5. In how many different ways can 3 white, 3 red, and 3 blue counters be arranged in 3 rows of 3 each? If there were n counters of each of n different colours, show that the number of ways in which the n2 counters could be arranged in n rows of n each is 1x cos (a+B) = y cos (a – B). Show that in (ii.), x+y= 2c cos a. 7. If be the circular measure of an angle less than a right angle, prove that sin 0>0-, and tan 00 + 03 03 8. A man, travelling due north along a straight road, observes the points A, B at which two distant objects P, Q lie respectively due east, and at an intermediate point C he measures the distances AC, BC, and the angles PCA, QCB. Show that he will have data from which he can calculate the length of PQ and its inclination to the direction of the road. Show that this inclination will be 30°, if each of the measured angles is 60°, and one of the measured distances is double of the other. 9. If 2 cos 0 = x+, express cos no and sin ne in terms of x. I x 10. An open box in the shape of a perfect cube lies on a horizontal plane with the diagonals of its base pointing towards the four cardinal points. Show that at noon, when the sun's altitude is tan-(2/2), the portions of the interior surface in sunlight and in shade are as 37 : 43. 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. Solve graphically the following problem :-A uniform beam AB, weight 100 lbs., is supported by two strings AC, BD, the former making an angle of 51° with the horizontal, and the latter being vertical. It is maintained at an angle of 5° to the horizon by a horizontal force P applied at B. Find, by graphical construction, the value of P in pounds. 2. Solve the following problem :-A uniform horizontal beam AB, weight 100 lbs., is placed with the end A resting against a rough wall AD, and is supported by a string CD. Find, by construction, the tension of CD in pounds, and the coefficient of friction in operation between the beam and the wall at A. AB=4', AC = 3′, AD = 2′.9". 3. Three forces, P, Q, R, make equilibrium; P is given in magnitude and position, Q in magnitude only, R in direction only, making an angle with P. Determine and R either by construction or by calculation. Show that there are generally two solutions, and that if P>Q there are limits to the angle 0, beyond which the question is impossible of solution. 4. If four forces in equilibrium act along the sides of a parallelogram, show that they are as the sides in which they act. If four forces in equilibrium act at right angles to the sides of a parallelogram (not rectangular) at their middle points, prove that they are as the sides on which they act. 5. A rectangular block ABCD, whose height is double its base, stands with its base AD on a rough floor, coefficient of friction. If it be pulled by a horizontal force at C till motion ensues, determine whether it will slip on the floor, or begin to turn over round D. 6. The masses of two particles at A, B are m, m': if P be any point, prove that the resultant of two forces represented by m. AP, m'. BP acts in GP, and is represented by (m+m')GP, where G is the centre of gravity of m, m'. Hence show that if G is the centre of gravity of the masses m, m', m" at any three points A, B, C, the three forces make equilibrium. m. AG, m'. BG, m". CG If OA, OB, OC are three diverging bars of the same material and same section, prove that if their centre of gravity is at O, the sines of the angles they make are as the squares of their lengths. 7. Find the slope of a smooth inclined plane, if the work done in drawing a heavy body up a given length of it, is equal to that done in drawing the body along an equal length of a rough floor (coefficient of friction). If the floor and plane are equally rough (μ<1), prove that more work is done in drawing it up the plane, than along an equal length of the floor, whatever be the slope. 8. If a body is projected vertically upwards, prove that the velocity at any point in its ascent is equal to its velocity when passing through that point in its fall. 9. Find the amount of vis viva lost in the direct collision of two inelastic balls, masses M, m, velocities u, v. If the balls are equal and going in the same direction, show that less than half their vis viva is lost. IO. A flexible heavy string, length 27, is moving over a smooth fixed small pulley, the two unequal portions of it hanging vertically. Prove that at the instant when its middle point is at a distance x below the pulley, the acceleration with which it is moving is x Find also the tension of the string at any assigned point of the descending portion at the same instant. II. A projectile thrown at a small elevation (3°), gives a range of 1000 yards on a horizontal plane. If the plane, instead of being horizontal, had an upward slope of 1°, what would be the range in yards, approximately? 12. A heavy bead, loosely strung on a smooth vertical circular wire, falls down it from rest at the highest point O. When at any assigned point, find the velocity with which its distance from O (in a straight line), is increasing. |