SEPTEMBER 1895. FROM QUADRATICS. I. Find the roots of the equation ax2 bx + co, and show what relation must exist between the coefficients in order (1) that the roots may be real and unequal, (2) real and equal, (3) that they may be imaginary. 4. Find the value of the recurring decimal 0.123535 +. 5. (a) Derive the formula for the number of permutations of m different things taken n at a time? (b) In how many different ways may a base-ball nine be arranged, the pitcher and catcher being always the same but the others playing in any position? 6. Define logarithm, characteristic, mantissa. Transform 3 a' d2 into a form adapted to computation. log 7. (a) Of what number is 2 the logarithm in a system whose base is 3? (b) Find log, 1000, having given log10 5 = 0.699. JUNE, 1896. TO QUADRATICS. 1. Resolve into lowest factors: (a)x — y; (b) m2 + 5m 50; (c) a'(bc) + b2(c-a) + c'(a - b). 2. Solve the equation (xa)(x — b) + (a + b)2 = (xa)(x+b). 3. Solve the simultaneous equations 5x + 3y = 65, 2y z=11, 3x + 4z = 57. - 4. The width of a room is two-thirds of its length. If the width had been three feet more, and the length three feet less, the room would have been square: find its dimensions. 5. Extract the square root of x* +16. 6. Simplify the following expressions: (a) 21/175 31/63 + 51/28; (b) x- y +¶ × x2-4a × (x2)¶−2r ÷ x4p—8r; 2T (c) x3p a yx; 7. Transform 31/5 +51/3 into an equivalent expres sion having a rational denominator. 2X bxx ax 3. Solve the simultaneous equations, x + 2y + 32 == 17, 3y+ z = 0, 3x + y — 5% = 15. 4. A. has $15 more than B, B has $5 less than C, and they have $65 between them. How much has each? 5. Expand (2x-2y)' by the binomial formula. 6. Simplify the following expressions: JUNE 1896. FROM QUADRATICS. 1. What relation do the roots of the equation x2 + px + q = o, bear to the coefficients p and q? What will be the value of ૧ if the roots are reciprocals of each othei? 2. Solve the equations: (a) x' — 3x2 = 88, 3. Solve the simultaneous equations x3 — y3 = 98, x y = 2. 4. (a) Deduce the formula for the sum of the first n terms in a geometric progression. (b) Find the limit of the sum of an infinite number of terms in the series 9 -6+4 2x2 etc. into a series of ascending powers of x, by the method of indeterminate coefficients. (b) What will be the nth term of this series? 6. Prove the following: (a) log mn log m+ log n; (b) log m= log m == ; (c) loga I = 0. r 7. (a) Give the numerical value of the sum loga a log10 .001 + 2/3 log, 8. (b) Show that log10 5 = I - log10 2. SEPTEMBER 1896. FROM QUADRATICS. 1. (a) For what value of c are the roots of the equation 3x2+4x+co, equal? (b) Construct the equation whose roots are 4. (a) Deduce the formula for the number of combinations of m different things taken n at a time. (b) In how many different ways may the letters in the word Yale be arranged? 5. If a b c d, prove that computation. (b) What is the base of a system of logarithms in which the logarithm of 81 is 4 ? 2x 7. Given log10 2 0.301, find x from the equation 100. |