SET X I. MATHEMATICS. (1.) 1. Multiply (x3 — x3y3 + y3) by (x3 + y3). (x+y) is divisible by (x + y) when (m) is an odd whole number. 2. State and prove the rule for placing the decimal point in the quotient when one decimal is divided by another. α If be a fraction in its lowest terms, examine the conditions that b α may b be expressed as a finite decimal, and from the form of (b) show how the number of decimal places is determined. 6. A mine full of water is emptied by two pumps working together in three days; if the pumps had been worked separately, one would have required 2 days more than the other to empty the mine; find the time in which each pump working alone would empty the mine. 7. Explain generally the use of the proportional parts as given in a table of logarithms. Given = log10 312044.4942103, logo 31203 4.4941964, construct the table for the proportional parts as given in tables corresponding to these logarithms, and hence find log10 31203.25. = 8. Given log10 20 1.3010300, find log10 000125. What is the logarithm of 243 in a system of logarithms. calculated to a base 3? Express to a base 10, with its proper characteristic, the logarithm of 9. On opposite sides of a base which is 120 yards long two isosceles triangles are constructed; the altitude of one triangle is double the altitude of the other, and the triangle that has the least altitude has a right angle for its angle opposite the base; find in square yards the area of the four-sided figure thus formed, and express the result also in acres, roods, &c. 10. If 30 cubic inches of gunpowder weigh one pound, what weight of gunpowder will be required to fill a cylinder of 8 inches internal diameter, and with a length of 2 feet? 11. Find the area of the six equal faces of an hexagonal pyramid, each side of the base being 6 feet and the perpendicular height of the pyramid being 8 feet. Find also the cubical content of the pyramid. How must a plane be drawn parallel to the base so as to divide the pyramid into two parts whose contents shall be equal to each other? MATHEMATICS. (2.) 1. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Write down the corresponding algebraical formula. 2. Find the centre of a given circle. If a circle be described on the radius of another circle, any straight line drawn from the point where they meet, to the outer circumference, is bisected by the interior one. 3. The angles in the same segment of a circle are equal to one another. A flagstaff BC on the top of a tower AB subtends equal angles at two points on the ground D and E, lying in a straight line with the foot of the tower. AD, AE being measured, and AB being known, find BC. 4. Describe an equilateral and equiangular pentagon about a given circle. 5. Inscribe an equilateral and equiangular quindecagon in a given circle. Show how a regular octagon, decagon, and dodecagon may be inscribed. 6. Triangles and parallelograms of the same altitude are to one another as their bases. 7. If an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line which bisects the angle. 8. The angle subtended at the centre of a circle by an arc which is equal in length to the radius is invariable. Express in degrees and circular measure the vertical angle of an isosceles triangle which is half of each of the angles at the base. 9. Investigate an expression which shall include all angles that have a given cosine. 10. Prove the following formulæ :— (1.) (sin A+ cos A) (tan A+ cot A) = sec A+ cosec A. (2.) sin 3A 3 sin A 4 sin3 A. 13. Two sides of a triangle are 1.5 and 13.5 respectively, and the included angle is 65°; find the remaining angles, having given log 2 = 3010300 and L cot 32° 30' 10.1958127, L tan 51° 28′ = 10.0988763, L tan 51° 29′ = 10.0991355. = 14. Find the area of a regular polygon inscribed in a circle, and prove that the square of the side of an inscribed pentagon equals the sum of the squares of the sides of a hexagon and decagon inscribed in the same circle. 2. If the numerator and denominator of a fraction be multiplied by the same quantity, show that the value of the fraction is not altered. Reduce to its lowest terms the expression If x + y = u, xy = v, express x+y' in terms of u and v. 4. Simplify the expression √2+1√2-1 √3+1 √3-1 Find a factor which will rationalise the expression a+b, i. e. get rid of the fractional indices. |