Standard +0.3 This is a standard discriminant problem requiring students to set the equations equal, form a quadratic, and apply b²-4ac < 0 for no intersection. While it involves careful algebraic manipulation with a parameter k, it follows a well-practiced technique taught explicitly in P1 curricula, making it slightly easier than average.
2 A curve has equation \(y = k x ^ { 2 } + 2 x - k\) and a line has equation \(y = k x - 2\), where \(k\) is a constant. Find the set of values of \(k\) for which the curve and line do not intersect.
\(kx^2 + 2x - k = kx - 2\) leading to \(kx^2 + (-k+2)x - k + 2 [=0]\)
\*M1
Eliminate \(y\) and form 3-term quadratic. Allow 1 error.
\((-k+2)^2 - 4k(-k+2)\)
DM1
Apply \(b^2 - 4ac\); allow 1 error but \(a\), \(b\) and \(c\) must be correct for *their* quadratic.
\(5k^2 - 12k + 4\) or \((-k+2)(-k+2-4k)\)
A1
May be shown in quadratic formula.
\((-k+2)(-5k+2)\)
DM1
Solving a 3-term quadratic in \(k\) by factorising, use of formula or completing the square. Factors must expand to give *their* coefficient of \(k^2\).
\(\frac{2}{5} < k < 2\)
A1
WWW, accept two separate correct inequalities. If M0 for solving quadratic, SC B1 can be awarded for correct final answer.
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $kx^2 + 2x - k = kx - 2$ leading to $kx^2 + (-k+2)x - k + 2 [=0]$ | \*M1 | Eliminate $y$ and form 3-term quadratic. Allow 1 error. |
| $(-k+2)^2 - 4k(-k+2)$ | DM1 | Apply $b^2 - 4ac$; allow 1 error but $a$, $b$ and $c$ must be correct for *their* quadratic. |
| $5k^2 - 12k + 4$ or $(-k+2)(-k+2-4k)$ | A1 | May be shown in quadratic formula. |
| $(-k+2)(-5k+2)$ | DM1 | Solving a 3-term quadratic in $k$ by factorising, use of formula or completing the square. Factors must expand to give *their* coefficient of $k^2$. |
| $\frac{2}{5} < k < 2$ | A1 | WWW, accept two separate correct inequalities. If M0 for solving quadratic, **SC B1** can be awarded for correct final answer. |
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2 A curve has equation $y = k x ^ { 2 } + 2 x - k$ and a line has equation $y = k x - 2$, where $k$ is a constant. Find the set of values of $k$ for which the curve and line do not intersect.\\
\hfill \mbox{\textit{CAIE P1 2021 Q2 [5]}}