VI. MATHEMATICS (2). [Full marks may be obtained for about three-fourths of this paper. Great importance is attached to accuracy.] The top and bottom layers of a pile of spherical shot are equilateral triangles, whose sides contain a and b shot respectively. Show that the number of shot in the pile will be positive if a, b, c; b, c, a, or c, a, b are in descending order of magnitude. 5. Explain how to solve in integers an indeterminate equation of the first degree in two variables. Find a number of two digits such that if it be multiplied by three, and added to four times the number formed by reversing the digits, the sum is 659. 6. If A and B be two acute angles, such that sin A = and sin B = 13, prove (without using tables) that A+2B is greater than 90° and less than 180°. 7. Explain what is meant by Mathematical Induction, and show by means of the identity cos(n+1)+cos(n − 1)0 = 2 cos no cos 0, 8. A, B are two stations a miles apart, and P a point moving in a straight line. At a given moment the angles BAP, ABP are observed to be a, B, and after a certain interval to be a+y, B-y. Find the distance described by the point in the interval. 9. Investigate expressions for the radii of the inscribed and circumscribed circles of a triangle in terms of the sides of the triangle. Find their numerical values in the case of a triangle in which a = 36, B = 73° 15', C = 45° 30'. IO. Sum the series sin a+sin(a+ẞ) + sin (a + 2ẞ) + ton terms. ... The right angle C of a triangle ABC is divided into n equal parts by lines which meet the hypotenuse AB in points, P1, P2, P3, ... Pn-1; show VII. MECHANICS. [Full marks may be obtained for about three-fourths of the paper. Mathematical instruments must be used for questions requiring a graphical method of solution.] I. Enunciate and prove the Polygon of Forces. Five forces OA, OB, OC, OD, OE acting at the point O are in equilibrium. If the forces in OA, OB, OD be respectively 4, 2, 3, find graphically the forces in OC and OE. Given 2. LAOB = 45°, LAOC = 120°, LAOD = 210°, LAOE = 270°. A rod ACB weighing 25 ozs. rests upon a smooth peg C, and its end A is attached to a fixed point O in the same horizontal line with C by means of a string OA. Find graphically the position of the centre of gravity of the rod and the magnitudes of the tension of the string and the pressure between the rod and peg. AC = 4, A0 = 2, OC = 2 inches. 3. A, B, C are three fixed smooth pegs in a vertical plane, A being 3 feet vertically below B and 4 feet horizontally to the right of C. A string 13 feet long passes round the three pegs and has its extremities attached to a weight W; find the tension of the string and the resultant pressure on each peg. 4. The weights of a system of heavy bodies are W1, W29 coordinates of their centres of gravity are x1, 1; X2, Y2; Find the coordinates of the centre of gravity of the system. ... and the respectively. A square board is divided into 36 equal squares by lines drawn parallel to the sides. A figure is formed by taking the 1st and 4th squares in the first column, the 1st, 2nd, and 4th squares in the second column, the 2nd, 3rd, 4th, and 5th squares in the third column, the 4th, 5th, and 6th squares in the fourth column, the 2nd, 3rd, 4th, 5th, and 6th squares in the fifth column, and the 3rd, 5th, and 6th squares in the last column. Show that the centre of gravity of the figure formed of these 20 squares divides the line AB in the ratio of 11 to 13. 5. A uniform rod rests on the rim of a plate with its middle point at the centre of the rim. The plate itself rests on a horizontal plane. A downward vertical force, just sufficient to disturb equilibrium, being applied to one end of the rod, show that the plate and rod will begin to move together, or the rod only, according as the weight of the rod is greater or less than rds of the weight of the plate; 4a, 5a being the diameters of the base and rim of the plate respectively, and 6a the length of the rod. 6. Define work, and show how it is measured. A right-angled triangle ABC turns stiffly in its own plane about the middle point of the hypotenuse AB. If forces P, Q, R, just sufficient to overcome the resistance, be applied at right angles to the sides BC, CA, AB at the angular points, all tending to turn the triangle in the same direction, show that the work done by them in turning the triangle through a right angle = }{Rc+(P− Q) (a - b)}; the forces remaining throughout the motion parallel to their original directions and constant in magnitude. 7. A heavy particle slides down a rough inclined plane; find the space described from rest in a given time. Two particles are projected with a velocity of 40 feet per second from points 88 feet apart, the one up and the other down, a rough plane (μ = 1) inclined to the horizon at an angle tan-14. Find when and where they will meet, and account for the double solution [g = 32]. 8. Within a smooth circular tube fixed in a vertical plane are two particles of mass P, Q connected by a string whose length is equal to half that of the tube. Find the acceleration of each particle in the direction of motion, and the tension of the string supposed tight, when the line joining the particles makes an angle @ with the horizon. 9. Two smooth imperfectly elastic balls, moving in one plane with given velocities in given directions, impinge obliquely on each other; determine the motion of each after impact. 10. A particle hangs from a fixed point in a wall by a string of length a, find the least velocity which must be given to it in order that it may make a complete revolution, without the string becoming slack. If the string come in contact with a nail in the wall situated in the horizontal line through the point of suspension and at a distance b from it, find the least initial velocity in order that the particle may make a complete revolution round the nail, without the string becoming slack. |