June 2022 Paper 2 Q3
3
(a) On the axes in the Printed Answer Booklet, sketch the curve with equation \(y = 3 \times 0.4^x\). [3]
(b) Given that \(3 \times 0.4^x = 0.8\), determine the value of \(x\) correct to 3 significant figures. [3]
| Scheme | Marks | AO |
|---|---|---|
 | M1 B1 A1 | 1.1 1.1 1.1 |
| [3] |
Notes
M1: decreasing concave up curve in 1st and 2nd quadrants which does not cut the \(x\)-axis; mark intent
B1: decreasing curve with intercept \((0, 3)\); may be in one quadrant only
A1: smooth curve from \((-0.5, a)\) through \((2.5, b)\), where \(4.5 \leqslant a \leqslant 5\) and \(0 \lt b \lt 0.5\)
| Scheme | Marks | AO |
|---|---|---|
| \(\log(3 \times 0.4^x) = \log(0.8)\) oe | M1 | 3.1a |
| \(x\log 0.4 = \log 0.8 - \log 3\) oe | M1 | 1.1 |
| 1.44 cao | A1 | 1.1 |
| [3] |
Notes
M1: taking logarithms in any base
M1: 3rd law of logs used correctly
A1: if M0M0 allow SC1 for 1.44 unsupported
Alternatively
| Scheme | Marks |
|---|---|
| \(0.4^x = \dfrac{0.8}{3}\) | M1 |
| \(x = \log_{0.4}\left(\dfrac{0.8}{3}\right)\) | M1 |
| \(x = 1.44\) cao | A1 |
M1: may see \(x\log 0.4 = \log\left(\frac{0.8}{3}\right)\) oe