Spectator 10 ft. from chair above ground; and line of 4. Make a freehand sketch of an ordinary armchair with cane seat and back, under the following conditions: which rests on ground; spectator's eye 6 ft. view approximately diagonal with chair. 50 OPTIONAL SUBJECTS. PURE MATHEMATICS. Examiner-C. LITTLE, ESQ., M.A. The figures in the margin indicate full marks. 1. Find the sum of the following series correct to the sixth decimal 12 place : 2. A newspaper issued on Saturday, June 4th, 1881, is numbered 10.962. Supposing it to be issued every weekday, without intermission, find the date of the first issue. Allowance should be made for leap years. 3. Simplify 12 15 15 5. If a, b and c are in arithmetical progression and a, b' and care in 12 geometrical progression, show from the definitions that Which is greater, b or b'? b a+c b' 2√ ac in a series of ascending powers of x, and write 12 7. Prove the following identities :— (1) 2 cosec 44 + 2 cot 4A = cot A-tan 4. (2) sin 34-4 sin A sin (60° + 4) sin (60° — A). 12 8. If the sides of a triangle be 7·152 inches, 8.263 inches, and 9.375 12 inches, find its area. 9. In the lines (1) 5x-y=7 (2) y=7x-5 (3) y+4x=2 12 find the length of (3) intercepted between the other two. 10. Prove that the parabolas, whose equations are y2 = 4ax, y2 = 4bx, 12 cut at an angle whose tangent is 11. If sin y=x sin (a+y), find the simplest value of 12 1. The sides AB, BC, CD and DA of the quadrilateral ABCD are bisected at E, F, G and H respectively. Prove that the resultant of the two forces represented in magnitude and direction by EG and HF is represented in magnitude and direction by AC. 16 2. Construct geometrically the directions of two forces 2P and 3P 17 which make equilibrium with a force 4P whose direction and point of application are known. 3. Prove that a system of forces in one plane cannot be in equilibrium 16 unless the sums of the moments about any three points in the plane are each equal to zero. 4. Two equal rods AB, BC are jointed at B, and have their middle points connected by an inelastic string, of such length, that when it is tightened, ABC is a right angle. If the system be suspended from 4, shew that the inclination of AB to the vertical is tan-13. Also find the tension of the string and the action at the joint. 5. A circular disc of radius a and weight W is placed within a smooth sphere of radius b and a particle of weight w is placed on the disc. If the coefficient of friction between the particle and the disc be μ, prove that the greatest distance from the centre of the disc at which the particle can rest is 17 17 6. Prove that the power necessary to move a cylinder of radius r and 17 weight W up a plane inclined at an angle a to the horizon by a crowbar of length l inclined at an angle 8 to the horizon is 7. Divide a given inclined plane into three parts so that a particle slid- 17 ing down it from rest may describe the three parts in equal times. Find in what time each part will be described if the height of the plane is 64 ft. and its length 288 ft. 8. A particle is projected from a point A with a given velocity v and strikes a plane, passing through A at the point B. If AB be a maximum prove that the locus of B is a parabola whose focus is A and latus rectum 2v2 16 9. A smooth wedge whose section is a right-angled triangle is placed 16 on a horizontal plane, so that the hypotenuse makes an angle a with the plane. A mass Q is drawn up the wedge by a mass P hanging over the vertex. Find the force necessary to prevent the wedge sliding along the table. 10. Two equal strings, whose lengths are 13 inches, fastened to two 17 points in the same horizontal line 10 inches apart support a particle weighing 39 oz If one string be cut, shew that the tension of the other is increased instantaneously by the weight of 143 ounces. initial acceleration of the particle. Also find the 11. Assuming the principle of virtual velocities, prove the parallelo- 17 gram of forces, by making displacements parallel to the resultant and at right angles to it. 12. A string of length has its ends fastened to two points A and B 17 in the same vertical line. If a bead P on the string rotates uniformly about AB so that BP is always horizontal, prove that its angular velocity is The same number of marks is given to each question. Full marks will be given for intelligent answers to SEVEN questions. 1. Explain why a gas is cooled by expansion. Describe a method of using expansion for the production of law temperatures. 2. Give a general description of the action of a gas engine. 3. At a place where water boils at 98°C, what quantity of steam must be passed into 5 kilos of water at 0°C, in which there are 2 kilos of ice, that the temperature of the whole may be raised to 30°C? Total Heat of Steam 4. Radiant Heat, Light and Electrical Radiations are said to be the same, only differing in wave length. Discuss this. 5. Describe an ordinary field glass and explain how an image of a distant image is formed by it. 6. Why is it believed that many elements found on the earth exist in the sun? How would you test if there is iron in the sun or not? 7. Explain how you would determine the magnetic declination. What is approximately its value near Calcutta ? 8. Give a sketch of some practical form of amperemeter or voltmeter and explain its action. How would you test if its readings were correct? 9. Describe a telegraphic system connecting two places-sending and receiving apparatus and line. 10. Supposing a current to be measured by the rate at which copper is deposited in a voltameter, explain how you would show that a current in a circular wire produces at the centre a magnetic force perpendicular to the plane of the wire and proportional to the strength of the current. 2. CHEMISTRY. Examiner J. A. CUNNINGHAM, ESQ., B.A., A.R.C.Sc.I. The figures in the margin indicate full marks. 1. How would you proceed to demonstrate the composition of water. Describe and explain the usual methods for the production of carbon dioxide and carbon monoxide. How are they distinguished ? Mention any uses to which they are put in commerce. 11 12 3. Describe and explain fully any method of determining the mole- 13 calar weight of a dissolved substance. 4. Give an account of bismuth and its characteristic compounds; 13 compare the latter with those of phosphorus. How would you determine the equivalent of bismuth? 5. Give examples of endothermic and exothermic reactions, and point 14 out the cause of the different degrees of chemical activity exhibited by the products of the two classes of reactions. 6. By what means has Moissen succeeded in isolating fluorine ? How was the equivalent of fluorine determined before the element was ob tained in the free state ? 122 7. What are the general characteristics of the peroxides? Describe 11 in detail the production and properties of any one of them. 8. Give a full account of the manufacture of (a) galvanised iron; and (b) bleaching powder. 14 |