JUNE 1899. FROM QUADRATICS. 1. (a) Determine by inspection the roots of the equation (b) Resolve the first member into factors—stating, in each case, the theorem you use. and verify your solution för a = b. (b) Show that the roots of the equation are imaginary using an auxiliary quadratic. 4. Given a b c d to show that 5. How many different three-figure integer numbers can be expressed by the nine digits, without repeating any figure in any one number? 6. Find the limit of the sum of the series as the number of terms increases without limit. 7. (a) Give the values of the log, 125; log, 5; (b) Given log(x'y') = a, to find log x and log y. following expressions: log, I. and log 2 X = b SEPTEMBER, 1899. FROM QUADRATICS. 1. Determine the roots of the equation − (x3 — 8 ) ( x2 — 1) (3x2 — 5x — 2) = 0. 2. (a) Solve the equation a2x2 + (a + b)2x + 2c2 = 2b2 + (a − b)2x — a2x2. (b) When are the roots real and unequal? When real and equal? When imaginary? 3. Find the value of x that will satisfy the following equation V 3x + 2 — √ 2x + 1 = √x +1 the radicals being taken in the sense of ordinary arithmetic. 4. Solve the simultaneous equations m, n, p, q, being multipliers chosen at pleasure. 6. How many diagonals has an ordinary polygon of n sides? 6. A and B buy stock, A buying twice as much as B. If A had paid $1000 more and B $1000 less, A would have paid three times as much as B. each invest? How much money did JUNE, 1899. (B) 1. Find 3 numbers in the ratio of 1:2:3 such that the sum of their squares is 350. 3. A man deposits in the bank 1 cent the first day, 2 cents the next day, 4 cents the third day, and so on for sixteen days: find the whole amount deposited. find the values of A B and C by the method of undeter mined coefficients. 5. Expand by the binomial theorem four terms. (x2a) GEOMETRY. June, 1899. (A). 1. Two triangles are similar if their homologous sides are proportional. 2. Define the limit of a variable. Prove that if two variables are always equal their limits are equal. Prove that the area of a circle is equal to one-half the product of its radius and circumference. 3. (a) In any quadrilateral if a line be drawn through the middle points of two adjacent sides, and a second line through the middle points of the other two sides, these lines will be parallel. (b) If the middle points of the opposite sides of a quadrilateral be joined the lines so drawn will bisect each other. 4. (a) If two circles are tangent internally and if the radius of the one be the diameter of the other, a chord of the larger drawn through the point of tangency is bisected by the smaller. (b) What example of a locus is found in the previous figure? |