SUBJECT V. PURE MATHEMATICS. Stage 1. 49 Before commencing your work, you must carefully read the following instructions: Put the number of the question before your answer. Every page of the paper supplied for the answers may be used. You are to confine your answers strictly to the questions proposed. All the work by which a result is obtained must be clearly shown in immediate connection with the question. No credit will be allowed for any result, however correct it may be, unless the work be shown, so as to enable the Examiners to satisfy themselves that the candidate has understood the question, and to see by what process the result has been obtained. Anything which you do not wish the Examiners to notice should be crossed out with the pen. The number of marks assigned to each question is given in brackets. Unless you expressly state the contrary, it will be assumed that you have read GEOMETRY in Euclid, and you will be expected to follow Euclid's sequence. Otherwise you must state what text-book you have used in Geometry. The examination in this subject lasts for three and a half hours. INSTRUCTIONS. Read the General Instructions on pages 3 and 4. The paper is divided into three sections. You are not to answer more than three questions in any one section, and not more than eight questions altogether. No candidate can be passed who fails to obtain marks in any two sections. A. 1. (a.) Show that the square root of 2 is very nearly 14142, and that it is intermediate to and 1 and is more 5 12 ,12 29 nearly equal to the latter than to the former of these numbers. (b.) Multiply 3.141592 by 0·097381, so as to obtain the first six decimal places of the product. (10) 2. Find the weight in kilogrammes of a cubical block of metal each of whose edges is 1.02 metres long, having given that a cubic centimetre of the metal weighs 8.95 grammes. Find also to the nearest centimetre the length of a diagonal of a face of the block. (10) 3. Divide 7437. 15s. into three parts proportional to the numbers 15, 12 and 8. A and B are partners in a business, and A contributed 10,000l. of the capital; at the end of a certain year there is a profit of 1,1517. for division. If A's share is 6757., find how much of the capital was contributed by B. (8) 4. A tradesman has 15 articles worth 7s. 24d. each, 12 articles of a like kind worth 6s. 81d. each, and 4 worth 8s. 1d. each; find the average value of the articles; and, if they were sold for 7s. 6d. each, find what would be the profit per cent. (8) 5. Two-thirds of a man's capital is invested at 4 per cent. per annum, the other third at 3 per cent. per annum; the income tax at 11d. per pound on the whole of the annual interest amounts to 38/. 10s.; find the capital invested. (8) 6. A railway on leaving a station S rises uniformly for half a mile at a slope of 1 in 160; it then runs on the level for three-quarters of a mile and afterwards falls for the next mile at a slope of 1 in 110. Find at how many yards distance from S it is at the same level with S, the distance being measured along the railway. B. (10) 7. Draw a line AB about 3 ins. long and mark a point P about 1ins. above its middle point; with your instruments draw a liue PN at right angles to AB. (a.) Define a right angle. (b.) Prove the correctness of your construction. (10) 8. If two triangles have two sides of the one equal to two sides of the other each to each, and the angles included by those sides equal, show that the triangles are equal in all respects. O is the centre and OA a radius of a circle; from any point B in OA a straight line BC is drawn at right angles to OA to cut the circle in C; join OC, from OC cut off a part OD equal to OB and join DA; show that the angle ODA is a right angle. (10) 9. ABC and DEF are two triangles, such that the side BC equals the side EF, and that the angles ABC, BCA are equal to DEF, EFD respectively; show that the triangles are equal in all respects. ABC and DBC are two triangles on the same side of BC, and the angles ABC, BCA are equal to DCB, CBD respectively; if AC and BD intersect in E, show that the triangles AEB, DEC are equal in all respects. (10) 10. Define a rectangle, a parallelogram,and a diameter (or diagonal) of a parallelogram. Show that the diameters of a parallelogram bisect each other. Show also that if the diameters of a parallelogram are equal the parallelogram is a rectangle. (12) 11. D and E are the middle points of the sides AB, AC of a triangle ABC; show that DE is parallel to BC and equal to half BC. If the middle points of the adjacent sides of a quadrilateral are joined in succession, show that the figure thus formed is a parallelogram and that its area is half the area of the quadrilateral. (14) 12. Show how to bisect a triangle by a line drawn through a given point in one side. If ABC is the triangle, and if the given point (P) is in BC, explain under what circumstances the bisecting line will be drawn to a point in BA (i.e., not in BA produced). Draw a triangle ABC whose sides AB, BC, CA are about 2, 3 and 3 inches long respectively; let CP be about one inch; bisect the triangle by a line drawn through P. (14) and finding the value of each of the quantities given above and their sum. 14. (a.) Show that (10) 4a2 b2 + 2 (a2 + b2) (a + b)2 — (a + b)1 = (a2 + b2)2. (b.) Verify the preceding identity in the case when a = 2, b = (c.) Express 3+ 2x2-3x as the product of three factors. 15. Find the greatest common measure of the numerator and denominator of the fraction 3 (12) numerical value of the fraction w' en x = I 16. Reduce the following expressions their simplest forms: Verify your result in (b) for x= 0, 2, 5, respectively. 17. Solve the following equations: x x + 5 the cases in which (12) (c.) a (x + y) + b (x − y) = a (x − y) + b (x + y) = 1. (14) 18. A and B are running in opposite directions along a straight road, and B's pace is four-fifths of A's pace. At a certain instant they face each other and are 200 yards apart; in the course of the next three minutes they pass each other, and at the end of the three minutes they are 1069 yards apart. Find A's pace in yards per minute. (14) SUBJECT V. PURE MATHEMATICS. Stages 2, 3, and 4, and Honours in Division I. Before commencing your work, you must carefully read the following instructions:You may take the Stage 2, or Stage 3, or Stage 4, or Honours, but you must confine yourself to one of them. Put the number of the question before your answer. Every page of the paper supplied for the answers may be used. You are to confine your answers strictly to the questions proposed. All the work by which a result is obtained must be clearly shown in immediate connection with the question. No credit will be allowed for any result, however correct it may be, unless the work be shown, so as to enable the Examiners to satisfy themselves that the candidate has understood the question, and to see by what process the result has been obtained. Anything which you do not wish the Examiners to notice should be crossed out with the pen. A table of logarithms must NOT be used in working these papers. All the logarithms needed will be found on page 66 of this paper. 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, The number of marks assigned to each question is given in brackets. Unless you expressly state the contrary, it will be assumed that you have read GEOMETRY in Euclid, and you will be expected to follow Euclid's sequence. Otherwise, you must state what text-book you have used in Geometry. The examination in this subject lasts for three and a half hours. INSTRUCTIONS. Read the General Instructions on pages 3 and 4. The paper is divided into three sections. You are not to answer more than three questions in any one section, and not more than eight questions altogether. No candidate can be passed who fails to obtain marks in any one section. Stage 2. A. 21. A straight line is divided into any two parts; show that the square on the whole line is equal to the sum of the squares on the parts together with twice the rectangle contained by the parts. Give a geometrical illustration of the identity (a - b)2 = a2 + b2 - 2ab, and show how to divide a given straight line into two parts such that the sum of the squares on the two parts may be the least possible. (20) |