• Two linear transformations are defined by

P:(x,y)  (-2 + 3y, 5x – 4y)
T:(x,y) (-3x – 4y, 6x – 5y)

          Find the

• inverse of T;
• image of (1,3) under the transformation P.

• Find the volume, in terms of p, of the solid formed when the area enclosed by the lines y = 1, 2y = x + 1 and x = 0 is rotated through two right angles about the y – axis. 

Very few candidates reportedly attempted this question and they performed poorly.  The matrix of transformation T =  .
    =   15 + 24  =  39.   T-1  =      .  Therefore, required inverse T-1 (x, y)  =            = 
 
(ii)    P =    .  Therefore the image of (1,3) under P =      =     =  (7, -7).
         
For part (b), since the volume generated was half of the total volume, the formular becomes V = . 2y = x + 1. Hence x = 2y – 1. When x = 0, y = ½ and when y = 1, x = 1. Thus, V =  =        =  cubic units.