| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | September |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Identify errors in student work |
| Difficulty | Moderate -0.3 This is an error-spotting question requiring students to identify algebraic mistakes (incorrectly expanding a square and wrong base conversion) and then solve a quadratic in log₃x. While it tests understanding of logarithm laws and quadratic factorization, the errors are fairly obvious and the solution method is standard substitution—slightly easier than a typical A-level question. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | E.g. \(\log x^2 = 2 \log x\); the student has ignored the brackets and used the power rule incorrectly. E.g. \(x = 3^2\); the student has done \(2^3\) | E1 |
| (i) | E1 |
Total: [2]
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | \((2\log x + 1)(\log x - 2) = 0\) | M1 |
| (ii) | \(\log x = -0.5\), \(\log x = 2\) | A1 |
| (ii) | \(x = 3^{0.5}\) or \(x = 3^2\) | M1 |
| (ii) | \(x = \frac{1}{3}\sqrt{3}\) and \(x = 9\) | A1 |
Total: [4]
(i) | E.g. $\log x^2 = 2 \log x$; the student has ignored the brackets and used the power rule incorrectly. E.g. $x = 3^2$; the student has done $2^3$ | E1 | Error identified with explanation
(i) | | E1 | Error identified with explanation
Total: [2]
(ii) | $(2\log x + 1)(\log x - 2) = 0$ | M1 | Attempt to solve quadratic in $\log x$ (soi)
(ii) | $\log x = -0.5$, $\log x = 2$ | A1 | Obtain two correct roots (BC)
(ii) | $x = 3^{0.5}$ or $x = 3^2$ | M1 | Attempt correct process to find $x$ at least once
(ii) | $x = \frac{1}{3}\sqrt{3}$ and $x = 9$ | A1 | Obtain both correct roots (Any equivalent exact form)
Total: [4]5 A student was asked to solve the equation $2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0$. The student's attempt is written out below.
$$\begin{aligned}
& 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\
& 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\
& \log _ { 3 } x - 2 = 0 \\
& \log _ { 3 } x = 2 \\
& x = 8
\end{aligned}$$
(i) Identify the two mistakes that the student has made.\\
(ii) Solve the equation $2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0$, giving your answers in an exact form.
\hfill \mbox{\textit{OCR H240/01 2018 Q5 [6]}}