| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Express in terms of substitution |
| Difficulty | Easy -1.2 This is a straightforward indices manipulation question requiring only basic index laws (multiplication, division, and power rules) and substitution. Each part involves routine algebraic manipulation with no problem-solving insight needed—students simply apply memorized rules to rewrite expressions in terms of y. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.06a Exponential function: a^x and e^x graphs and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) \(3^{3x} = (3^x)^3 = y^3\) | B1 | Condone \((y)^3\). Ignore once correct answer seen. |
| (b) \(\frac{1}{3^{x-2}} = \frac{1}{3^x \times 3^{-2}} = \frac{9}{y}\) | M1 A1 | M1: Correct application of addition/subtraction law, e.g. \(\frac{1}{3^x \times 3^{-2}}\) or \(\frac{1}{3^x \div 3^2}\) or sight of 9 or \(\frac{1}{y}\). A1: \(\frac{9}{y}\) or \(9y^{-1}\) but NOT expressions still containing \(\div\) or fractions within fractions or \(3^2y^{-1}\). |
| (c) \(\frac{81}{9^{2-3x}} = \frac{9^2}{9^{2-3x}} = 9^{2-(2-3x)} = 9^{3x} = 3^{6x} = y^6\) | M1 A1 | M1: Simplifying indices to form \(9^{...}\) or \(3^{...}\) (not as denominator), e.g. \(9^{3x}\), \((9^x)^3\), \(3^{6x}\), \((3^x)^6\), or \(k \times y^6\), \(k \neq 0\). Also allow unsimplified equivalents e.g. \(\frac{1}{y^{-6}}\). A1: \(y^6\) only (not \(\frac{1}{y^{-6}}\)). Condone \((y)^6\). |
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** $3^{3x} = (3^x)^3 = y^3$ | B1 | Condone $(y)^3$. Ignore once correct answer seen. |
| **(b)** $\frac{1}{3^{x-2}} = \frac{1}{3^x \times 3^{-2}} = \frac{9}{y}$ | M1 A1 | M1: Correct application of addition/subtraction law, e.g. $\frac{1}{3^x \times 3^{-2}}$ or $\frac{1}{3^x \div 3^2}$ or sight of 9 or $\frac{1}{y}$. A1: $\frac{9}{y}$ or $9y^{-1}$ but NOT expressions still containing $\div$ or fractions within fractions or $3^2y^{-1}$. |
| **(c)** $\frac{81}{9^{2-3x}} = \frac{9^2}{9^{2-3x}} = 9^{2-(2-3x)} = 9^{3x} = 3^{6x} = y^6$ | M1 A1 | M1: Simplifying indices to form $9^{...}$ or $3^{...}$ (not as denominator), e.g. $9^{3x}$, $(9^x)^3$, $3^{6x}$, $(3^x)^6$, or $k \times y^6$, $k \neq 0$. Also allow unsimplified equivalents e.g. $\frac{1}{y^{-6}}$. A1: $y^6$ only (not $\frac{1}{y^{-6}}$). Condone $(y)^6$. |
---2. Given $y = 3 ^ { x }$, express each of the following in terms of $y$. Write each expression in its simplest form.
\begin{enumerate}[label=(\alph*)]
\item $3 ^ { 3 x }$
\item $\frac { 1 } { 3 ^ { x - 2 } }$
\item $\frac { 81 } { 9 ^ { 2 - 3 x } }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2020 Q2 [5]}}