(a) Without using mathematical tables or calculator, evaluate (frac{frac{3}{2}log 27 – 3log 5sqrt{5}}{log 0.6})
(b) Two linear transformations A and B in the (O_{xy}) plane, are defined by :
(A : (x, y) (x + 2y, -x + y))
(B : (x, y) (2x + 3y, x + 2y)).
(i) Write down the matrices A and B; (ii) Find the image of the point P(-2, 2) under the linear transformation A followed by B.
Explanation
(a) (frac{frac{3}{2} log 27 – 3 log 5sqrt{5}}{log 0.6})
= (frac{frac{3}{2} log 3^{3} – 3 log 5^{frac{3}{2}}}{log (frac{3}{5})})
= (frac{3}{2} times 3 log 3 – 3 times frac{3}{2} log 5}{log 3 – log 5})
= (frac{frac{9}{2}(log 3 – log 5)}{log 3 – log 5})
= (frac{9}{2} = 4frac{1}{2})
(b)(i) (A = begin{pmatrix} 1 & 2 \ -1 & 1 end{pmatrix}, B = begin{pmatrix} 2 & 3 \ 1 & 2 end{pmatrix})
(ii) Transformation of A followed by B is BA.
(BA = begin{pmatrix} 2 & 3 \ 1 & 2 end{pmatrix} begin{pmatrix} 1 & 2 \ -1 & 1 end{pmatrix})
= (begin{pmatrix} -1 & 7 \ -1 & 4 end{pmatrix})
Image of (-2, 2) is given by
(begin{pmatrix} -1 & 7 \ -1 & 4 end{pmatrix} begin{pmatrix} -2 \ 2 end{pmatrix})
= (begin{pmatrix} 16 \ 10 end{pmatrix})