Find by the aid of the tables: (1) A mean proportional between 211 and •000256. (2) The amount of 14201. 10s. in fourteen years at 5 per cent. compound interest, payable quarterly. 5. Divide (x2-y2)3 — z6 by x2—y2 — z2. x2 y2 = and + + = 1. a2 b2 (3) x1 — 3x3 —6x2+28x-24=0 which has equal roots. 7. When is an infinite series said to be convergent or divergent? Show that the series 1 1 1 1 + + + + &c., is convergent if a be 24 3 4* greater than unity, and divergent if x=1. Ꮖ Show when the series for log, (1+x) is conver gent. 8. Assuming the law for the conversion of a continued fraction into a series of converging fractions, show that the difference between any two consecutive converging fractions is a fraction whose numerator is unity and the denominator the product of the denominators of the convergents. Find fractions converging to 26. 9. Divide a given straight line into two parts so that the rectangle contained by the whole and one part shall be equal to the square of the other part. If AB be divided in C so that the rectangle contained by AB and BC is equal to the square of AC, show that if in AC, CD be taken equal to CB, the rectangle contained by AC and AD will be equal to the square of CD. 10. Inscribe a circle in a given equilateral and equiangular pentagon. ABC is an equilateral triangle; in AB take AD and BE each equal to one-third of AB, through D draw DF parallel to BC meeting AC in F, and through E draw EG parallel to AC meeting BC in G. Show that a circle may be inscribed in the pentagon FDEGC. 11. If from any 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. If ABC be the triangle, BE the perpendicular from B on AC, BO the diameter of the circle, and if from B, BD be drawn parallel to the tangent to the circle at A cutting AC in D, prove the rectangle contained by BO and BE is equal to the rectangle contained by AC and BD. PURE MATHEMATICS. 10A.M. to 12 A.M. REV. CANON HEAVISIDE, M.A. I. What is the circular measure of an angle? When this measure is referred to, examine the length of the arc subtending the unit of angle in a given circle. The radius of a circle is 10 feet, express in degrees the angle subtended at the centre of this circle by an arc of 3 feet in length. 3. In any plane triangle, A an obtuse angle, prove cos A = b2+c2-a2 2bc If (a), (b), (B) had been given to solve a triangle, where b is less than a, and if (c) (c) be the two values found for determining the third side, prove b2+cc1 =a2. 4. AB, AC are two railroads inclined at an angle of 50° 20', a locomotive engine starts from A along AB at the rate of 30 miles an hour; after an interval of one hour, another locomotive engine starts from A along AC at the rate of 45 miles an hour, find the distance of the engines from each other, three hours after the first started. 5. When an object is viewed from an elevation above it, what is the angle of depression? An observer in a balloon observes the angle of depression of an object on the ground due south to be 35° 30′; the balloon drifts due east at an unaltered elevation, for 24 miles, when the angle of depression of the same object is observed to be 23° 14', find the height of the balloon. 6. A regular hexagon, each side of which is 10 feet, makes a revolution about a line which joins the points of bisection of two of its opposite sides, find the whole surface of the solid thus generated. PURE MATHEMATICS. REV. CANON HEAVISIDE, M.A. 1. In a right-angled spherical triangle given the hypothenuse, 74° 20′, and one angle 34° 15', find the side opposite to the given angle, proving the formula used in the solution. 2. The equation to a straight line referred to X y rectangular co-ordinates is = 1, find the dis 3 tance of the origin from the line. 4 If y2=4m(x-m) be the equation to a parabola, where is the origin? find the equation to the normal drawn at the extremity of the latus rectum of this parabola. 3. Îf u = ƒ (x), examine the condition, that u may have a minimum value when du d2u = 0 and remains dx dx2 finite. Through a given point between two lines inclined at a given angle draw a straight line forming with these given lines the least possible triangle. 4. dy = dx 2a-y is the equation to the cycloid y from an extremity of its base, find the radius of curvature at any point, and show that it is equal to twice the normal to the curve drawn at the same point. 5. Integrate the following functions: (1) √ √ a2 + x2 (2) Sシューズ x2dx 6. If A be the area of a plane curve referred dA to rectangular co-ordinates, prove z=y. Find the equation to a curve such that its area is equal to twice the rectangle contained by its co-ordinates." MIXED MATHEMATICS. REV. CANON HEAVISIDE, M.A. 1. If four forces act upon a point to keep it at rest, state generally how any one of the forces must be related to the resultant of the other three. If three forces acting on a point are in equilibrium, show that each force is proportional to the sine of the angle contained by the other two. If OP, OP1 represent two equal forces acting upon O at right angles to each other, and OQ |