If the forces in the preceding question be turned round the points a, B, y, &c., in the same sense, and through the same angle, show that their resultant is then a couple, whose moment is proportional to the square root of the area of the polygon enclosed by their lines of action. MR. F. PURSER. 6. A conic touches the axes and the three lines x+y+1=0, x+2y+3=0, y+2x+4=0; apply Brianchon's theorem to determine the point of contact on x + y + I = 0. 7. Show that if a chord of a conic pass through a fixed point, the product of the tangents of the semi-angles subtended by the segments at either focus is constant. 8. Find the equation in r, p, of the curve whose polar equation is r = a cos e + b, and apply it to obtain the radius of curvature at any point. 10. Two equal uniform beams, AB, AC, moveable about a hinge at A, are placed upon the convex circumference of a circle in a vertical plane; find their inclination to each other when they are in their position of equilibrium. MR. W. R. ROBERTS. II. Two heavy uniform rods are freely jointed at a common extremity, and are connected at their other extremities with two smooth hinges in the same horizontal line; find the magnitudes and directions of the pressures on the hinges. 12. Obtain a formula of reduction for the integral 14. Find the locus of the centre of a conic which passes through four fixed points. 15. Two vertices of a given triangle move along fixed right lines; find the locus of the third vertex. C. MR. PANTON. 1. Find the equation of the circle osculating a central conic at a given point. 2. If O be any point, and O' the middle point of the chord intercepted on the polar of O with respect to the conic' show that when either of these points describes a locus whose equation is known, the equation of the locus of the other can be derived from this by substituting (a) Apply this method to find the locus of the middle point of a chord of the above conic which touches a conic given by the general equation. 3. The equation of a curve of the third degree being written in its most general form, express in terms of the coefficients the condition that the three asymptotes should pass through a common point. 4. Find the integral x2 dx (x − 1)2 (x2 + 1) * 5. An elliptic cylinder rests in limiting equilibrium between a rough vertical and an equally rough horizontal plane, the axis of the cylinder being horizontal, and the major axis of the ellipse inclined to the horizon at an angle of 45°. Determine the coefficient of friction in terms of the eccentricity of the ellipse. (a). If the vertical plane be smooth, what is the value of μ? (b). If the horizontal plane alone be smooth, is equilibrium possible ? MR. F. PURSER. 6. From any point P on the lemniscate r2 = 4a2 cos 20, tangents PT, PT' are drawn to the rectangular hyperbola r2 cos 20 a2, touching the curve in TT'; prove that the circle circumscribing the triangle PTT' touches the hyperbola. 7. State and prove the formula expressing the area of a spherical triangle in terms of two sides a, b, and the included angle C. (a) If C vary, a, b remaining constant, show that the maximum area is given by sintana tan b. 8. Find the expression for the angle made by a curve with the radius vector from the origin at any point. (a) The polar equations of two curves referred to the same origin are r=f(0), and r=f(0) + const.; show that if the normal at any point of the first be known, the normal at the codirectional point at the second may be found by a simple geometric construction. 9. Prove that the radius of curvature of a parabola at any point is double the length of the normal to the curve drawn to meet the directrix. 10. Two unequal uniform beams, connected by a light rope attached to their middle points, rest in a vertical plane, an extremity of each beam resting on a rough horizontal plane; if the coefficient of friction be gradually diminished, which beam will slip first? MR. W. R. ROBERTS. 11. A heavy uniform beam, AB, rests with one end, B, against a smooth inclined plane, while the other end, A, is connected with a rope which passes over a pulley and supports a given weight; find the position of equilibrium. |