2. If a, ß, y,...., be the n roots of a positive integral equation of the nth degree, then will ƒ (x) = P(x − a) (x — B).........(xx) be Prove that a3 (b + c)3 + b3 (c + a)3 + c3 (a + b)3 + 3 a2 b2 c2 + 3abc (a + b + c) (bc + ca + ab) = 2 (bc + ca + ab)3. Express Pn + 1 qn — Pn qn + 1 = +1. 51 as a continued fraction, and find the first five convergents to it. 4. Three points A, B, C are taken in a straight line, and circles described on AB, BC, CA, as diameters; shew that the diameter of the circle which touches them all is equal to where AC = c, AB = a, BC = b. abc 5. Give Cardan's method for the solution of a cubic equation. Solve x8+12x+12= 0. 6. Find the values of x for which f (x) has a maximum or a minimum value. Two chords of an ellipse are drawn through one focus at right angles to one another; find when their sum is a maximum and when a minimum. 7. Change the independent variables from r, e to x, y, in the expression 1 d2u d2u 1 du. + + r2 do2 dr2 r dr 8. Explain what is meant by the mean value of a function (), when x may have all values between a and b. A chord is drawn through the focus of an ellipse, all directions being equally likely; shew that its probable length is the axis minor. 9. Interpret the equation ABу + μya + vaẞ= 0, a, B, y being the trilinear co-ordinates of any point. Find the equations of the tangents to the curve passing through the points A, B, C. If these tangents form the triangle DEF, and if they intersect the opposite sides in G, H, K; then will GHK be a straight line and AD, BE, CF will intersect in one point which is the pole of this straight line with respect to the conic. 10. Find the co-ordinates of the centre of the conic $(a,B,y) = 0. Hence shew that, if las + mB2 + ny2 a2 b2 C2 represent a parabola, then + + m n 11. Define the radius of absolute curvature at any point of a curve in space, and find its magnitude. Determine that at any point of a helix. 12. What is meant by the Indicatrix at any point of a surface? If p1, p2 be the principal radii of curvature at any point of a surface, p that of any normal section inclined at an angle to one of the principal sections, shew that APPLIED MATHEMATICS. 1. Define a couple, the moment of a couple, the axis of a couple. Find the resultant of two couples acting in different planes. M 2. Find the general equations of equilibrium of a flexible inextensible string acted on by any forces. Prove from these equations that when such a string hangs in equilibrium under the action of gravity only, (1) the whole curve lies in one plane, and (2) if the string be uniform, the difference between the tensions at any two points is proportional to the vertical height between those points. 3. What is meant by the potential of an attracting body with respect to any point? Find its value in the case of a homogeneous spherical shell, and deduce the attraction of such a shell on any particle within or without it. 4. A body moves under the action of a force tending to a fixed centre. Prove, first analytically, that in all cases the orbit is a plane curve; and secondly, if the force vary inversely as the square of the distance, assuming the general polar differential equation to the orbit, investigate the elements of the orbit in terms of the initial radius and the velocity and direction of projection. If there be a small error 8V in the estimated velocity of projection, find the resulting error in the mean distance. 5. A particle is placed in a circular tube of radius a, which revolves about a diameter which is vertical with uniform angular velocity w. Find the position of relative equilibrium, and show that the time of a small oscillation when slightly displaced from that 2πωω √ a2 w1 — g2 w4 pressure on the curve. 92 Find also the resultant 6. Show that when a body describes an orbit under the influence of a central force tending to a point S, the force at any point P is given by the formula 2h: lt QR SP2 QT2 where Q is a point on the orbit adjacent to P, QR is a subtense to the tangent parallel to SP and QT is perpendicular on SP. Deduce the formula P = h2u2 (+x). 7. Investigate a formula for the position of the centre of pressure of a plane area exposed to the action of fluid. An isosceles triangle of height h is placed in an elastic heavy fluid with its base uppermost and horizontal and plane vertical. If I be the pressure at any point of the base, find the centre of pressure. 8. Find the equation which gives the velocity of efflux of water out of of a small orifice. Find the time in which a hemispherical bowl will be half emptied through a small hole at the lowest point. 9. Define a prism. Investigate the condition that a ray may pass through a prism with minimum deviation. If the amount of minimum deviation in passing through a prism of given angle be known, find the refractive index. 10. Explain the phenomenon of the primary rainbow, and investigate the order of the colours. 11. Describe the transit instrument. What is level error? Calculate the effect on the time of transit of a known star of a given amount of a level error. 12. What is the First point of Aries? Explain Flamsteed's method of determining it. V. HYDROSTATICS, OPTICS, AND PLANE 1. Define a fluid, the pressure at any point of a fluid. Find the general differential equation which gives the pressure at any point of a fluid in equilibrium under the action of any forces. Show that for equilibrium, surfaces of equal pressure must be also surfaces of equal temperature, Hence show that the atmosphere of the earth cannot be in equilibrium. 2. Define the metacentre. Find its position. A right cone of semi-vertical angle a and weight is floating in fluid of twice its own specific gravity. Find whether the equilibrium is stable. If it be unstable, find the least weight that can be attached to the vertex to render it stable. 3. A flexible but inextensible surface of revolution with its axis vertical contains heavy fluid. Find the tensions at any point perpendicular to the principal sections. Find the condition that the two tensions may be equal at any point. 4. Find the horizontal pressure in any direction on a surface exposed to the action of fluid. A hemispherical bowl is filled with fluid. Find the resultant pressure on a quadrant of it. 5. State the laws of refraction and reflection of light. What is the critical angle of a medium? What happens to light incident internally at a greater angle than the critical angle? 6. Find the position of the geometrical focus of a pencil of light refracted at a spherical surface. Trace the changes in position of the geometrical focus as the point of light travels from an infinite distance up to the refracting surface. 7. Define a prism. Show that in passing through a prism of material less dense than the surrounding medium the light is always bent towards the edge. 8. Explain the method of obtaining a pure spectrum. What is the received explanation of the fixed lines in the Solar spectrum? 9. Describe the common astronomical telescope with Ramsden's eye piece. Explain the advantages of this eye piece over (1) a single lens, (2) over Huyghens' eye piece. What disadvantage has it compared with the latter. 10. Define the terms right ascension, declination, sidereal time, mean solar time, apparent solar time. What instruments are required for determining the first two of these. 11. What is the collimation error of a transit instrument. Describe the method of determining it by the collimating eye piece. How are the effects of level error eliminated from this determination in a reversible instrument. |