SECTION XXI. MISCELLANEOUS QUESTIONS. dxn-1 + 1. If P be any rational function of x, in which the highest power of x is less than n; and if A, B, C, D, ... K be the values of dP 42P 43P da-P P, • dx' dx22 dx3 when x = a, then P A B + a)n-1 a)n-? D K + 1.2... (n − 1)(x - a)' 2. Divide a given paraboloid into two parts in the ratio of m:n, by a plane inclined at a given angle to the axis. PQP 3. Prove that the chord of curvature = QR 4. Solve the equation px? = (2x) + 2. 5. Prove the following formula for small arcs, I sin x = lx + jl cos x. 6. Find the part of a sphere cut out by three planes passing through its centre, and inclined to each other at angles of 120°. 7. Two straight lines are inclined to each other at a given angle, find the area of all the circles which can be described Y touching each other and the two given lines, the position of the centre of the last circle being given. 8. Find that point in the surface of a spherical triangle. from which, if straight lines be drawn to the angular points, the pyramid thus formed shall be a maximum. when x is large, and shew the utility of this formula in the solution of the following problem. 10. In a pack of fifty-two cards, containing an equal number of red and black cards, determine the probability that in drawing any even number of cards, there shall be an equal number of red and black cards; supposing that the probability of drawing any even number is the same. A numerical result is required. 11. In the series of quantities A,, A2, A3, 2. 47 ta 3 and the remaining ones be derived according to the following . if A, law : A,. A, .Az = (A, + A, + A2), 2nnt ta that A. (sin m m-P m-9 m-1 12. If x = m tan (z – nx) where x is small compared with 2, sin 2x very nearly - P. be the negative terms of an equation of m dimensions, then will the greatest root of this equation be less than the sum of the two greatest of the quantities P., P. 14. Having given the first two terms of the expansion of (a? + 12 + 2ab cos 0)-- in a series of the form A. + A, cos 0 + A, cos 20 + shew how from them the first two terms of the expansion of (a? + b2 + 2ab cos 0)-+ may be determined. 15. If S, represent the sum of the ordinates in the quadrant of a circle whose radius is 1, S, represent the sum of their squares, cubes, . Tx = 16. Integrate the following differentials and differential equations: dx dx dx x+ + I X4 VT vā d°у dy (x2 + y2) dx + x2ydy 0, (1 + x) ta 0, dx2 da 1 + p? + q? and also the following equations of differences: f (x2) – f(x) = m and 1, Umts = kl. 17. There are two urns A and B, the former containing three white and the latter three black balls; a ball is taken from each at the same time and put into the other, and this operation is repeated three times; what is the probability that A will contain three black and B three white balls? 18. Develope sin (a + Bx + yra) in a series of the form A + Bx + Cx2 + Dx3 + 19. If the true centre of the moon's orbit move uniformly in a circle about the mean centre, the result is a change of the moon's place of the form m sin 2{() - 0) – A}, where A is the moon's anomaly 20. If m be the maximum lunar nutation in N. P. D., 2 the longitude of the moon's ascending node, shew that when the longitude = l', the nutation = m cos (C – 1). 21. The coefficient of an in the expansion of n3 + 11n (1 + 2 + 2x2 + 3x3 + ad inf.)2 is equal to 6 + B . . 22. If A,, A, .. ...B A, and B1, B, ....B, be two series of positive numbers arranged in order of magnitude, of which A, and B, are respectively the greatest, shew that A, A, A A A + + B, B, B. B. greater than if the denominators B,, B, B, be arranged in any other order under A,, A, . A, 23. Two vessels, of which the capacities are a and b, are filled, the one with wine and the other with water; equal quantities c are taken from each and poured into the other, and this operation is repeated n times. Find the quantities of wine and water remaining in each vessel. 1 24. Expand into a series of the form A + B cos x 1 + e cos x + C cos 2.0 + ...., and explain the law of the coefficients. 25. If a circle be described on AM the axis-major of an ellipse, and if an ordinate to the axis meet the ellipse in P and the circle in Q, S being a point in the major-axis, the areas ASP, ASQ are in a constant ratio. 26. If ax + 6Y + cZ 07 where X = ax + apk, + Any a, X + 6, Y + cZ os Y = bx + 6,4, + bz, Z = cx + Call + C29 then X? + Y + Z2 {az (bc, b,c) + 6,(a,c – ac) + c, (ab, – ab)}”. (bc, – b,c)2 + (ac – ac,)2 + (ab, – ab)2 27. Solve the functional equation ar b2ar. 28. If AB be an elliptic quadrant, CP, CD semi-conjugate diameters, PF perpendicular to CD, K the point, in which CD produced meets the circumscribing circle, and KMQ a line perpendicular to the major-axis meeting the ellipse in Q, then will arc BP - arc AQ = CF. 29. The square of the area of any one of the faces of a triangular pyramid, is equal to the sum of the squares of the other three, minus twice the rectangle contained by the product of every two and the cosine of their inclination. 30. A mortgage is taken on an estate worth N acres of it; land rises n per cent. in price, and in consequence the mort-' gage is only worth N, acres, and it is then paid off. During the continuance of high prices another mortgage is taken, which is worth N acres as before; prices return to their former level, and the mortgage is worth N, acres ; shew that N – N,:N, - N=1:1 + 100 n 31. The ages of a man and his wife are respectively 85 -- m and 86 – n; find the present worth of an annuity A to be paid to the wife after the death of her husband, supposing one male out of every 85, and one female out of every 86, to die annually; and n greater than m. 32. If a straight line PSP, revolve about a fixed point S; and if perpendiculars PM, P,M, be drawn upon a fixed straight line passing through S, find the equation to the curve 1 1 2 in which PM" PM" 33. If P, Q be any two points in a curve referred to a pole S, and PM, QN perpendiculars on a fixed straight line TMN, find the equation of the curve, when the area SPQ is to the area PQMN as I to n, and construct the curve when n = 1. an. 34. Three given quantities, (a +x), (a +z)+h, (a +z)+h', approximations to the root a of an equation, being substituted for the unknown quantity, give results n, n +d, n + 8; shew that z will be very nearly found from the equation, z? (ho — h'd) + 2 (h28 — h'28) + nhh' (h' 128) + nhà (1 - 1) = 0. 35. If each of the angles of a spherical triangle, the sides of which are very small compared to the radius of the sphere, be diminished by a third part of the spherical excess, the angles so diminished may be taken as the angles of a plane triangle, the sides of which are equal in length to those of the spherical triangle. 36. Having given the three edges of a parallelopipedon, and the angles they make with each other, find its solidity. |