I. If a father was four times as old as his son seven years ago, and will be twice as old as his son in seven years more, what is the present age of each? 5. Simplify the following expressions : SEPTEMBER 1897. TO QUADRATICS. 1. Resolve into simplest factors: I (a) x3 — y2; (b) 8x′ + —;; (c) 110 — x − x'; I X = + a b' a b a+b' 4. Find the fraction such that if I be added to the numerator it becomes ; but if I be added to the denominator it becomes 4. JUNE 1897. FROM QUADRATICS. 1. (a) Determine by inspection the sum and also the product of the roots of the equation, 3x2 and state the theorem used. 5x + I 0, (b) What must be the value of b, in order that the roots of the equation 2x2+ bx + 1% = 0, shall be equal? X 2. Solve the equation = a'b(x + c) x + c 3. Solve the simultaneous equations, x2 + 3xy = 28, xy + 4y2 = 8; and associate the values of x and y. 4. (a) Find the sum of n terms of the series (b) Find the value of the repeating decimal 0.43232 +. How 5. A club consists of 8 Seniors and 6 Juniors. many different committees of five may be selected from this club, each committee to consist of 3 Seniors and 2 Juniors? 4. Prove the following: (a) log (m ÷ n) = log m log n; (b) log m= ; (c) logao = a > I. log m r 7. (a) Find the numerical value of 1⁄2 log: 9 loga a. (b) Solve amx þux c, for x. ∞, when 2 log, 3 SEPTEMBER 1897. FROM QUADRATICS. 1. (a) Solve the equation ax + bx + c = 0. (b) From this result find tests for determining the character of the roots. 2. Solve the equations: (a) 3x3 + x! +x 7x18 24. 3. Solve the simultaneous equations : 30; (b) x2 7x+1/x2 ay of x by the method of undetermined coefficients. terms. Get five 5. Write down formulae for the number of permutations of m different things taken (a) u at a time, (b) taken all at a time. (c) How many different sums of money can be formed with a cent piece, a nickle, a dime, a quarter, and a half dollar? 6. (a) Define loga m, where m and a are numbers. (b) Prove loga o∞, when a < 1. (c) Find the numerical value of x from 2 = 128. 7. (a) What is the logarithm of % in a system whose base is 2? SHEFFIELD SCIENTIFIC SCHOOL. TRIGONOMETRY. JUNE 1885. 1. Describe the different ways of measuring angles in Trigonometry. How many degrees are there in the angle π ? 2. Define the sine, tangent, and secant of an angle and represent each by a straight line. 3. How many angles are there having the same function, sine, tangent, etc.? Name five angles having the cosine 2. 5. Given tan 9 = (m÷ n) to find sin 4 and cos 9. 6. Show that cos 29 = cos3 sin = 1 2 sin' ; also sin 1⁄2 = √ 1⁄2 ( 1 cos ). 120° 10', the side a, 7. In a triangle ABC, given A (opposite A) = 4256.6 ft., the side b (opposite B) = 2267.8 ft., to find the remaining parts of the triangle. more than one solution possible with these data? Is there |