A. 41. (a.) Show that, for real values of x, the expression (x-1) (x2) cannot lie between 7+ 4/3 and (b.) If 1, a, ẞ are the three roots of the equation as (35.) 42. Find the ratio, which the number of numbers of five different digits of which zero is not one, bears to the number of numbers of five different digits of which zero is one. 43. Define Harmonical Progression. (30.) Show that the geometrical mean of two numbers is also the geometrical mean of the arithmetical and harmonical means of those numbers. If a, b, c, d are in harmonical progression, show that a2 d2 : b2 c2 3a2d2 b2c2. : 44. Find the coefficient of a in the continued product (30.) and deduce the coefficient of xr in (x − a)", when n is a positive integer. Find the greatest coefficient in the expansion of (1 + x)", and show that the sum of the squares of the coefficients is 12n n n (35.) 45. AB is a given straight line, and P, Q are two given points on the same side of it; find the point in AB at which the line joining P and Q subtends the greatest angle. (40.) 46. A, B, C are three points in order in a straight line, such that AB equals one-half of BC; with B as centre and radius equal to half of AB describe a circle; take P any point in the circumference, join AP and produce it to Q, making PQ equal to AP; join QB, QC; show that BQC is a right angle. (40.) If the tangents of the angles of a triangle are in arithmetical progression, show that the squares of the sides are in the ratios, x2 (x2 + 9) : (3 + x2)2 : 9(1 + x2), where x is the least or greatest tangent. 48. In any triangle ABC, show that (30.) a3 cos B cos C + b3 cos C cos A+ c3 cos A cos B. (35.) 49. The difference between the areas of the hexagon and pentagon circumscribed about a circle is 5 square feet; show that the square of the radius can be found in the form cos e cos 36° and calculate 0, and the length of the sin (36° —0)' (40.) 51. If r, R be the radii of the inscribed and circumscribed circles of the triangle ABC, show that Show also that the square of the distance between the centres of the circles is R2 — 2Rr. (35.) 52. A, B, C are three points in a horizontal plane; the angle BAC is a right angle, and the length of AC is 1,000 feet; P and Q are points vertically over A and B, and the line joining P and Q is horizontal; the angle of vertical elevation of P at C is 52° 40′, and that of Q at C is 34° 43'; find the distance PQ. C. (40.) 53. Show that the angles at the base of an isosceles spherical triangle are equal. The equal sides of an isosceles triangle are greater than quadrants; show that the angles at the base are greater than right angles. (30.) 54. Define the polar triangle of a given spherical triangle, and show that the sides and angles of the polar triangle are respectively the supplements of the angles and sides of the given triangle. If A, B, C are the angles of a spherical triangle, show that A+B-C is less than two right angles. (30.) 55. ABC is a spherical triangle having a right angle at C; write down the formula connecting a, b, and c, and also that connecting A, B, and c, and prove them. Given A = 124° 43′ and B = 60°, find c. 1 (35.) 56. Assuming the expression for the cosine of an angle of a spherical triangle in terms of the sides, find the formula for the tangent of half an angle in terms of the sides. Find from the formula the greatest angle of the triangle whose sides are 24° 52′ 6", 36° 25', and 58° 42′ 54′′. (35.) 57. ABC is a spherical triangle, E is the middle point of BC, and AD is drawn at right angles to BC; show that tan ED. sin (B+ C) = tan 12 a. sin (B - C). Putting out of the question all cases in which ED exceeds a quadrant, find under what circumstances D and B are on the same side of E, and under what circumstances they are on opposite sides of E. (40.) Read the General Instructions on page 1. You may not answer more than ten questions. 62. Find the coefficient of amyn in the expansion of (1 + x)2(1 + y)2 (1 + x + y)2 in powers of x and y. (45.) 63. If n be a positive integer and a greater than unity, show that x2+1 n + 1 ] x-(n+1) and less than n 1 x-n - x=(n+1) (55.) 64. If x, y, z are the sums to infinity of the continued fractions 65. A, B, C, D are the angular points taken in order of a quadrilateral inscribed in a circle, and each of three sides is double the side CD; show that the diagonal BD divides the diameter of the circle drawn through A into parts one of which is to the other as 5 to 3. (45.) 66. In the triangle ABC perpendiculars from B, C on the opposite sides intersect each other at D, and when produced meet the circumscribing circle in E, F; show that AD, AE, AF are all equal. (45.) 67. Show how to describe a circle to touch two given intersecting straight lines and a given circle. (45.) 68. ABC is a triangle having an acute angle at C; draw BD at right angles to AC; in DA, produced if necessary, take DP equal to DC, also in BA, produced if necessary, take BQ equal to BC and join PQ; show that PQA equals C + B. Show that the theorem is true in both cases, viz., when P is in DA and when it is in DA produced. (40.) 69. AB is a straight line and C a point in it nearer B than A; draw CD at right angles to AB; find, by a geometrical construction, a point P in CD, such that AP may have a given ratio to BP. Between what limits must the given ratio lie? 70. ABC is an equilateral triangle on a horizontal plane, and P, Q, R are points vertically over A, B, C respectively; if QB equals twice PA and RC equals three times PA, show that there is no point in the plane PQR at which the angle of vertical elevation of R exceeds QAB. (50.) 71. The distances of a point, P, in the plane of a triangle from the angular points A, B, C are respectively l, l, m, and P and C are on opposite sides of AB; show that 12 = ab cos C + a sin B. √(41o — c2). (40.) 72. (a.) Find an expression for all the angles whose tangents have the same value as tan 0. (b.) Find all the values of which satisfy the equation- 73. State De Moivre's theorem and prove it when the index of the power is a positive whole number. Find all the values of {−√(− 1)}, and the sum of the products of the values taken two and two. 75. A given lune is divided into two isosceles triangles, and the area of one of them is n times the area of the other; show that where A denotes the angle of the lune, and one of the equal sides. Find the ultimate value of cos 0, when A becomes indefinitely small; and hence show that the surface of a segment of a sphere is to the surface of the sphere as the height of the segment is to the diameter of the sphere. (55.) |