9 A curve has equation \(y = 3 \times 12 ^ { x }\).
- The point ( \(k , 6\) ) lies on the curve \(y = 3 \times 12 ^ { x }\). Use logarithms to find the value of \(k\), giving your answer to three significant figures.
- Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x\), giving your answer to two significant figures.
- The curve \(y = 3 \times 12 ^ { x }\) is translated by the vector \(\left[ \begin{array} { l } 1
p \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the curve \(y = \mathrm { f } ( x )\) passes through the origin ( 0,0 ), find the value of the constant \(p\). - The curve with equation \(y = 2 ^ { 2 - x }\) intersects the curve \(y = 3 \times 12 ^ { x }\) at the point \(T\). Show that the \(x\)-coordinate of \(T\) can be written in the form \(\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }\), where \(q\) is an integer. State the value of \(q\).
[0pt]
[5 marks]
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