Moderate -0.3 This is a standard discriminant problem requiring students to set the equations equal, form a quadratic, and apply b²-4ac ≥ 0. It's slightly easier than average because it's a direct application of a well-practiced technique with straightforward algebra and no additional complications.
2 A line has equation \(y = x + 1\) and a curve has equation \(y = x ^ { 2 } + b x + 5\). Find the set of values of the constant \(b\) for which the line meets the curve.
Eliminate \(x\) or \(y\) with all terms on one side of an equation
\((b^2-4ac=)\ (b-1)^2-16\)
M1
\(b\) associated with \(-3\) & \(+5\) or \(b-1\) associated with \(\pm 4\)
A1
\((x-2)^2=0\) or \((x+2)^2=0\), \(x=\pm 2\), \(b-1=\pm 4\) (M1A1); Association can be an equality or an inequality
\(b\geqslant 5,\ b\leqslant -3\)
A1
Total: 4
**Question 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2+bx+5=x+1 \rightarrow x^2+x(b-1)+4\ (=0)$ | M1 | Eliminate $x$ or $y$ with all terms on one side of an equation |
| $(b^2-4ac=)\ (b-1)^2-16$ | M1 | |
| $b$ associated with $-3$ & $+5$ or $b-1$ associated with $\pm 4$ | A1 | $(x-2)^2=0$ or $(x+2)^2=0$, $x=\pm 2$, $b-1=\pm 4$ (M1A1); Association can be an equality or an inequality |
| $b\geqslant 5,\ b\leqslant -3$ | A1 | |
| **Total: 4** | | |
2 A line has equation $y = x + 1$ and a curve has equation $y = x ^ { 2 } + b x + 5$. Find the set of values of the constant $b$ for which the line meets the curve.\\
\hfill \mbox{\textit{CAIE P1 2018 Q2 [4]}}