Standard +0.3 This is a straightforward numerical methods question requiring students to rearrange a logarithmic equation into f(x)=0 form, identify a sign change interval, and apply an iterative method (likely interval bisection or Newton-Raphson). While it involves logarithms, the solution process is standard and mechanical with no conceptual challenges beyond basic A-level technique, making it slightly easier than average.
Use law for the logarithm of a power, a quotient, or a product correctly at least once
M1
Use \(\ln e = 1\) or \(e = \exp(1)\)
M1
Obtain a correct equation free of logarithms, e.g. \(1 + x^2 = e^{x^2}\)
A1
Solve and obtain answer \(x = 0.763\) only
A1
[4 marks total]
[For the solution \(x = 0.763\) with no relevant working give B1, and a further B1 if 0.763 is shown to be the only root.]
[Treat the use of logarithms to base 10 with answer 0.333 only, as a misread.]
[SR: Allow iteration, giving B1 for an appropriate formula, e.g. \(x_{n+1} = \exp(\ln(1 + x_n^2) - 1)/2)\), M1 for using it correctly once, A1 for 0.763, and A1 for showing the equation has no other root but 0.763.]
Use law for the logarithm of a power, a quotient, or a product correctly at least once | M1 |
Use $\ln e = 1$ or $e = \exp(1)$ | M1 |
Obtain a correct equation free of logarithms, e.g. $1 + x^2 = e^{x^2}$ | A1 |
Solve and obtain answer $x = 0.763$ only | A1 |
[4 marks total]
[For the solution $x = 0.763$ with no relevant working give B1, and a further B1 if 0.763 is shown to be the only root.]
[Treat the use of logarithms to base 10 with answer 0.333 only, as a misread.]
[SR: Allow iteration, giving B1 for an appropriate formula, e.g. $x_{n+1} = \exp(\ln(1 + x_n^2) - 1)/2)$, M1 for using it correctly once, A1 for 0.763, and A1 for showing the equation has no other root but 0.763.]