Linear relationship between log variables

3. \begin{figure}[h] \end{figure} Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(\log _ { 10 } x\) The line passes through the points \(( 0,4 )\) and \(( 6,0 )\) as shown.

1. Find an equation linking \(\log _ { 10 } y\) with \(\log _ { 10 } x\)
2. Hence, or otherwise, express \(y\) in the form \(p x ^ { q }\), where \(p\) and \(q\) are constants to be found.

2. \begin{figure}[h] \end{figure} Figure 1 shows the linear relationship between \(\log _ { 6 } T\) and \(\log _ { 6 } x\) The line passes through the points \(( 0,4 )\) and \(( 2,0 )\) as shown.

1. 1. Find an equation linking \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
   2. Hence find the exact value of \(T\) when \(x = 216\)
2. Find an equation, not involving logs, linking \(T\) with \(x\)

1. (i) The variables \(x\) and \(y\) are connected by the equation

$$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$ Sketch the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\) Show on your sketch the coordinates of the points of intersection of the graph with the axes.
(ii) \begin{figure}[h] \end{figure} Figure 2 shows the linear relationship between \(\log _ { 3 } N\) and \(t\).
Show that \(N = a b ^ { t }\) where \(a\) and \(b\) are constants to be found.

\(\mathbf { 1 }\) & \(\mathbf { 1 }\) \hline \end{tabular} \end{center} Two quantities are related by the equation \(Q = 1 \cdot 25 P ^ { 3 }\). Explain why the graph of \(\log _ { 10 } Q\) against \(\log _ { 10 } P\) is a straight line. State the gradient of the straight line and the intercept on the \(\log _ { 10 } Q\) axis of the graph. In the binomial expansion of \(( 2 - 5 x ) ^ { 8 }\), find a) the number of terms,
b) the \(4 ^ { \text {th } }\) term, when the expansion is in ascending powers of \(x\),
c) the greatest positive coefficient. A curve \(C\) has equation \(y = \frac { 1 } { 9 } x ^ { 3 } - k x + 5\). A point \(Q\) lies on \(C\) and is such that the tangent to \(C\) at \(Q\) has gradient - 9 . The \(x\)-coordinate of \(Q\) is 3 . a) Show that \(k = 12\).
b) Find the coordinates of each of the stationary points of \(C\) and determine their nature.
c) Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis. The diagram below shows a triangle \(A B C\) with \(A C = 5 \mathrm {~cm} , A B = x \mathrm {~cm} , B C = y \mathrm {~cm}\) and angle \(B A C = 120 ^ { \circ }\). The area of the triangle \(A B C\) is \(14 \mathrm {~cm} ^ { 2 }\). Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places. Prove that \(f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 13 x - 7\) is an increasing function. The diagram below shows a curve with equation \(y = ( x + 2 ) ( x - 2 ) ( x + 1 )\). Calculate the total area of the two shaded regions. \section*{END OF PAPER}