(b) Simplify (5×√(x4/3)-3/2
6. Two trains run toward each other from A and B, respectively, and meet at a point which is 15 miles farther from A than it is from B. After the trains meet, it takes the first train 2⅔ hours to run to B, and the second 3⅜ hours to run to A. How far is it from A to B?
| 7. Solve | { (1/a + 1/b)x + (1/a - 1/b)y = 4 |
| { | |
| { x/(a + b) + y/(a - b) = 2 |
8. (a) Deduce a test for finding when the roots of the equation ax2 + 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 3x2 + 4x - 10 = 0 |
| 2º 5x2 + 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.