(a) Without using mathematical tables or calculator, evaluate \(\frac{\frac{3}{2}\log 27 - 3\log 5\sqrt{5}}{\log 0.6}\) (b) Two linear transformations A and...
Question 1 Report
(a) Without using mathematical tables or calculator, evaluate \(\frac{\frac{3}{2}\log 27 - 3\log 5\sqrt{5}}{\log 0.6}\)
(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.
Answer Details
(a) Work on the numerator:
\[\tfrac{3}{2}\log 27 = \tfrac{3}{2}\log 3^3 = \tfrac{9}{2}\log 3,\qquad 3\log 5\sqrt{5} = 3\log 5^{3/2} = \tfrac{9}{2}\log 5\] \[\text{Numerator} = \tfrac{9}{2}\log 3 - \tfrac{9}{2}\log 5 = \tfrac{9}{2}\log\tfrac{3}{5} = \tfrac{9}{2}\log 0.6\]Dividing by \(\log 0.6\):
\[\frac{\tfrac{9}{2}\log 0.6}{\log 0.6} = \frac{9}{2} = 4.5\](b)(i) The matrices are:
\[A = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix},\qquad B = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}\](ii) Apply \(A\) to \(P(-2, 2)\):
\[A(P) = (\,-2 + 2(2),\ -(-2) + 2\,) = (2,\ 4)\]Then apply \(B\) to \((2, 4)\):
\[B(2,4) = (2(2) + 3(4),\ 2 + 2(4)) = (16,\ 10)\]The image of \(P\) under \(A\) followed by \(B\) is \((16, 10)\).
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