# Question 1 (part a(i)):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Exponential (growth) shaped curve in any position | B1 | Tolerant on slips of pen at either end |
| Intersections at $\left(\ln\left(\frac{5}{2}\right), 0\right)$ and $(0, -3)$ | B1 | Allow $\ln\left(\frac{5}{2}\right)$ and $-3$ marked on correct axes; condone $(0, \ln\left(\frac{5}{2}\right))$ and $(-3,0)$; do not allow $(0.92, 0)$ unless seen elsewhere |
| Asymptote equation $y = -5$ | B1 | Curve must appear to have asymptote at $y=-5$; having $-5$ marked on axis alone insufficient; extra asymptote with equation scores B0 |

# Question 1 (part a(ii)):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape or reflection of (a)(i) curve in $x$-axis, appearing both above and below $x$-axis with correct cusp curvature | B1ft | Must appear both above and below $x$-axis |
| Both intersections from (a)(i) followed through, including decimals | B1ft | Curve must appear both above and below $x$-axis |
| Asymptote $y=5$ or follow-through on $y=-C$ from (a)(i) | B1ft | Curve must appear to have asymptote at $y=C$ |

# Question 1 (part b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \ln\left(\frac{5}{2}\right)$ or follow-through on $x$-intersection from part (a) | B1ft | Accept awrt 0.92 |

# Question 1 (part c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $2e^x - 5 = -2$ or $-2e^x + 5 = 2 \Rightarrow x = \ln(\ldots)$ | M1 | Allow squaring: $(2e^x-5)^2=4 \Rightarrow e^x=\ldots \Rightarrow x=\ln(\ldots), \ln(\ldots)$ |
| $x = \ln\left(\frac{3}{2}\right)$ or exact equivalent e.g. $x=\ln 1.5$ | A1 | Decimal answer awrt 0.405 |
| $x = \ln\left(\frac{7}{2}\right)$ or exact equivalent e.g. $x=\ln 3.5$ | B1 | Decimal answer awrt 1.25 |
| If both answers given in decimals only with no working, $x \approx 1.25, 0.405$ | SC 100 | |

---