4. Prove that if a and b are positive numbers, (a ÷ b) + (ba) > 2. 5. Write the expansion of (a - b)" by the binomial formula. (b) 723 · b)m × (a —b). 3 ( 1 ) }; (c) 5 ( 2 ) } ÷ 5 (2)3; (d) (ab°cm)m. 3 21 4 JUNE 1892. FROM QUADRATICS. 1. Solve the equation 9x+6x= 19. 2. Determine by inspection the nature of the roots of the following equations, that is, whether the roots are real and equal, real and unequal, or imaginary: (a) 2x2 5 = 0; (b) 3x2. 5x = 2; (c) 9x2 12x+4= 0. 3x + 3. Form a quadratic equation whose second member shall be zero, whose known term in the first member shall be 4, and one of whose roots shall be — . 4. Ascertain the square root of 14 + 61 5. 5. Find the formula for the sum of a decreasing geometric progression of an infinite number of terms. 6. Write the formula expressing the number of combinations of m different things taken n at a time. x, by the method of undetermined coefficients. (Four terms will suffice.) SEPTEMBER 1892. FROM QUADRATICS. 1. Determine by inspection the roots of the equation ax (bx 2)(x2 — 9) = 0. 2, Solve the equation x2 +1.3x=7.7. 3. What is the criterion by which you determine whether the roots of an equation of the form ax + bx + c = 0 are real and unequal, real and equal, or imaginary? 4. Solve the equation 1 2 3X- 17+x=1 5 + 4x. 5. Find the sum of an infinite number of terms of the series I x + x2. x3 + etc., if x < 1. 6. Show that the logarithm of an infinitesimal is infinite, and negative or positive, according as the base of the system in which it is taken is greater or less than one. 7. Find the limit of without limit. (x2 + 1)(x − 2) when x increases 2x - X 4. Given 2x 3 < x + 5, and 11 + 2x < 3x + 5, to find the limits between which x lies. = 5. Show that if a b c : d, then a + b: a d: c d. 6. What term in the development of (a + contain a? 211 I does not a 7. Simplify the following expressions: (a) 21 3 X 31 3;. (b) 41/8 ÷ 21/2; (c) (33 — 3 ̄†)'; (d) √/18 — √ ́8; (e) |