5. Sum the Geometrical Series, a, ar, arm, &c. to n terms; and find the limit of the sum of the series, when r < 1. Illustrate what is meant by “the limit of the sum,” and shew that alas, as. a2n-1=ana. 6. The (2n + 1)th term of an Arithmetical Series is p, and the (2m + 1)th is q: find the (m + n)th : and/ work the corresponding problem when the series is Geometrical. 7. Shew that the Quadratic equation, ax2+bx+c=0 will have one root, two real unequal roots, or two imaginary roots according as b2= 4 ac, greater than it or less. 8. Sum n terms of the series, 1. 4 + 2. 5+3. 6+ &c. 9. Standing on the seashore, with the eye 6 feet above sea-level, an observer watched the disappearance of a vessel's topmast which was known to be 96 feet above the water-line. Shew that the vessel is about 15 miles distant from him, at the instant of disappearance. 10. Prove that, there being n quantities, a, b, c, k, 1 kn a > (abc.. n : 11. Given 203 + px2 +qx +r=0, and its roots a,b,c. Prove a? + b2 + c2 =p2 – 2q: and generalize the statement for any equation. 12. Given 204 – 6x8 + 5x2 +14x –4=0. Shew that there is one positive root between 0 and 1, one between 3 and 3. 5, and one negative root between – 2 and -1. TRIGONOMETRY AND ALGEBRA.-SECOND YEAR. APRIL 18.-3 TO 6 P. M. 1. Shew that all angles that have the same sine and cosine, must differ by some multiple of 27, and prove that tan 0 =tan (nt +0). 2. Given sin (x+y)=rin x cos y + cos x sin y, deduce the formulae for sin (3C – y) and cos (30 – y). 3. Prove cot 28 – cos 20=cot28 cos28 ; and sin (A +30)+sin (A +150) =cos A. 4. Shew the area of a triangle = 1 bc sin A= } ab sin C= &c. aʼsin B sin C 2 sin A 5. Write the expression for the area of a triangle in terms of the sides a, b, c, and deduce the area of the isosceles triangle when b=a. 6. Shew that the perimeter of the regular polygon of n sides described about a circle, is to the perimeter of the regular polygon of n sides inscribed in the same circle as 1 is to cos ; and deduce from n this and your knowledge of Geometry the ratio of their areas. 7. Prove that the distance between the centres of the inscribed circle of a triangle ABC and the escribed one that touches the A side a=a sec 2 8. A man is walking along a straight level road towards a tower right in front of him. He observes the angle of elevation of the top of the tower to be ao and, after walking d yards, the angle of elevation is 3°. Find the height of the tower and state the logarithmic equation for its calculation. 9. There is a company of 3 sergeants, 5 corporals, 15 private soldiers. How many different guards can be set, consisting of 1 sergeant, 2 corporals, and 5 privates ? 10. If n factors (2c + a), (20+ b), (ac+c), (x + k), be multiplied together, express he law of the coefficients of the descending powers of x. This law is employed in the partial proof of an important theorem. ...... a a 11. Given a table of logarithms to a base a, shew with proof how a similar table to a base b could be derived from them. 12. Shew that any number whatever can be reduced to the form, 2m +2° +2° + &c., where m, n, p, &c. are in a descending order of magnitude, and mention any curious application of the fact. ADDITIONAL MATHEMATICS.-SECOND YEAR. APRIL 20.-3 TO 6 P. M. 1. If a solid angle be contained by any number of plane angles, these shall be together less than four right angles. 2. State and prove the fundamental property with respect to sine and cosine of any angles, on which Demoivre's Theorem is based, and hence prove the first case of the theorem. 3. Write the expansion of , and deduce, by the appropriate substitutions, the exponential values for sin x and cos . 4. If x+yV-1= log (a +6 V-1), prove' 6 log Va2 + 62. tan y a 5. A, B, C, D, are consecutive angles of a regular polygon of * sides, side = a. Join AC and BD, intersecting in P. Prove that P and all points similarly found lie in the circumference of a circle con 21 centric with the polygon, and whose radius = a cot n 6. At a station, A, the angle of elevation of an object in a hori. zontal plane and bearing due N. is ao. Insurmountable obstacles prevent the observer from measuring a base line either towards or straight back from the object. He therefore measures either 1 feet S. W. from A, or 2 feet S. E. from A ; and finds the angle of elevation of the object at the end of either of these distances to be ßo. Shew how the height and distances of the object are found. 7. A and B throw 3 dice alternately, A having first throw. There is a stake m dollars ; and the condition is, that he who first throws as many as two faces the same, wins. Find in what proportion they ought to contribute to the stake that their expectations may be equal. 8. A person borrows a sum of money at a yearly interest of m per cent; and pays it by annual instalınents, paid at the end of the year, of the first year's interest together with an nt nth part of the sum borrowed. Shew that the money will be repaid in the number of log (1 + nr) where r = 100 • m years denoted by log (1 + r) • . PHYSICS. Examiner .J. G. MACGREGOR, D. Sc. THIRD YEAR CLASS. APIRL 18TH, 10 A. M.—1 P. M. N. B.-Questions marked with an asterisk have the higher values. A.—Three of the following : 1. Define mean speed, instantaneous speed, mean velocity, instan. taneous velocity.—Shew how to resolve a given velocity into two components in given directions. *2. Shew that the acceleration of a point moving with uniform speed in a circle is directed towards the centre, and is equal to the quotient of the square of the speed by the radius of the circle. 