Standard +0.3 This is a straightforward application of the discriminant method: substitute the line equation into the curve, rearrange to get a quadratic, then show b²-4ac < 0. While it requires understanding that no real solutions means no intersection, it's a standard C1 technique with minimal algebraic manipulation and no conceptual subtlety.
condone one error; allow M1 for \(x^2 - 8x = -17\) [oe for \(y\)] only if they go on to completing square method
use of \(b^2 - 4ac\) with numbers substituted (condone one error in substitution) (may be in quadratic formula)
M1
or \((x-4)^2 = 16 - 17\) or \((x-4)^2 + 1 = 0\) (condone one error)
\(b^2 - 4ac = 64 - 68\) or \(-4\) cao [or \(16 - 52\) or \(-36\) if \(y\) used]
A1
or \((x-4)^2 = -1\) or \(x = 4 \pm \sqrt{-1}\); [or \((y-2)^2 = -9\) or \(y = 2 \pm \sqrt{-9}\)]
\([< 0]\) so no [real] roots [so line and curve do not intersect]
A1
or conclusion from completing square; needs explicit correct conclusion and correct ft; allow '\( < 0\) so no intersection' o.e.; allow '\(-4\) so no roots' etc; allow A2 for full argument from sum of two squares \(= 0\); A1 for weaker correct conclusion; some may use condition \(b^2 < 4ac\) for no real roots; allow equivalent marks, with first A1 for \(64 < 68\) o.e.
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 5x + 7 = 3x - 10$ | M1 | or attempt to subst $\frac{y+10}{3}$ for $x$ |
| $x^2 - 8x + 17 [= 0]$ o.e. or $y^2 - 4y + 13 [= 0]$ o.e. | M1 | condone one error; allow M1 for $x^2 - 8x = -17$ [oe for $y$] only if they go on to completing square method |
| use of $b^2 - 4ac$ with numbers substituted (condone one error in substitution) (may be in quadratic formula) | M1 | or $(x-4)^2 = 16 - 17$ or $(x-4)^2 + 1 = 0$ (condone one error) |
| $b^2 - 4ac = 64 - 68$ or $-4$ cao [or $16 - 52$ or $-36$ if $y$ used] | A1 | or $(x-4)^2 = -1$ or $x = 4 \pm \sqrt{-1}$; [or $(y-2)^2 = -9$ or $y = 2 \pm \sqrt{-9}$] |
| $[< 0]$ so no [real] roots [so line and curve do not intersect] | A1 | or conclusion from completing square; needs explicit correct conclusion and correct ft; allow '$ < 0$ so no intersection' o.e.; allow '$-4$ so no roots' etc; allow A2 for full argument from sum of two squares $= 0$; A1 for weaker correct conclusion; some may use condition $b^2 < 4ac$ for no real roots; allow equivalent marks, with first A1 for $64 < 68$ o.e. |
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