cepted without the triangle, between the perpendicular and the obtuse angle. Describe a square whose area shall be three times the area of a given square. 6. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle. Prove that the base of any segment of a circle makes equal angles with the diameter drawn through one extremity of the base, and with the perpendicular let fall from that extremity upon the tangent at the other extremity. 7. In equal circles, equal angles stand on equal circumferences, whether they be at the centre or circumference. Describe a circle cutting the sides of a given square in eight points, such that they shall be the angular points of a regular octagon. 8. Two straight lines AB, CD within a circle, one of which passes, and the other does not pass, through the centre, intersect in O; shew that the rectangle AO, OB is equal to the rectangle CO, OD. Segments of circles are described on a given base, and from a fixed point in the base produced a tangent is drawn to each segment. that the points of contact all lie on the circumference of a circle. Shew 9. Give constructions (without proof) for (1) inscribing a circle in a given triangle, (2) describing a circle about a given square, (3) inscribing an equilateral and equiangular pentagon in a given circle. Shew that any equilateral figure inscribed in a circle must also be equiangular. IO. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally. Shew that two straight lines drawn from two angular points of a triangle to the middle points of the opposite sides enclose with those two sides a quadrilateral, whose area is one-third of the area of the triangle. II. If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. [Name your authority in all cases where you do not follow Euclid.] II. ARITHMETIC. (Including the use of Common Logarithms.) [N.B.-Great importance will be attached to accuracy in numerical results.] I. Add together 87 of 3, 91 of 48, 34 off, and 68 of 15. 217 2. Subtract 3 of 41% from 3. Multiply together of 1, by 2% of 6 of 101. Divide 25 by 338. 4. 5. Add together 023 of a £, 946 of a shilling, and 3'48 pence, and subtract the sum from 26 of a guinea. Give the answer in pence and the decimal of a penny. 6. Multiply 36 2894 by 00893. 7. Divide 99'994 by 2890. 8. Divide 42 547 by '00542. 9. Express 3 ozs. 7 dwts. 12 grs. as the decimal of a lb. troy. IO. What is the difference between '038 of a mile and of a furlong? Express the answer as the fraction (vulgar) of a furlong. 11. Find, by Practice, the value of 3 tons 5 cwt. 2 qrs. 21 lbs. at £12 per ton. 12. Divide £56 between A, B, C, and D in the ratio of the numbers 3, 5, 7, and 9. 13. In what time will £345 amount to £454. 25. 1d. at 2 per cent. per annum simple interest? 14. Two horses can plough in a given time as much as 3 oxen, but the cost of 4 oxen is only equal to that of 3 horses, the daily cost of a horse being 35. A certain field can be ploughed by 3 horses in 8 days. What would be the cost of ploughing it by oxen in 6 days? 15. The value of a certain house in 1880 has increased 35 per cent. since 1877. The house was rated in 1877 at two-thirds of its value, and in 1880 it is rated at three-fifths of its value, the rate in the remaining the same. Compare the rate paid in 1877 with that paid in 1880. 16. When is one number prime to another? Find the greatest common measure and the least common multiple of 7560, 27720, and 108108. 17. Find the length of the edge of a cubical block of stone containing 46 cubic yards 513 cubic inches, and the number of square inches in its entire surface. 18. One gallon of spirit which contains 11 per cent. of water, is added to 3 gallons containing 7 per cent. of water, and to this mixture half a gallon of water is added. Find the per-centage of water in the mixture. 19. A buys 3 per cent. stock at 89%. He receives one half-year's dividend, and afterwards sells his stock at 949, and finds that he has gained £54. What sum did he originally invest? 20. Find the true discount on £142. Is. 9d. due 18 months hence at 3 per cent. per annum. 21. Find, correct to a farthing, the present value of £10,000 due 8 years hence at 5 per cent. per annum compound interest. Find log 10, and calculate to six decimal places the value of (201 22. and 2. 3 Logarithms required for Questions 21 and 22. log 2=3010300 294 × 125 'III. ALGEBRA. [N.B.-Great importance will be attached to accuracy in numerical results.] I. Find the Greatest Common Measure of Find the value of log 67683=48304796 a (a − 1 ) x2 + (2a2 − 1) x + a (a + 1) (a2 - 3a+2) x2 + (2a2 − 4a + 1 ) x + a (a − 1). x3 − x2+3x+5, when x=1+2=1; and prove that (√3 + 1)2 − 2 (√2 − 1)2 = √59 – 24√6. 7. A Bill before Parliament was lost on a division, there being 600 votes recorded. Afterwards, there being the same voters, it was carried by twice as many votes as it was before lost by, and the new majority was to the former as 5: 4. How many members changed their minds? 8. Find the sum of n terms of the Arithmetical progression 17, 117, 211,...... y=5x If G is the Geometric mean between two quantities A and B, shew that the ratio of the Arithmetic and Harmonic means of A and G is equal to the ratio of the Arithmetic and Harmonic means of G and B. 9. Sum to six terms and to infinity 12+9+62 + ....... and insert two geometric means between 1 and - I. 10. If y is the sum of two numbers, of which the first varies directly, and the second inversely as x, and if y=7, when x=2, and y= −1, when x=1, shew that II. Given that the number of combinations of n things taken 4 together is to the number of combinations of n- I things taken 5 together as 5: 3, find n. 12. Assuming the form of the continued product of any number of binominal factors, deduce the truth of the binomial theorem for a positive integral value of the index. Shew that three consecutive terms of the expansion of (1+x)" can be in continued proportion only when n+1=0. 13. Expand (1 − 4x)-₺ may be thrown into the form to five terms; and shew that the general term | 2r+ I (|)”. IV. PLANE TRIGONOMETRY. (Including Solution of Triangles.) [N.B.-Great importance will be attached to accuracy in numerical results.] I. In plane trigonometry state how a right angle is usually divided by English and French mathematicians respectively. If the circumferences of the quadrants of two circles be divided similarly to the right angles they subtend, what would be the radius of a circle divided according to the French scale, in which the length of the arc of one grade would be equal to the length of the arc of one degree on a circle whose radius was 18 feet? Express in the French scale (1) the sum of the angles of a quindecagon, (2) one of the angles when the quindecagon is equiangular. 2. According to the ratio definition, point out which of the elementary trigonometrical functions are never less than unity, and which may be either less or greater than unity. Prove (sin A)2 + (cos A)2 = 1, and express the numerical values of sin 135° and tan 150° with their proper signs. 3. If (A+B) is an angle in the first quadrant, obtain by means of a geometrical figure the formula cos (A+B) = =cos A cos B - sin A sin B, and deduce from it the expression for cos (A – B). Find sec (A+B) in terms of sec A and sec B, and prove sec 105° - √√√2(1 + √3). |