2026 HCI P2 Q1

Junior College 2

9 marks

The curve $C$ has equation $y=\dfrac{kx^2-kx+68}{x-2}$, where $k$ is a constant.
1. If the equation of the oblique asymptote of $C$ is $y=2x+2$, state the value of $k$.[1]
  Use the value of $k$ found in part (a)(i) for the rest of the question.
2. Sketch $C$, showing clearly the coordinates of any axial intercepts and stationary points, and the equations of any asymptotes.[3]
3. Find the range of values of $r$, where $r>0$, such that the equation \[ (x-8)^2+\left(\frac{kx^2-kx+68}{x-2}-10\right)^2=r^2 \] has no real solutions for $x$.[2]
The graph of $y=\mathrm{f}(x)$ is transformed onto the graph of $y=\mathrm{g}(x)$ in a sequence of three transformations. The graph of $y=\mathrm{f}(x)$ is first transformed onto the graph of $y=\mathrm{f}(x)+1$, followed by a reflection in the $x$-axis and is finally transformed onto the graph of $y=\mathrm{g}(x)$ by transformation $T$. A point $P$ with coordinates $(a,b)$ lies on the graph of $y=\mathrm{f}(x)$. $P$ is mapped onto a point $Q$ with coordinates $(a-1,m)$ where $Q$ lies on the graph of $y=\mathrm{g}(x)$. