Standard +0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (using substitution for the right side and a standard trigonometric identity for the left), then apply initial conditions. While it involves multiple steps and careful algebraic manipulation, it follows a completely standard pattern with no novel insight required, making it slightly easier than average.
Separate variables correctly and attempt integration of at least one side
B1
Obtain term in the form \(a\sqrt{(2x + 1)}\)
M1
Express \(1/(\cos^2 \theta)\) as \(\sec^2 \theta\)
B1
Obtain term of the form \(k \tan \theta\)
M1
Evaluate a constant, or use limits \(x = 0, \theta = \frac{1}{4}\pi\) in a solution with terms \(a\sqrt{(2x+1)}\) and \(k \tan \theta\), \(ak \neq 0\)
M1
Obtain correct solution in any form, e.g. \(\sqrt{(2x+1)} = \frac{1}{3}\tan \theta + \frac{1}{3}\)
A1
Rearrange and obtain \(x = \frac{1}{4}(\tan \theta + 1)^2 - \frac{1}{2}\), or equivalent
A1
7 marks
Separate variables correctly and attempt integration of at least one side | B1 |
Obtain term in the form $a\sqrt{(2x + 1)}$ | M1 |
Express $1/(\cos^2 \theta)$ as $\sec^2 \theta$ | B1 |
Obtain term of the form $k \tan \theta$ | M1 |
Evaluate a constant, or use limits $x = 0, \theta = \frac{1}{4}\pi$ in a solution with terms $a\sqrt{(2x+1)}$ and $k \tan \theta$, $ak \neq 0$ | M1 |
Obtain correct solution in any form, e.g. $\sqrt{(2x+1)} = \frac{1}{3}\tan \theta + \frac{1}{3}$ | A1 |
Rearrange and obtain $x = \frac{1}{4}(\tan \theta + 1)^2 - \frac{1}{2}$, or equivalent | A1 | 7 marks