8. (_a_) Deduce a test for finding when the roots of the equation _ax_^2 + _bx_ + _c_ = 0 are: 1º real and unequal; 2º real and equal; 3º imaginary; 4º numerically equal with contrary signs. (_b_) Apply the tests to find the nature of the roots of the equations 1º 3_x_^2 + 4_x_ - 10 = 0 2º 5_x_^2 + 6 = 0 9. Given a square whose side is 2. The middle points of its adjacent sides are joined by straight lines forming a second square inscribed in the first. In the same manner, a third square is inscribed in the second, a fourth in the third, and so on indefinitely. Find the sum of the perimeters of all the squares. Substitute for any of the above.--A person has $6,500, which he divides into two portions and lends at different rates of interest, so that the two portions produce equal returns. If the first portion had been lent at the second rate of interest, it would have produced $180; and if the second portion had been lent at the first rate of interest, it would have produced $245. Find the rates of interest. * * * * * =Plane Geometry.=--Candidates will be required to give accurate definitions of the terms used in _plane geometry_, to demonstrate any proposition of plane geometry as given in the ordinary text-books and to solve simple geometrical problems either by a construction or by an application of algebra. The following questions were used at a recent examination: 1. Theorem: The three medians of any triangle intersect in a common point which is at two-thirds of the distance from each vertex to the middle of the opposite side. 2. Theorem: If two triangles have their three sides respectively equal, the triangles are equal in all respects. 3. (_a_) How many circles can be drawn tangent to three given straight lines? (_b_) Problem: To draw a circle through a given point and tangent to two given straight lines. 4. Theorem: If two parallel right lines be divided into corresponding parts, proportional each to each, and straight lines be drawn through the corresponding points of division, these straight lines will pass through a common point. 5. Exercise: Find the locus of all points, the sum of the squares of the distances of any one of which from two fixed points is equal to a given square. 6. Problem: Given two circles, to construct a third circle equivalent to their difference. 7. Exercise: If the radius of a circle is 5, find the area of the segment subtended by the side of a regular hexagon. 8. Theorem: The areas of two triangles which have an angle of the one equal to an angle of the other, are to each other as the products of the sides including those angles. 9. Problem: Through a given point on one side of a triangle to draw a right line which shall divide the triangle into two equivalent areas.