(c) an exterior angle of a concyclic quadrangle is equal to the opposite interior angle. 9. (a) The square on a side of a right-angled triangle is equal to the rectangle contained by the hypothenuse and the projection of the side on it. (b) From (a) deduce common expressions for (i) the square on the hypothenuse, (ii) the square on the altitude to the hypothenuse. 10. P is any point in the base BC of an isosceles triangle ABC. Prove AB2-AP2=BP PC. C. 11. A ball of 2 feet radius subtends an angle of 43'. Find its distance from the eye. a b c 12. Prove (a) R=4A (b) cos (A-B)=cos A⋅ cos B+sin A sin B, and deduce values for cos(A+B), sin(A+B), and sin (A-B). 13. In ▲ ABC, a=7, b=6, c=8. D is the mid point of AB, E is on AC twice as far from C as from A. Find the sides, angles, and circumdiameter of A ADE. SENIOR MATHEMATICS. 1. Sum the series (a) 1+2x+3x2+....+nx2-1, (b) Σn (3n—1) to ʼn terms. 2. Use the principle of undetermined coefficients to (a) separate 2x2/(1—x)(2+x)2 into partial fractions. (b) expand (1-x) to 4 terms. 3. (a) Write out an expansion for a in powers of (i) x, (ii) a-1. (b) From (a) deduce the logarithmic series. 4. Prove (a) the cosine formula, (b) sin 30 etc., in terms of sin 0. = (c) the area of a triangle = abc/4R=√s(s—a) (s—b) (s—c). 5. (a) Find a logarithmic formula for obtaining an angle of a triangle in terms of the sides. (b) The sides of a triangle are 14, 21, 27. Find the largest angle, using (i) natural functions, (ii) logarithmic functions. 6. Show that 2 tan-1+tan-14= 7. 7. (a) The homologous sides of similar triangles are proportional. (b) The perpendicular from any point of the circumference upon a chord is a mean proportional between the perpendiculars from the same point upon the tangents drawn at the end points of the chord. 8. (a) Given the base and the vertical angle of a triangle, construct the triangle so that its sides may be as 4 is to 1. (b) Describe a circle to pass through a given point and to touch two given intersecting straight lines. How many solutions are there? 9. The diameters of a regular tetrahedron are mutually perpendicular. 10. Show by comparing corresponding laminae that pyramids are equal if their bases and altitudes are equal. 11. (a) The volume of a zone of a sphere is πh(3r2 +32 +h2). (b) Find the volume cut off from a sphere of 13 inches radius by a plane 12 inches from its centre. JUNIOR PHYSICS. A. Dynamics. (Any five questions). 1. A stone, having a mass of 800 grams, is dropped over the edge of a cliff and strikes the ground 4 seconds later. Calculate the height of the cliff, and the velocity, momentum and kinetic energy of the stone just before it strikes the ground. 2. A particle, having a mass of 700 grams and moving with a velocity of 5.4 metres per second, is brought to rest in 30 seconds by a steady force. Calculate the distance traversed in this time, the magnitude of the force, and the work done by the force. 3. (a) A rope, having its two ends tied to two supports at the same level, is loose enough so that its two halves make an angle of 120 degrees with each other when an object is attached to the middle point. Calculate the tension of the rope if the mass of the object is 100 kilograms. (b) Tell what is meant by a vector quantity, naming several examples, and how to find the vector quantity equivalent to two given vector quantities. 4. Explain three devices which would enable you to lift an object weighing 400 lbs. if you could only exert a force of 100 lbs. 5. Explain the action of a siphon, and calculate how deep a tank of salt water could be emptied by means of a siphon placed over the wall of the tank if the density of the salt water is 1.2. 6. (a) Prove that the pressure at a point within a fluid is the same in all directions, or describe an experiment showing the same. What is the magnitude of this pres \ sure? (b) Describe experiments illustrating the pressure of the atmosphere, the expansibility of a gas, and the existence of molecular forces in liquids. B. Electricity, Magnetism and Sound. 7. (a) What is the electrostatic unit of potential? (c) What is the relation between the volt and the electrostatic unit? 8. A Daniell cell is composed of a zinc plate in a dilute solution of sulphuric acid and a copper plate in a saturated solution of cupric sulphate. Give the two chief reasons. for the use of the cupric sulphate. 9. A resistance of 120 ohms is in parallel with one of 180 ohms. What resistance must be put in series with this system to allow 12 amperes to flow when 110 volts are applied? 10. Describe an experiment to demonstrate that sound waves reflect from surfaces according to the same law as light waves. C. Heat and Light. (Any three questions). 11. On passing 20 grams of steam into 1000 grams of water originally at 10 degrees centigrade, the resulting temperature is 22.3 degrees. Calculate the latent heat of steam. |