JUNE 1894. FROM QUADRATICS. 1. Solve the equation 12x2 + 5x + 1 = 0. 2. Determine by inspection the roots of the equation x2 - 2ax (b+ a)(b − a), and state the theorem employed. 4. (a) Derive the formula for the number of arrangements of m different elements taken n at a time. (b) How many combinations can be made of 10 different things taken in sets of 7? 5. Prove the following propositions: (a) log (mn) I log m+log n; (b) log mã = loge NX loga e. log m 11 (c) loga N = 6. (a) Given log10 3 0.47712 to determine the number of figures in 3100. (b) Determine from the datum of (a) log10 V0.3. (c) Indicate the manner of finding the value of x from the equation 5* = 8, when you have given a table of logarithms. 7. Expand 2 3x2 4x into a series of ascending powers of x to five terms, by the method of undetermined coefficients. 3. Resolve 8x2 + 18x 5 into two factors. 4. Solve the simultaneous equations, x' + y3 504, and x2- xy + y2 84. 5. Find the sum of the series 3+1 + + etc., to infinity. 6. (a) Derive the formula for the number of combinations of m things taking n at a time. (b) Show that the number of combinations of m things taken n at a time is the same as the number taken m at a time. 7. (a) Prove that loga b X logь a I (b) Given log 0.11 = 1.04139, to find log /0.11. 11 3. Solve the simultaneous equations (a + c)x bc, x+ya+b. by 5. Expand (2x3±1)5 by the binomial formula. 6. Simplify (a) 21/128/18; (b) (xm+n)m-n + (x") +m ÷ (xm)n+m; (c) 7. Which is the greater, the cube root of two, or the seventh root of five? Prove your answer. SEPTEMBER 1895. To QUADRATICS. I. Resolve into lowest factors: (a) y-y-6; (b) x' + x3y + xy2 + y3; (c) (2a - b)2 — (2b+a)2; (d) x4n 4. Prove that the sum of the squares of any two different integers is greater than twice their product. 5. Extract the square root of 9x-12x2+10x2 — 4x + 1. 6. Simplify the following expressions. JUNE 1895. FROM QUADRATICS. 1. (a) Find by inspection the sum and the product of the roots of the equation ax2 + bx + c o, and state the theorem used. (b) For what value of a will the equation ax2 + 6x + 3 = o have equal roots? 2. Solve the equation (4a3 — b')(x2 + 1) 4a3 + b2 3. Solve the simultaneous equations x2 + xy = 12, xy — 2y2 = I. = 2X. 4. Derive a formula for the amount of P dollars in n years at r per cent, compound interest, interest being payable annually. I X 5. Convert into a series of ascending powers of 2+3x x, by the method of undetermined coefficients. Five terms will suffice. 6. Prove the following: (a) log (mn) = log m— log n; (b) log mPp log m; (c) log, ex loge 10 = 1. 7. (a) Write equivalents for loga a; loga 1; loga (1 ÷ a); logo .001; log, 8. I (b) Indicate the solution of 3* = IO. |