7. In the quadratic ax+be+c=0 deduce the relation amongst a be that the roots may be 8. In a set of simultaneous equations what is the result of having (i) more equations than unknowns (ii) less 66 66 Given 2x-y+3=3x+2y-5=x+4y+m=0 to find m. 9. A rectangular plot with sides a and b is bordered within with a walk of uniform width. How wide is the walk when it occupies one half the plot? 10. Prove that generally an expression giving the sum of an A.P. involves both n and n2; and state the particular cases in which it may not involve one or the other of these. Find the sum of n terms of an A. P. whose nth n 1 term is 2 How many terms will make 1580 ? 11. Explain the difference between a "permutation" and a "combination." Prove that nPr=nCrXr! How many different guards of 5 men could be made from 12 men; and how often relatively will any one man be on guard? 12. Write out the expansion of (1+x)". When x=1 what conclusion follows with respect to all the combinations that can be made from n articles ? 13. What is the characteristic of a logarithm? Show how to find it. If log. 20-3010, log. 6=0.7782. Find log. 15, log., log. (0·0083), log. (7·2). 1. ABC is an isosceles triangle and CD cuts the base AB in D. Prove that CA2 CD2 = AD.DB, and show in what sense this relation is universally true. 2. Divide a line into two parts so that the rectangle contained by the parts may be equal to the differences of the squares upon the parts. 3. If a chord be drawn from the point of contact of a tangent line to a circle the angles between the chord and line are equal to those contained in the alternate segments into which the chord divides the circle. Tangents are drawn at the vertices B and C of a triangle inscribed in a circle. Show that the angle between these tangents is π-24. 4. The angle of intersection of two secants to a circle is measured by half the sum of the intercepted arcs or half their difference, according as the secants intersect within the circle or without it. In a given circle an inscribed equilateral triangle has its vertices A, B, C and an inscribed regular pentagon has vertices A, D, E, F, G. Express in radians the angle between the lines AC and DF; between AC and GF. 5. Two triangles having two sides in the one proportional to two sides in the other, and the angles contained by these sides in each equal, are similar. Describe the "line of lines" in the sector and its uses. 6. If two triangles have an equal angle their areas are proportional to those of the rectangles under the sides containing the equal angle in each. 7. (a) Express each trigonometric function in terms of the sine. (b) Find when 2 sin2 + 1/2 cos 02. 8. Obtain the formula a2 = b2 + c2 −2bc cos 0. The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the four sides. 9. Define a plane-a ruled surface. The intersection of two planes is a straight line. 10. AA', BB, CC, DD are the diagonals of a cube whose edge is e. (b). Find the length of the perpendicular from BX upon AA', and the perpendicular distance of X from one of the faces of the cube. |