3. Enunciate the three Laws of Motion and give explanatory comments. а *4 Define work done, foot-pound, Potential Energy.-Enunciate and prove the law of the Conservation of Energy for a single particle. *5. Shew how to determine the resultant of two forces acting in opposite directions at different points of a rigid body. B. -Seven of the following : 6. What is meant by the pressure at a point of a fluid ?-Prove that at all points of any horizontal surface in a heavy fluid, which is at rest, the pressures are the same. *7. What are the fundamental hypotheses of the Kinetic theory of gases ?-Shew that Boyle's Law may be dcduced from them. 8. Describe the Mercury Thermometer.—Shew what precautions must be taken in making thermometers, that their indications may be comparable. a 9. A body is slowly heated from a very low temperature to a very high one. Describe the changes it undergoes, pointing out what becomes of the heat supplied. *10. How has heat been shown to be a form of energy ? 11. I give you a piece of iron on a glass handle. How will you determine whether or not it is (a) magnetized, (b) electrified ? *12. Pieces of iron and bismuth are introduced into magnetic fields. What changes do they produce in the lines of force of these fields ?-Hence show that they must possess the properties in virtue of which they are called paramagnetic and diamagnetic respectively. 13. How would you shew by experiment the relation between the quantity of an inducing charge and the quantity of the total induced charge? *14. Given the law of the direction of the force exerted by a current on a magnetic pole in its neighbourhood, find the law of the mutual attraction or repulsion of wires though which currents are flowing. *15. Explain how it is that the pitch of a note from a flute is changed by opening or closing apertures in its side. *16. Explain the formation of an image in a plane mirror.—Shew that a straight line in the object is straight also in the image. FOURTH YEAR CLASS. APRIL 18TH, 10 A. M.--1 P. M. N. B.-Answer only ten questions. Those with an asterisk have the higher values. 1. Prove that the total pressure on the surface of a body immersed in a heavy liquid is equal to the weight of a column of the liquid whose section is the area of the surface, and whose length is the depth of the centre of mass of the surface beneath the free surface of the liquid, the pressure at the free surface being zero. *2. Account for the elevation or depression of a liquid in a capillary tube ; and shew that it is inversely proportional to the diameter of the tube. *3. Shew that it may be deduced from the Kinetic theory of gases, that if two gases are at the same temperature and pressure the number of molecules per unit of volume is the same. 4. State the laws of variation of (a) the pressure of a gas with its volume at constant temperature ; (b) the pressure with temperature at constant volume, and (c) the volume with temperature at constant pressure ; and shew how you would verify any one of these laws by experiment. *5. How has heat been shown to be a form of energy ? 6. Define latent heat of fusion. Shew how you would determine it in any case by experiment. a *7. A divergent pencil of rays is incident directly on a convex spherical mirror. Find the relation between the radius of the mirror and the distances from it of conjugate force. 8. To an observer whose eye is vertically above an object at the bottom of a pool of water, the object appears to be a feet beneath the surface. Find the real depth, the index of refraction of the water being po *9. A small object is placed on the principal axis of a convex lens. Determine the character of the image, and draw a diagram shewing the course of the rays by which it is produced (a) when the object is between the principal focus and the lens, and (b) when it is beyond the principal focus. 10. The deviation produced by a prism in a ray which is in a plane perpendicular to the edge of the prism is in all cases away from the edge. 11. Describe the structure of the ordinary spectroscope, explain. ing the use of its various parts. 12. Define and illustrate by diagrams, altitude and azimuth, right ascension and declination, and celestial latitude and longitude. 13. What evidence is there to show that the diurnal motion of the stars is due to the rotation of the earth. *14. By what observations is the path of the sun in the celestial sphere determined ? By what additional observations may the form and dimensions of his path in space be determined ? 15. How is it proved that elements such as sodium, iron, &c., exist in the sun. DYNAMICS. APRIL 18TH, 3—6 P. M. N. B.-Answer only ten questions. Those with an asterisk have the higher values. 1. Given the displacement of a point Q relative to a point P, and that of P relative to a point 0, find that of Q relative to 0. 2. Define mean curvature, and curvature at a point. --Show that the curvature of a circle is measured by the reciprocal of the radius. 3. Shew that the normal component of the acceleration of a moving point is equal to the quotient of the square of its speed by the radius of curvature of the path. *4. A point is moving with uniform speed v in a circle of radius r. Shew that its angular velocity about a point in the circumference is a 2 r. 5. Obtain an expression for the range of a projectile on an inclined plane; and show that, with an initial velocity of given magni. tude, the same range may in general be attained by two paths. *6. Shew that two component simple harmonic motions of the same period give as resultant, in general, elliptic harmonic motion. |