Moderate -0.8 Part (a) is a routine sketch of an exponential function requiring only basic knowledge that 3^0=1 and the general shape. Part (b) is a standard substitution exercise (let u=3^x to get a quadratic) followed by taking logarithms—a textbook C2 technique with no novel problem-solving required. This is easier than average A-level content.
8. (a) Sketch the graph of
$$y = 3 ^ { x } , \quad x \in \mathbb { R }$$
showing the coordinates of any points at which the graph crosses the axes.
(b) Use algebra to solve the equation
$$3 ^ { 2 x } - 9 \left( 3 ^ { x } \right) + 18 = 0$$
giving your answers to 2 decimal places where appropriate.
All three criteria correct: (1) correct shape for \(x \geq 0\), at least touches positive \(y\)-axis; (2) correct shape for \(x < 0\), must not touch \(x\)-axis or have turning points; (3) \((0,1)\) stated or marked on \(y\)-axis
Forms a quadratic of correct form in \(3^x\) or in "\(y\)" where "\(y\)" \(= 3^x\)
\(y = 6,\ y = 3\) or \(3^x = 6,\ 3^x = 3\)
A1
Both \(y=6\) and \(y=3\)
\(3^x = 6 \Rightarrow x\log 3 = \log 6\) or \(x = \frac{\log 6}{\log 3}\) or \(x = \log_3 6\)
dM1
A valid method for solving \(3^x = k\) where \(k > 0, k \neq 1, k \neq 3\)
\(x = 1.63092...\)
A1cso
awrt 1.63
\(x = 1\)
B1
\(x=1\) stated as solution from any working
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph of $y = 3^x$ | B1 | At least two of three criteria correct |
| | B1 | All three criteria correct: (1) correct shape for $x \geq 0$, at least touches positive $y$-axis; (2) correct shape for $x < 0$, must not touch $x$-axis or have turning points; (3) $(0,1)$ stated or marked on $y$-axis |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(3^x)^2 - 9(3^x) + 18 = 0$ or $y^2 - 9y + 18 = 0$ | M1 | Forms a quadratic of correct form in $3^x$ or in "$y$" where "$y$" $= 3^x$ |
| $y = 6,\ y = 3$ or $3^x = 6,\ 3^x = 3$ | A1 | Both $y=6$ and $y=3$ |
| $3^x = 6 \Rightarrow x\log 3 = \log 6$ or $x = \frac{\log 6}{\log 3}$ or $x = \log_3 6$ | dM1 | A valid method for solving $3^x = k$ where $k > 0, k \neq 1, k \neq 3$ |
| $x = 1.63092...$ | A1cso | awrt 1.63 |
| $x = 1$ | B1 | $x=1$ stated as solution from any working |
---
8. (a) Sketch the graph of
$$y = 3 ^ { x } , \quad x \in \mathbb { R }$$
showing the coordinates of any points at which the graph crosses the axes.\\
(b) Use algebra to solve the equation
$$3 ^ { 2 x } - 9 \left( 3 ^ { x } \right) + 18 = 0$$
giving your answers to 2 decimal places where appropriate.
\hfill \mbox{\textit{Edexcel C2 2014 Q8 [7]}}