THURSDAY, JUNE 7, from 2 to 4.30 P.M. III. 6. Mathematics. [N. B. Candidates are reminded that in order to pass in this subject they must satisfy the Examiners in the first part of this Paper. No credit will be given for any answer, the full working of which is not shewn.] 2. Multiply together y+z, z+x, x+Y. 3. Find the G. C. M. of 23-93x-308 and 3-21x2+131x—231; and the L. C. M. of 12x2y (x3+y3), 18xу2 (x3—y3) and 21 x2 y2 (x2+x2 y2+y1). 6. A man buys 570 oranges, some at 16 for a shilling and the rest at 18 for a shilling: he sells them all at 15 for a shilling and gains three shillings: how many of each sort does he buy? 8. The difference of two numbers multiplied by their product is 30, and the difference of their cubes is 117; find the numbers. 9. Insert n arithmetical means between a and b. If twice the sum of the whole series is three times the 11. If a: bcd, prove that (la3 + mb3) (+) = (ld3 + mo3) (+). 3. 12. If a, b, c, d are in geometrical progression, prove that [N. B. Candidates are reminded that in order to pass in this subject they must satisfy the Examiners in Euclid I, II.] 1. Euclid I, II. 1. Define a straight line, an acute-angled triangle, a rhomboid. 2. Bisect a finite straight line. 3. If a straight line is divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. 4. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side of the line; and likewise the two interior angles upon the same side of it together equal to two right angles. 5. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. 6. If the square described upon one of the sides of a triangle is equal to the squares described upon the other two sides, the angle contained by these two sides is a right angle. 7. Describe a square equal to a given rectilineal figure. 2. Euclid III, IV, VI. 8. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. 9. Describe a circle about a given triangle. 10. If an angle of a triangle is divided into two equal angles by a straight line which also cuts the base; the segments of the base shall have the same ratio which the other sides of the triangle have to one another. The triangle ABC is right-angled at C. The internal and external bisectors of the angle BAC meet the side BC in the points D and E. Shew that AB touches the circle described on DE as diameter. 11. If from a point without a circle two straight lines are drawn, one of which cuts the circle and the other meets it; then if the rectangle contained by the whole line which cuts the circle and the part of it without the circle is equal to the square of the line which meets it, the line which meets shall touch the circle. Describe a circle which shall touch a given straight line and pass through two given points which lie both on the same side of the straight line. 12. Equal triangles, which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional. 13. If perpendiculars are drawn from any point on the circumference of a circle to the sides of an inscribed triangle, their feet are in the same straight line. SATURDAY, JUNE 9, from 2 to 4.30 P. M. III. 6. Mathematics. [N.B. No credit will be given for any answer, the full working of which is not shewn.] 1. Plane Trigonometry and use of Logarithms. 1. Trace the changes in sign and magnitude of cos A — sin A as A varies from zero to four right angles. sine. Find an expression for all angles which have a given 2. Prove that (1) cos (A+B) = cos A cos B-sin A sin B; 4. If a, b, c, A, B, C are the sides and angles of a plane triangle, the radius of the inscribed circle, ▲ the area, s half the sum of the sides; shew that 5. If in a plane triangle sin A+ sin B = 2 sin C, shew that 6. Given log 2 = 30103, log 3 = 47712; find log 72, and log √15. 7. In a triangle ABC, A = 30°, a = 13, b = 17 ; find the possible values of B, having given log 1.311394, log 1.723045, L sin 40° 50′ 9.81548. 8. A tower stands on a horizontal plane, and is observed from two points on the plane distant 20 yds. and 45 yds. from its base respectively. The angle of elevation of its summit in the former case is double that in the latter: find the height of the tower. 2. Mensuration. 1. A garden in the form of a rectangle, whose sides are 36 and 28 yards respectively, has a path 5 feet wide running round inside its edge, and 8 equal circular flower-beds of radius 3 feet. How many square feet of grass will be needed to cover the remainder? 2. The volume of a cone is the same as that of a hemisphere on an equal base. Compare the height of the cone with the radius of the hemisphere. 3. If a cubic foot of water weighs 1000 ounces, what weight of water will fill a cylindrical pipe 96 yards long whose internal diameter is 31⁄2 inches? 4. What will it cost to paint the curved surface of the frustum of a cone, the radii of whose ends are 8 feet and 8 inches respectively, and whose height is 13 ft. 9 in., at 18. 6d. the square foot? FRIDAY, JUNE 8, from 5.30 to 8 P.M. III. 7. Mechanics and Mechanism. [N. B. More credit will be given to a few questions answered fully than to a greater number answered imperfectly. The answers are to be illustrated by diagrams or drawings, where these can be introduced.] 1. Define the terms-mass, weight, momentum, acceleration, work. 2. Shew how to find the line of action of the resultant of two unlike parallel forces. ABCD is a square: forces of magnitudes 2, 4, 2, 4 act along AB, BC, CD, DA respectively: find the magnitude and line of action of their resultant. |