Standard +0.8 This FP1 question requires polynomial long division to find asymptotes, then solving a discriminant condition for tangency (requiring the quadratic x² - 2x + λ = 0 to have exactly one solution when y=0), followed by two sketches with different parameter values. It combines multiple techniques (division, discriminant analysis, curve sketching) and requires understanding how the parameter affects the curve's behavior, making it moderately challenging but still within standard FP1 scope.
9 The curve \(C\) has equation
$$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$
where \(\lambda\) is a constant. Show that the equations of the asymptotes of \(C\) are independent of \(\lambda\).
Find the value of \(\lambda\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
Sketch \(C\) in the case \(\lambda = - 4\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
One asymptote is $x = -1$ | B1 |
(Allow $x \to -1, y \to x - 3$) |
$y = x - 3 + (\lambda + 3)/(x + 1)$ | M1 |
$\Rightarrow y = x - 3$ | A1 |
$\lambda = 1$ | B1 |
Axes plus both asymptotes drawn | B1 ft |
**DEF:** Here Z denotes 'correct shape, orientation and approximately correct location' |
RH branch with Z | B1 ft |
LH branch with Z | B1 |
$\lambda = -4$: RH branch with Z | B1 |
LH branch with Z | B1 |
Intersections with x-axis $(1 + \sqrt{5}, 0), (1 - \sqrt{5}, 0)$ | B1 |
9 The curve $C$ has equation
$$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$
where $\lambda$ is a constant. Show that the equations of the asymptotes of $C$ are independent of $\lambda$.
Find the value of $\lambda$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
Sketch $C$ in the case $\lambda = - 4$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.
\hfill \mbox{\textit{CAIE FP1 2008 Q9 [10]}}