2. ́(b) Find yn 2 if y=e4 sin3x. =xytan-1(3xy). (a) Explain the function of the first and second derivatives in the theory of maximum and minimum. (b) Find the area of the greatest triangle formed by the ordinate, abcissa and radius vector to any point on the central ellipse. 3. (a) Develop the equation xf+yf =0 (b) Show +zar Əx ду Əz for the tangent to the curve f(x, y)=0. that the curves x2+y2 = 2a2 log x+c cut the ellipses x2 12 a2 + =1 orthogonally. 4. (a) In polars prove (i) tan =rf'. (ii) p=r2\\/r2 +ƒ'2. (b) Find the p, r equation of the Lemniscate of Bernouillir2=a2 cos 20. 5. Find the equation of the evolute of the standard ellipse. (b) The paraboloid of revolution has one-half the volume of its circumscribing cylinder. 8. By means of a differential equation show that the law of coefficients in the expansion of (sin-1x)2 an, and write out the expan is an+2 (n+1)(n+2) sion as far as x8. INTERMEDIATE. SYNTHETIC SOLID GEOMETRY. 1. Show how to draw a common perpendicular to two noncomplanar lines, and prove that it is the shortest segment from one line to the other. 2. (a) Express any function of a dihedral angle of a three-faced corner in terms of the face angles. (b) Find the dihedral angle of a regular octahedron. 3. Find R, p, r for the regular icosahedron in terms of an edge. 4. If an equilateral cone and an equilateral cylinder be inscribed in the same sphere, the whole surface of the cylinder is a mean proportional between those of the cone and sphere. 5. Find the volume of a sphere as generated by the translation of a variable circle. 6. The distance from the centre of a circle to the centre of figure of an arc which subtends an angle at the centre is C/0, where C is the chord of the arc. 7. Show how two given angles and a given line may be projected into two required angles and the line at infinity. 8. (a) The minor axis of any elliptic section of a cone is a mean proportional between the diameters of the circular sections at the extremities of its major axis. (b) A cone has vertical angle 60°. Show how to cut it that the section may be a parabola with latus rectum 12 inches. 9. The area of an equilateral spheric triangle is onefourth that of the surface of the sphere. Show that its angle is 120°, and find its side. SPHERICAL TRIGONOMETRY AND ASTRONOMY. INTERMEDIATE OR FINAL. 1. Obtain the formula cos a = cos b cos c+sin b sin c cos A, and render it logarithmic by means of an auxiliary angle. 2. Give Gant's proof of L'huillier's theorem. 3. In following the loxodrome at angle from lat. to lat. ', if λ be the change of longitude show that ecote φ' φ an (45° - £) cot (45° + 2). = tan 4. Express the value of the earth's radius in terms of the geometric latitude of the place. 5. Explain how the length of the siderial year can be found by observing two meridian passages of the same star. 6. If the equinox occurs at 3"40"16" p.m. on Mar. 20th, and the R.A. of Aldebaran is 4"2938, find the mean time when the star comes to the meridian on Aug. Ist. 7. Find the Golden number, the epact, the Sunday number, and the date of Easter for the year 1935. 8. Explain how to find the distance of the sun by (1) a transit of Venus, (2) opposition of Mars, (3) the constant of aberration. |