7. Prove that when a function has a maximum or minimum value its differential coefficient either vanishes or is infinite. Is the converse of this proposition true? Determine, and distinguish between, the maximum and minimum values of the expression 2x3-15x+36x-23. 8. If o be the angle between the tangent and radius vector at any point (r, 0) of a curve, prove that Prove that, in the case of the curve, r2=a2 cos20, $=1π+20. 9. Find the asymptotes of the following curves: 10. Shew how to find the envelope of the system of curves represented by the equation f(x, y, a) = 0, a being a variable parameter. Find the envelope of the lines 12. Shew how to find the area of a curve in polar coordinates. Find the area of the curve r2 = a2 cos20. 13. A plane cuts the axes of reference in the points A, B, C, at the distances a, b, and c from the origin. Prove that its equation is If G be the centre of gravity of the area of the triangle ABC, and D the foot of the perpendicular from the origin (0) on the plane, find the coordinates of G and D, and the cosine of the angle GOD, the axes being rectangular. 14. Find the areas of the three circles in which the sphere (x − a)2 + (y - ẞ)2 + (≈ − y)2 = c2 is intersected by the coordinate planes. If this sphere, remaining unchanged in size, move about so that its centre is always at a constant distance from the origin, prove that the sum of the areas of the three circles of intersection will be constant. ANSWERS AND HINTS FOR SOLUTION. DIRECT, MAY, 1873. Pp. 1-7. 5. Bisect the given line AB at C, through which draw DCE perpendicular to AB; make DC, CE each equal to AC or CB, and join AD, DB, BE, EA. 6. Use Euc. I. 20. 18. 3 tons 8cwt. 1 qr. 14lbs. 8.96 oz. 19. 36·1875. 22. 120 sq. yds. 8 sq. ft. 23. £1449. 24. 21 per cent. 25. £113. 4s. 74d. 26. 4 horses. 1. 1, 4. Algebra, Logarithms, and Mensuration. 2. (A) 2 (2b – 3c); (B) ao — a1b2 + a2bˆ — b®; (C) 4x2 - 12ax +9a2; (D) a2 + ab √/3+b2; 3 (x2 - 7ax + 12a2) 3. (1 − x) (1 − x*)* ; 2 (x2 + 7ax+12a2) * 1 x2)2; * 4.73; express the different terms as powers of 2 and 3 with fractional indices, and use For.* p. 35, ll. 9, 10. 5. (A) 15; (B) x=16, y=71, z=12. *Arnett's Mathematical Formula (Tomlin, Cambridge; Simpkin, Marshall and Co., London). 6. 4 mls. per hour; 8 mls. per hour. 7. (A) 17; (B) 5, side is 23; For. p. 40, 1. 1. l. - 1; (c) 2, 6, 4. The left-hand therefore x-2 divides out. See 8. 598 sq. ft. 10'8" 598 sq. ft. 128 sq. = in. 9. See For. pp. 10, 51; 3, 9, 27, 1, 1, 27; √(10); 8286.97. 10. 3 rd. 21 pl. 1 sq. yd. 3 sq. ft. 108 sq. in. 11. 2400 √(3) c. ft. 12. 3600; 60 √(3). Geometry and Trigonometry. 2. Shew that the bisector of the angle bisects at right angles the straight line joining the extremities of the two equal straight lines. 4. The angle at the centre of the circle and the angle opposite to it in the original triangle are equal to two right angles; the result follows by Euc. III. 20. 5. The sides of the inscribed square respectively bisect the four squares, which make up the described square. Use Euc. VI. C. 9. See For. pp. 75, 76; 120o, 108°. 10. 180° +π° 180°-T 14. For. p. 80, 1. 2; we have 11. For. 77. 8. 16. Find C, and use For. p. 84, 1. 5; b=89.646. A A (90° 17. For. p. 84, 1.11; also, cot =tan 90° See also For. p. 83; 74° 18. Produce AD, making parallel to AC. sin BAD 2 sin ABE sin A 2 tan61°46'. 32' 44", 48° 59′ 16′′. since z ABE=180° – A; AE=b+c2-2bc cos ABE b sin A therefore sin BAD = √(b2+c2+2bc cos 4) DIRECT, AUGUST, 1873. PP. 8-14. Euclid, Book I. 5. Draw a diagonal, and use Euc. 1. 8. 6. Draw the diagonal through A, and use Euc. I. 19. Algebra, Logarithms, and Mensuration. 1. x 8x+29x3-79x+108x-45; 10611; (a) 9 (x+a) (x− a); (B) (x−a)(x−b); (y) (x−11)(x−7). 2. (a) x+3x+15x-11; (B) x + y2+z"-xy-yz-zx; (Y) x2 - 3xy + y2. 3. x-3; x10x+75x-309x+882x-1323; 4 or 5. 4. Add and subtract 1, and divide; see For. p. 37. 5. 98-8 √(15)+6 √(5)−40 √(3); 5 √(5)—3 √(3); 16·4248. 6. Take a − 1, a, a + 1, as the three numbers. |