11. Prove that (1.) sec2A sec2B tan2A tan2B. (2.) sin (A + B) sin (A – B) = cos2B — cos2A. (3.) tan-1 +tan-1 1 1 + a 2 1 -1 1 -α +tan-1 = Nπ. 13. Two sides of a triangle are as 5: 9, and the included angle is a right angle. Find the other angles, having given log 23010300, L tan 19° 26' = log 34771213. 9.7446051, L tan 19° 27' = 9.7448497. 14. Find an expression for the radius of the circle described about a given triangle. If the altitude of an isosceles triangle is equal to its base, the radius of the circumscribing circle is g of the base. 15. Prove that the area of the square described about a circle is double that of the square inscribed in the same circle. and if a+b+c= 0, show that a3 + b3 + c3 = 3abc. c) (x − a) (a - b) (ac) (x − a) (x - b) to its lowest terms: and find the highest common divisor of ax5+ bx2 + cx3 ax2 - bx —c, and 3. If two square numbers be added together, the double of the result is also the sum of two square numbers. Extract the square root of 9 (~2+12) - 24 (~ + 1) + 34, 4. Reduce the following expressions to their simplest forms :(1.) 4√147 – 3√75 – 2√§. y x = 1. 4x+6+4√x2 (3.) 22-4x+6 + 4 √ ∞2 − 7x + 11 = 3x. 6. If 3 yards be taken from one side of a rectangle whose perimeter is 14 yards, and added to the other side, its area will be doubled. Find the lengths of the sides. 7. How many superficial feet of inch plank can be sawn out of a log of timber 20 feet 7 inches long, 1 foot 10 inches wide, 1 foot 8 inches deep? The unit of length used by engineers is the chain containing 100 links: each link being 0.66 of a foot. Show that an acre contains 10 square chains, assuming 1760 yards to a linear mile, and 640 acres to a square mile. 8. How many coins, each 13 inch in diameter, can be arranged on a circular table 20 inches in diameter, in the form of a regular hexagon? Find the area of that portion of the table not covered by the coins, and compare the length of a string which just passes round all the coins with the circumference of the table. 9. Assuming that a cubic foot of water weighs 1000 ounces, and that a given volume of iron weighs 7.21 times as much as the same volume of water: find the weight of a bombshell, the exterior and interior diameters being 10 and 8 inches respec tively (T = 22). 10. Define a logarithm, and explain the terms base, characteristic, mantissa, and modulus. What arithmetical calculations are facilitated by the use of logarithms? State the properties of logarithms by which this is effected. 11. Given log 2 = 3010300 and log 3 = 4771213, find the logs 29 of √, √, and (31) ̃3: and find the number whose log is 12. The sum of £9. 08. 10d. is allowed for papering a room 27.7 feet long, 19.55 feet wide, and 12.4 feet high: how much per yard must be given for a paper 2.7 feet wide? MATHEMATICS. (2.) 1. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the line which is made up of the half and the part produced. 2. If two circles touch one another externally, show that the straight line joining their centres passes through the point of contact. Two circles touch one another externally: show that the square on the common tangent is equal to the rectangle contained by their diameters. 3. If in a circle two straight lines cut one another which do not both pass through the centre, they do not bisect one another. 4. If from any point without a circle there be drawn two straight lines, one of which cuts the circle and the other meets it, and if the rectangle contained by the whole line which cuts the circle and the part of it without the circle be equal to the square on the line which meets the circle, this straight line touches the circle. 5. Show how to inscribe a circle in a given triangle. A circle is inscribed in an isosceles triangle; show that the triangle formed by joining the points of contact is also isosceles. 6. Show how to find a mean proportional to two given straight lines. 7. Similar triangles are to one another in the duplicate ratio of their homologous sides. 8. Show that the circumference of a circle always bears a constant ratio to the diameter, and explain what is meant by the circular measure of an angle. Express in circular measure an angle of 240°, in degrees the angle whose circular measure is 9. Establish the formulæ and express 2 π. 3 11. If A+B+C = 180° show that tan A+ tan B+tan C = tan A. 12. Find an expression for all the angles which have a given 13. One of the sides of a right-angled triangle is of the hypotenuse find the other angles, having given 14. Two sides of a triangle are 4 feet and 6 feet in length respectively, and the included angle is 30°. Find the area of the triangle. 15. Show that the length of a side of an equilateral triangle inscribed in a circle is to that of a side of a square inscribed in the same circle as 3 to √2. |