Moderate -0.3 This is a standard tangent-to-curve problem requiring students to set the line equal to the curve, use the discriminant condition (b²-4ac=0) for tangency, solve for m, then find coordinates. It's slightly easier than average as it follows a well-practiced procedure with straightforward algebra, though it does require understanding the tangency condition and completing multiple steps.
3 A line with equation \(y = m x - 6\) is a tangent to the curve with equation \(y = x ^ { 2 } - 4 x + 3\).
Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.
Substitute both their \(x\) values into their equation in line 1
DM1
Or substitute both their \((x, y)\) into \(y = mx - 6\)
When \(x = 3\), \(m = 2\); when \(x = -3\), \(m = -10\)
A1
If A0, DM1, A0 scored, SC B1 for one point correct WWW
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 4x + 3 = mx - 6$ leading to $x^2 - x(4+m) + 9$ | *M1 | Equating and gathering terms. May be implied on the next line. |
| $b^2 - 4ac$ leading to $(4+m)^2 - 4 \times 9$ | DM1 | SOI. Use of the discriminant with their $a$, $b$ and $c$ |
| $4 + m = \pm 6$ or $(m-2)(m+10) = 0$ leading to $m = 2$ or $-10$ | A1 | Must come from $b^2 - 4ac = 0$ SOI |
| Substitute both their $m$ values into their equation in line 1 | DM1 | |
| $m = 2$ leading to $x = 3$; $m = -10$ leading to $x = -3$ | A1 | |
| $(3, 0), (-3, 24)$ | A1 | Accept 'when $x=3, y=0$; when $x=-3, y=24$'. If final A0A0 scored, **SC B1** for one point correct WWW |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 2x - 4 \rightarrow 2x - 4 = m$ | *M1 | |
| $x^2 - 4x + 3 = (2x-4)x - 6$ | DM1 | |
| $x^2 - 4x + 3 = 2x^2 - 4x - 6 \rightarrow 9 = x^2 \rightarrow x = \pm 3$ | A1 | |
| $y = 0$, $24$ or $(3, 0)$, $(-3, 24)$ | A1 | |
| Substitute both their $x$ values into their equation in line 1 | DM1 | Or substitute both their $(x, y)$ into $y = mx - 6$ |
| When $x = 3$, $m = 2$; when $x = -3$, $m = -10$ | A1 | If A0, DM1, A0 scored, **SC B1** for one point correct WWW |
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3 A line with equation $y = m x - 6$ is a tangent to the curve with equation $y = x ^ { 2 } - 4 x + 3$.\\
Find the possible values of the constant $m$, and the corresponding coordinates of the points at which the line touches the curve.\\
\hfill \mbox{\textit{CAIE P1 2021 Q3 [6]}}