The answer
shown (gradient argument from the fixed point \((0, -2)\))
O-Level A-Math 2020 Paper 1 Question 5 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 5 of the O-Level A-Math 2020 Paper 1. It tests modulus graph, in the Modulus functions & graphs area. It is worth 4. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
The line \(y = mx - 2\) always passes through the fixed point \((0, -2)\) whatever the value of \(m\). The graph \(y = |7 - 2x|\) is a "V": its vertex is at \(\left(\tfrac72, 0\right)\), the left branch (\(x \leqslant \tfrac72\)) is \(y = 7 - 2x\) with gradient \(-2\), and the right branch (\(x \geqslant \tfrac72\)) is \(y = 2x - 7\) with gradient \(+2\).
Two distinct cuts means the line meets one branch each side of the vertex. - Right branch. \(mx - 2 = 2x - 7 \Rightarrow x = \dfrac{5}{2 - m}\). This is a valid point on the right branch (\(x \geqslant \tfrac72\), and positive) only when \(m < 2\). (When \(m = 2\) the line is parallel to the right branch and never meets it.) - Left branch. \(mx - 2 = 7 - 2x \Rightarrow x = \dfrac{9}{m + 2}\). This lies on the left branch (\(x \leqslant \tfrac72\)) only when \(m \geqslant \dfrac47\).
The boundary \(m = \dfrac47\) is exactly the gradient of the line joining \((0, -2)\) to the vertex \(\left(\tfrac72, 0\right)\), namely \(\dfrac{0 - (-2)}{\tfrac72 - 0} = \dfrac47\); there both branch-intersections coincide at the vertex, giving only one point. For two distinct points we therefore need \(m > \dfrac47\) together with \(m < 2\), i.e. \(\dfrac47 < m < 2\).
Answer: shown (gradient argument from the fixed point \((0, -2)\))
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level A-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a modulus graph question from Modulus functions & graphs, worth 4.
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