SEPTEMBER 1898. SOLID AND SPHERICAL. 1. Find the locus of all points in space equi-distant from two given points, 2. The angle a line makes with its projection on a plane is the least angle it makes with any line drawn in the plane through its intersection with the plane. 3. A pyramid is formed by cutting a given pyramid by a plane parallel to its base; show that the volumes of the two pyramids have the same ratio as the cubes of their altitudes. 4. The greater angle of a spherical triangle lies opposite the greater side. 5. If the surface of a sphere is four times that of a second sphere, what is the ratio of the volumes of the spheres? TRIGONOMETRY JUNE 1898. 1. Express an angle of 25° in radians. 2. State five fundimental relations between two or more of the six simple trigonometric functions. 3. Given sin p = m, to find expressions for cos 9, tau 4, cot P, sec , cosec P. 5. Transform the first member of the following formula into the second: 1. Find (a) the logarithm of 0.02564. (b) the number whose logarithm is 9.7100 - 10. 2. Calculate the value of x = 3 0.152 X 0.025 3. Find (a) the log cos of 65° 16′.2. 25 X 0.035 (b) the smallest angle whose logarithmic tangent is 9.4092 IO. SEPTEMBER 1898. 1. Express an angle of 2.81 radians in degrees. 2. Give the simplest equivalents for sin (x-7); cos (x — 1⁄2); tan (x — 11⁄2 7); cot (27 x). 3. Derive the formula - cos p+cos q = 2 cos 1⁄2 (p + q) cos 1⁄2 (p − q). 2 tan a 4. Derive the formula, tan a = I tan' 2α 5. Transform the first member of the following formula 1+tan' into the second: I tan' q = sec 29. 6. Knowing that A and B are two angles of a triangle and a and b the sides respectively opposite them, derive 1. Find (a) the logarithm of 0.61675. (b) the number whose logarithm is 8.83017 10. 2. Given s 1⁄2 (a+b+c), a = 2567.4, b = 2546.2, c = 2345.6, to calculate 1⁄2 A from the formula ALGEBRA JUNE 1898. TO QUADRATICS. 1. Resolve into simplest factors, (a) 81x — x2 Χ · 30; (c) 2x2 + 5x 2. Solve the equation 12; (d) ac - y; (b) bd ad + bc. 3. Solve the simultaneous equations x + 2y + 2z 2x + y + z = 7, 3x + 4y + z = 14. II, 4. Divide 91 into two such parts that the quotient of the greater part divided by the difference between the parts may be 7. 5. Expand (I 2x) by the binomial formula. 6. Simplify the following expressions: 3. Solve the simultaneous equations, x + y = 2a, and (ab)x = (a + b)y. 4. The combined wages of 6 carpenters and 2 painters for one day are $28. If for another day's work the same sum is paid to 5 carpenters and 4 painters, what are a day's wages of each? 5. Extract the square root of 4x′ — 8xʼy3 + 4xy° + y^. 6. Simplify the following expressions: |