JUNE 1898. FROM QUADRATICS. 1. (a) What relations exist between the roots and the coefficients in the equation x' + px + q = o? (b) Construct the equation whose roots are 3. Solve the simultaneous equations, x3 — y3 = =208, and 4. Derive a formula for finding the amount of P dollars placed at compound interest for n years at r per cent. per annum, interest being compounded annually. 5. How many different combinations of three colors are possible, using the colors of the rainbow, viz., red, orange, yellow, green, blue, indigo, and violet? 6. Prove the following: (a) log mu= log mlog n; log m3 = p log m. 7. (a) What is the logarithm of 1 in any system? of the base of the system? of the reciprocal of the base? (b) If the base is > 1, for what numbers are the logarithms positive, and for what numbers are they negative? SEPTEMBER 1898. FROM QUADRATICS. 1. (a) Give tests for determining the character of the roots of the equation ax2 + bx + c = o. (b) What must be the value of c if the roots shall be equal in the equation 3x2 2x + co? 4. (a) Deduce a formula for finding the sum of n terms in a geometric progression. (b) Find two geometric means between P and q. 5. Expand I 2x2 I + 2x 3x2 into a series of ascending pow ers of x by the method of indeterminate coefficients, 6. (a) Define a logarithm, its characteristic, its mantissa. (b) Given logь a = c. Express the same relation between the quantities involved, without using logarithmic notation. 7. (a) Find a method for changing the logarithm of a number from one base to another. (b) Get log, 100, having given log10 5 = 0.699. Find the value to two decimal places of 21/721/343 +71/28 61/63 5. If the hands of a clock are together at 12 o'clock, at what time between 2 and 3 o'clock will they be together? 2110:92 4. Find the seventh term and also the sum of an infinite number of terms of the geometrical series 5. The arithmetrical mean of two numbers is 14 and their difference is 6. Find the numbers. ACADEMIC DEPARTMENT. GEOMETRY (A) 1. Two triangles which have their sides perpendicular each to each are similar. 2. Define the terms, locus of points and limit of a variable, and give an example of each. State (without proof) the converse of the following proposition. A line drawn perpendicular to a tangent to a circle at the point of tangency passes through the center of the circle. 3. The sum of two opposite angles of a quadrilateral inscribed in a circle is equal to the sum of the other two angles, and is equal to two right angles. 4. Construct with ruler and compass a circle passing through a given point A, and tangent to a given line at a given point B, and prove the construction correct. 5. The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including those angles. |