57. In a right-angled spherical triangle, the hypotenuse is 105° 20′ and one angle is 35° 4′; find the side opposite to the given angle, and the remaining angle. (35.) 58. Let A be the pole of a given great circle, within which a point B is taken, the arc of the great circle joining A and B being known; with centre B and any given spherical radius describe a small circle cutting the great circle; find the angle between the two circles at their point of intersection in terms of AB and the given radius. (45.) Honours Examination. INSTRUCTIONS. Read the General Instructions on the first page. 61. If a2 y2+b2x2-a2 la2 y (E-x)—b2 x (n—y)=y (a1 y2+b1 x2)+ show that (a )2+(bn)3=(a2—b2)3* 62. If p and q are whole numbers prime to each other, and if one of them is even, show that the difference of their squares can be a perfect square only when p+q and p-q are perfect squares. 63. Show that the series obtained by expanding (1+x)" by the binomial theorem is convergent when a is numerically less than unity. 9 10' If n=-5 and x=- after what term is the ratio of any term 64. Of three equal circles the first and second cut each other in points A and B, the third touches the first at B and cuts the second at C; show that the straight line joining AC passes through the centre of the second circle, and is parallel to the line joining the centres of the first and third circles. 65. AB is the diameter of a semicircle, CD the radius at right angles to AB; from any point E in AB a line EF is drawn parallel to CD, and meeting the circumference in F; show how to draw a straight line AG, to meet CD produced in G, such that the square on AG shall equal the sum of the squares on AE, EB, EF. 66. Two sides AB, AC of a given triangle ABC are produced to E and F; DB and DC are drawn bisecting the exterior angles EBC, FCB; show that Area ABDC× cos (B+C)=area BDC×cos (B—C). 67. About a given circle describe a hexagon whose sides are equal, and of a given length. Within what limits must the given length lie for the construction to be possible? 68. The bisectors of the interior angles of a quadrilateral ABCD, are produced to form a second quadrilateral EFGH, the lettering being such that if AB and CD produced meet in P the points P, G, E, are in a straight line; show that 69. (a.) If a+B+y=180°, show that cos (8+y-a)+cos(y+a−ẞ) + cos (a+B−y)=1-4 cos a cos ẞ cos(a +B). (b.) Find cos a, in terms of a and b, from the equations: cos cos x=cos a, cos o cos x=cos b, cos (0+)=cos a cos b. What is the geometrical meaning of these equations, and how does attention to it explain the double sign in the deduced value of cos a? 70. If ▲ be the area of the triangle ABC and ▲, the area of the triangle formed by joining the centres of the escribed circles of the triangle ABC; show that 71. Assuming the exponential expressions for sin and cos e, show and state within what limits the series is convergent. 72. If sin x=n sin (a+x), show, by the use of exponential forms, that x=n sin a+n2 sin 2a+n3 sin 3a+..... 1 2 1 3 and that the series is convergent if n is less than unity. 1 2 1 (a.) cos sin + cos2 0 sin 20+ cos e sin 30+......ad inf. (b.) sec a sec (a+ß)+sec(a+ß) sec(a+23)+....+sec (a+n−1ẞ)sec (a+nß). 74. From the vertex C of a spherical triangle an arc of a great circle is drawn to meet the base AB in D; if the segments of the angle at C are C, and C2, severally opposite to AD and DB, show that cotan CD sin C=cotan a sin C2+cotan ↳ sin C, 75. Show that the area of a spherical triangle is to that of half the surface of the sphere on which it is described as the spherical excess is to four right angles. A, B, C are three angular points of a cube inscribed in a sphere, no two of the points being in the same edge of the cube; show that the area of the spherical triangle ABC is one fourth of the area of the sphere. 76. Find the lengths of the diagonals of a parallelopiped in terms of the lengths of the edges and of the acute angles forming the solid angle at one end of the longest diagonal. Explain the results obtained for the lengths of the diagonals when the three angles become indefinitely small. SUBJECT V. PURE MATHEMATICS, STAGES 4 and 5. EXAMINER, THOMAS SAVAGE, Esq., M.A. GENERAL INSTRUCTIONS. If the rules are not attended to, the paper will be cancelled. You are permitted to answer questions from the Fourth Stage, or from the Fifth Stage, or from the Honours paper, but you must confine yourself to one of them. Put the number of the question before your answer. The figures in descriptive geometry should not only be constructed with ruler and compasses, but the construction should in all cases be explained and its accuracy demonstrated. You are to confine your answers strictly to the questions proposed. Your name is not given to the Examiner, and you are forbidden to write to him about your answers. The examination in this subject lasts for three hours. Fourth Stage.-Subjects: Plane, Solid, and Descriptive Geometry and Geometrical Conics. INSTRUCTIONS. You are not permitted to attempt more than eight questions, and of these not more than two should be on the same subject. The values attached to the questions differ little from one another. 1. In a right angled triangle, prove that the perpendicular from the right angle divides the hypothenuse into segments, whose ratio is the duplicate of the ratio of the sides containing the right angle. From any point P in the circumference of a circle, a perpendicular PN is drawn to the diameter AB; and P is also joined with either of the extremities of CD, the diameter perpendicular to AB. If this line, produced if necessary, meet AB in M, prove that the ratio of AN to NB is the duplicate of the ratio of AM to MB. 2. If two triangles of equal area have an angle of the one equal to an angle of the other, prove that the sides about the equal angles are reciprocally proportional. If one of the diagonals of a quadrilateral inscribed in a circle be bisected by the other, prove that the sides of the quadrilateral, taken in order beginning at one end of the bisected diagonal, are proportional. |