(b) Interpret (a+b+c)2—(a+b)2+c2=2(a+b+c)c as a theorem in geometry. 8. (a) The square on a side of a right-angled triangle is equal to the rectangle on the hypotheneuse and the projection of the side on it. (b) Write out the expressions for the square on a side of a triangle in terms of the sides, according as the opposite angle is acute, right, or obtuse; and show how these expressions are connected by continuity. 9. Find the value of C. cos 136° 42′Xtan 151° 33: cos 143° 15′, correct to three decimal places, using contracted methods. 10. (a) Deduce the sine formula, and write down one form of the cosine formula. (b) State when you would use the sine formula, and when the cosine formula, in the solution of triangles. 11. (a) Prove sin(A+B)=sin A cos B+cos A.sin B. (b) Prove sin 3A-3 sinA-4 sin3A. 12. Find the distance in miles, to one decimal place, between two points on the same meridian on the earth's surface, if their latitudes are 10° 29′ South and 43° 51′ North, respectively; given that the radius of the earth is 3960 miles. 2. Separate. 3x2+x-2 into its partial fractions. (x-2)2(1-2x) 3. Show from continued fractions that 355/113 is a very close approximation to =3.14159; and determine from the convergents the maximum possible error. 4. A corporation borrows $30,000 at 4% interest; and is to repay principal and interest in 30 equal annual instalments. Find the value of an instalment. 5. Prove that the area of a triangle is 6. Prove that (a) √/s(s—a) (s—b) (s—c) Sin no 2 sin @ cos (n-1)+sin(n-2)0, and deduce sin 36 in terms of sin 0. (b) -1 tan1+2 tan-1}+tan-1=7. 7. Solve AABC when a=222, b=318, c=406, using logarithms throughout. 8. Two chords are drawn in a circle. Find a point on the circle from which perpendiculars to the chords are proporthe lengths of the chords. tional to 9. In any quadrilateral, skew or plane, prove that the sum of the squares on the sides is greater than the sum of the squares on the diagonals by four times the square on the join of the middle points of the diagonals. 10. Show that the medians of a tetrahedron pass through the centre, and show how the centre divides a median. 11. Prove the prismoidal formula and deduce the expression for the volume of a frustum of a pyramid. JUNIOR PHYSICS. A. Dynamics and Properties of Matter. (Any four questions). 1. Calculate, in foot pounds, the work done in slowly pulling a 20-ton car 100 yards down a grade of one in two hundred, when the coefficient of friction is 0.01. 2. An adjustable stand for a lantern is made by hinging together at the edges four light square boards so that their unhinged edges form a rhombus. One of these boards is Screwed to the table and the height of the stand is regulated. by adjusting the length of a chain that forms the longer diagonal of the rhombus. In a certain case the chain is shortened until it makes an angle of 30° with each of the boards and then a lantern of mass 20 lbs. is placed centrally on the upper board. Neglecting the weight of the boards, (a) determine the tension of the chain, and (b) find the compressive stress in each of the two inclined boards. Take cos 60°-sin 30°=0.500 and sin 60°-cos 30°=0.866. 3. In target practice over a frozen harbour a spent ninepound shell sliding along the ice at 400 feet per second beds itself in a box of sawdust (mass 150 lbs.). (a) Find the velocity imparted to the box. (b) Find how far the box is driven over the ice if the coefficient of friction be 0.04. 4. A brake shoe is pressed against the tire of a wagon wheel with a force of 500 lbs. wt. If the coefficient of friction be 0.40, find the power absorbed (in horse power) when wagon goes down hill at 3 mile per hour. the 5. A flat piece of oak (density 0.97) 6 cm. X6 cm. X2 cm. floats in a vessel of water. (a) How far below the water level is the bottom face of the block? (b) If oil (density 0.75) be poured over the water until the oil surface is level with the top of the block, find the change in the depth of the lower surface of the block beneath the surface of the water. B. Experimental Subjects. (Answer any six of the following seven questions). 6. Five cells of copper sulphate are joined in series. How long will it take a current of five amperes to deposit thirty grams of copper? Atomic weight of copper is 63.4. Would a solution of sugar be a good conductor of electricity? Explain. 7. Explain production of colors with white light by reflection, using Newton's rings apparatus. What (roughly) is the wave length of red light, of violet light, of a sound wave of frequency 512? What determines the pitch of a sound? 8. A wire running east and west and which is free to move carries a current which flows towards the west. What effect will the earth's magnetic field have upon it? What determines the magnitude of the force acting to move the wire? 9. It is desired to project an image of a lantern slide so as to just cover a screen six by eight feet; if the screen is twelve feet from the lens and the slide three by four inches, where must the slide be placed, and what focal length lens is required? Prove your result graphically, showing lens, focal points, optical centre, object, image and principal rays. 10. What effect on the color of a red body would appear if placed in different parts of the spectrum? Explain the relation between radiating and absorbing power. What do you understand by a transverse wave, by a longi |