MONDAY, 24TH NOVEMBER. [10 A.M. TO 1 P.M.] ARITHMETIC AND ALGEBRA. J. T. HATHORNTHWAITE, M.A.; Professor KERO LAXUMAN CHHATRE [The black figures to the right indicate full marks.] 1. If 17 guineas be lost by the sale of 460 lbs. of raw silk at £1 6s. 4d. per lb., what was the prime cost per pound—and loss per cent. ? 2. What rate of interest does a person obtain by purchasing shares in a railway at Rs. 720, the annual dividend being 7 per cent., and the original value of a share Rs. 500? 3. What sum must be insured to cover Rupees 3,091 in case of loss, the premium being 2 per cent., the policy duty 3 annas per cent. and commission 5 annas per cent. ? 4. A flagstaff 16 feet high stands at the top of a tower whose height is 36 feet; find the distance from the foot of the tower at which the flagstaff subtends the greatest angle. 5. Solve the following 2x 1 – 查æ 8 6 8 6 12 (i) + x + y + 4 = (iii) 1+2xy + 3x2 y2 x3+ y3 2 y2 x + 2y2 + ∞ + 1 S 6. Two master-bricklayers undertake to lay the founda- 10 tion of a new court, each taking a part and beginning together. If they had worked together till the whole was fini hed, it would have taken only of the time it actually took to finish it; and B would have done enough to occupy A three months, and A enough to occupy B twelve months, which is 36 yards more than A actually did. How many yards were there in all ? 7. Express in the form of the sum of two simple surds 10 the roots of the following equations: (i) x 2 ax2b2 = 0 · 4 (1 + n2) a2 x2 + n2 a1 = 0 8. Given log 1. 6989700; log }=1.5228787; find 3 4 3 the logs of √3, √2, § √ ·05 and 13 √ (1·6) 3 × √ (21·6)* 9. A country trebles its population in a century: what is the increase per year per million: given log 3 0.4771213 log 101·12.0047512 = - 2.0047941 10. Write down the general terms of: 8 6 6 -} (i) (a2 + x2) 11. If the mth term of an A. P. be n and the nth term 10 m, how many terms must be taken so as to give the sum } (m + n ) ( m + n − 1 ) ? and what will be the last of them? - Determine m and n in terms of a and b so that 10 12. ma+nb m + n may be the arithmetic mean between m and n and the geometric mean between a and b. MONDAY, 24TH NOVEMBER, [2 P.M. TO 5 P.M.] EUCLID AND TRIGONOMETRY. J. T. HATHORNTHWAITE, M.A.; Professor KERO LAXUMAN CHHATRE, [The black figures to the right indicate full marks.] 1. Define Parallel straight lines. If a straight line falling upon two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another. 10 2. Show that in any triangle the sum of the squares on 10 two sides is equal to twice the square on half the base, together with twice the square on the straight line drawn from the vertex to the middle point of the base. 3. Show that each of the triangles made by joining the 12 extremities of adjoining sides of a regular pentagon is less than a third and greater than a fourth of the whole area of the pentagon. 4. Equal parallelograms which have one angle of the one 10 equal to one angle of the other have their sides about the equal angles reciprocally proportional; and if the sides about the equal angles be reciprocally proportional the parallelograms are equal to one another. 5. If two parallel planes be cut by another plane, their 8 common sections with it are parallel. 6. Prove that any regular polygon inscribed in a circle is 10 a mean proportional between the inscribed and circumscribed regular polygons of half the number of sides. 7. If be the circular measure of a positive angle less than 10 a right angle, show that sin @ is greater than 03 8. If an angle of a triangle be divided into two parts such 10 that the sines are in the ratio of the sides adjacent to them respectively, prove that the difference of their cotangents is equal to the difference of the cotangents of the angles opposite to these sides. 9. In the ambiguous case the area of one triangle is m 10 times that of the other; show that if b be the greater of the given sides and a the less, 10. A rope-dancer wishes to ascend a tower 100 feet high 10 by means of a rope 193 feet long. If he can do so, find at what inclination he must be able to walk up the rope, having given log 2 = 0:30103 log sin 30° 49′ = 9.70761 TUESDAY, 25TH NOVEMBER. J. T. HATHORNTHWAITE, M. A.; Professor KERO LAXUMAN CHHATRE. [The black figures to the right indicate full marks.] 1. Enunciate and prove the proposition called the Polygon 5 of Forces, and state in what cases its converse is true. 2. If three forces acting upon a rigid body in one plane keep it in equilibrium, their directions, if not parallel, shall meet in one point, and each force shall be proportional to the sine of the angle between the directions of the other two. 8 3. A wire without weight is bent into the shape of an arc 10 of a circle. To one extremity is fixed a heavy particle, and the other extremity attracts the article with a force equal to its weight. The arc is placed with its plane vertical on a rough horizontal table. Determine the angle subtended by the arc at its centre in order that there may be equilibrium. 4. Show how to find the centre of gravity of any number of heavy particles in one plane. 5. Explain the conditions of equilibrium of a body on a horizontal base. How far may a person tilt back his chair without upsetting it? Will the angle of tilting be larger if he stretch his feet out or if he gather them under the chair? 5 8 6. Find the ratio of P to Win the 3rd system of pulleys. 10 In such a system prove that if the tensions of the strings increase in G. P., so do the weights of the pulleys. 7. In the system of pulleys in which each hangs by a 10 separate string, prove that there is a mechanical advantage if the weight to be supported exceed the weight of the heaviest pulley. 8. Find the conditions of equilibrium of three levers, AB. 8 BC, CD, placed horizontally, the middle lever BC resting with each end in contact with the ends of the side levers AB, CD, and the system being subject to the forces P, W, acting vertically at A, D, respectively. 9. Describe the toothed wheel. Prove that there is equilibrium on a pair of toothed wheels when the moments of the power and the weight about the centres of their respective wheels are as the perpendiculars from the centres of the wheels on the direction of the pressure between the teeth in contact. 8 5 10. Investigate the conditions of equilibrium in the Wedge. 11. A six-sided figure has each diagonal parallel to a pair 13 of opposite sides. Prove that its centre of gravity is the point of intersection of two straight lines bisecting pairs of opposite sides, and hence prove that the three lines bisecting pairs of opposite sides of such a figure will meet in one point. 12. A cylinder of weight w and radius is supported 10 with its axis horizontal on an inclined plane by a beam of |