3. The sum of two numbers is 200 and their difference is equal to % of the less. Find the numbers. 4. Simplify, and find to two decimal places the value of 21/3+ 151/27/13 JUNE 1897. (b) 1. Solve the equation 1 + 1/8x3 3x2 = 2x. 2. Solve x'y' x'y' 29:21, x + y = 7. : 3. Find the seventh term, and the sum of the first seven terms of the series 4. How many different numbers of three figures each can be formed from the digits 1, 5, 6, 8, 9? 5. Expand by the binomial theorem to three terms, (1 — 2x2)4. 6. Change into a continued fraction and find the 59 255 first three convergents. SUPPLEMENT FOR 1898. SHEFFIELD SCIENTIFIC SCHOOL. GEOMETRY JUNE 1898. PLANE. I. (a) Two lines perpendicular to the same line are parallel. (b) If two lines are parallel, and one of them is perpendicular to a third line, then the second line is also perpendicular to the third line. 2. (a) Of two unequal chords in a circle, the lesser chord is farther from the center. (b) of two chords of a circle unequally distant from the center, the nearer is the greater. 3. Define similar polygons, similar sectors, similar segments. If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. 4. The areas of similar segments have the same ratio as the squares of the radii. 5. Show that the apothegm of an equilateral triangle inscribed in a circle is equal to one-half the radius of the circle. SEPTEMBER 1898. PLANE. 1. Each vertex of a triangle is joined by a straight line to the middle point of the opposite side; show that these three lines meet in a point. 2. The bisector of an interior or exterior angle of a tri. 'angle divides the opposite side into segments proportional to the other two sides. 3. The area of a trapezoid equals the altitude times the line joining the middle points of the non-parallel sides. 4. (a) The circumferences of two circles have the same ratio as the radii.. (b) The areas of two circles have the same ratio as the squares of the radii. 5. The area included between the circumferences of two circles having the same center equals the area of a circle whose radius is the length of a tangent to the smaller circle drawn from any point on the circumference of the larger circle. JUNE 1898. SOLID AND SPHERICAL. 1. (a) Every plane passed through a line perpendicular to a plane, is perpendicular to that plane. (b) If through any point of the intersection of two perpendicular planes a line is drawn perpendicular to one of the planes, the line will lie in the other plane. 2. Any oblique prism is equivalent in volume to a right prism, whose base is a right section of the oblique prism and whose altitude is equal to a lateral edge of the oblique prism. 3. A lune is the same portion of the surface of the sphere on which it lies as the angle of the lune is of four right angles. 4. Find the volume ot a sphere circumscribing a cube whose edge is six inches. |