Moderate -0.3 This is a standard logarithmic linearization problem requiring students to take ln of both sides, identify the gradient and intercept from given points, then solve for constants. It involves routine algebraic manipulation and logarithm laws with straightforward arithmetic, making it slightly easier than average but still requiring multiple connected steps.
3
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The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { - x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0,1.3\) ) and ( \(1.6,0.9\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.
Form a numerical expression for the gradient of the line
M1
Obtain \(b = 1.28\)
A1
[5]
State or imply $\ln y = \ln A - x\ln b$ | B1 |
State $\ln A = 1.3$ | B1 |
Obtain $A = 3.67$ | B1 |
Form a numerical expression for the gradient of the line | M1 |
Obtain $b = 1.28$ | A1 | [5]
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The variables $x$ and $y$ satisfy the equation $y = A \left( b ^ { - x } \right)$, where $A$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points ( $0,1.3$ ) and ( $1.6,0.9$ ), as shown in the diagram. Find the values of $A$ and $b$, correct to 2 decimal places.
\hfill \mbox{\textit{CAIE P2 2008 Q3 [5]}}