Moderate -0.5 This is an error-spotting question requiring students to identify that dividing by sin x (Step 3) loses solutions where sin x = 0. While it tests understanding of algebraic manipulation and solution loss, it's a standard AS-level topic with a well-known pitfall, making it slightly easier than average but not trivial since students must articulate the error clearly.
Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0° \leq x \leq 360°\)
The steps of Jessica's working are shown below.
\(\tan x = \sin x\)
Step 1 \(\Rightarrow\) \(\frac{\sin x}{\cos x} = \sin x\) Write \(\tan x\) as \(\frac{\sin x}{\cos x}\)
Step 2 \(\Rightarrow\) \(\sin x = \sin x \cos x\) Multiply by \(\cos x\)
Step 3 \(\Rightarrow\) \(1 = \cos x\) Cancel \(\sin x\)
\(\Rightarrow\) \(x = 0°\) or \(360°\)
The teacher tells Jessica that she has not found all the solutions because of a mistake.
Explain why Jessica's method is not correct.
[2 marks]
Question 5:
5 | Demonstrates a clear
understanding that sin x = 0 is a
solution, and that this has not been
properly taken into account. | AO2.3 | R1 | sin x = 0 leads to a solution, but when she
cancelled sin x she effectively assumed it
was not equal to 0 and hence lost a
number of solutions.
Explains that cancelling sin x is not
allowed if it is zero / only allowed if
it is non-zero | AO2.4 | E1
Total | 2
Q | Marking Instructions | AO | Marks | Typical Solution
Jessica, a maths student, is asked by her teacher to solve the equation $\tan x = \sin x$, giving all solutions in the range $0° \leq x \leq 360°$
The steps of Jessica's working are shown below.
$\tan x = \sin x$
Step 1 $\Rightarrow$ $\frac{\sin x}{\cos x} = \sin x$ Write $\tan x$ as $\frac{\sin x}{\cos x}$
Step 2 $\Rightarrow$ $\sin x = \sin x \cos x$ Multiply by $\cos x$
Step 3 $\Rightarrow$ $1 = \cos x$ Cancel $\sin x$
$\Rightarrow$ $x = 0°$ or $360°$
The teacher tells Jessica that she has not found all the solutions because of a mistake.
Explain why Jessica's method is not correct.
[2 marks]
\hfill \mbox{\textit{AQA AS Paper 1 Q5 [2]}}