Moderate -0.3 This is a straightforward logarithmic transformation question requiring students to recognize that ln(ay) = ln(a) + x·ln(b) gives a linear relationship, then use two points to find the gradient (ln b) and intercept (ln a). The algebra is routine and the question explicitly tells students the relationship is linear, removing any problem-solving element. Slightly easier than average due to its mechanical nature, though the exponential context and need to work backwards from ln values provides minor challenge.
\includegraphics{figure_6}
The variables \(x\) and \(y\) satisfy the equation \(ay = b^x\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((0.50, 2.24)\) and \((3.40, 8.27)\), as shown in the diagram.
Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure. [4]
Carry out a correct method for finding lna or lnb or a or b
DM1
Condone sign slip.
Obtain value a = 0.3
A1
(0.30109…)
Obtain value b = 8
A1
(7.99895…)
Allow A0A1 if both values ‘correct’ but not
rounded to 1 sf.
Allow 4/4 for 0.3y=8x with correct working
shown.
Answer
Marks
Guidance
Question
Answer
Marks
6
Alternative Method for Question 6:
Carry out a correct method for finding lnb or b
Answer
Marks
Guidance
(Need to link the gradient to lnb at some point)
*M1
Condone sign slips but must be using the given
coordinates correctly.
8.27−2.24
lnb= (=2.079....)
3.4−0.5
Answer
Marks
Guidance
Obtain value b = 8
A1
Correct method to find lnaor a
DM1
(lna=−1.200...).
Condone sign slip
Answer
Marks
Guidance
Obtain value a = 0.3
A1
Allow A0A1 if both values ‘correct’ but not
rounded to 1 sf.
Allow 4/4 for 0.3y=8x with correct working
shown.
4
Answer
Marks
Guidance
Question
Answer
Marks
Question 6:
6 | Form a pair of equations in a and b | *M1 | Condone sign slips but must be using the given
coordinates correctly.
lna+8.27=3.4lnb
e.g.
lna+2.24=0.5lnb
ae2.24 =b0.5
or
ae8.27 =b3.4
Carry out a correct method for finding lna or lnb or a or b | DM1 | Condone sign slip.
Obtain value a = 0.3 | A1 | (0.30109…)
Obtain value b = 8 | A1 | (7.99895…)
Allow A0A1 if both values ‘correct’ but not
rounded to 1 sf.
Allow 4/4 for 0.3y=8x with correct working
shown.
Question | Answer | Marks | Guidance
6 | Alternative Method for Question 6:
Carry out a correct method for finding lnb or b
(Need to link the gradient to lnb at some point) | *M1 | Condone sign slips but must be using the given
coordinates correctly.
8.27−2.24
lnb= (=2.079....)
3.4−0.5
Obtain value b = 8 | A1
Correct method to find lnaor a | DM1 | (lna=−1.200...).
Condone sign slip
Obtain value a = 0.3 | A1 | Allow A0A1 if both values ‘correct’ but not
rounded to 1 sf.
Allow 4/4 for 0.3y=8x with correct working
shown.
4
Question | Answer | Marks | Guidance
\includegraphics{figure_6}
The variables $x$ and $y$ satisfy the equation $ay = b^x$, where $a$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(0.50, 2.24)$ and $(3.40, 8.27)$, as shown in the diagram.
Find the values of $a$ and $b$. Give each value correct to 1 significant figure. [4]
\hfill \mbox{\textit{CAIE P3 2024 Q6 [4]}}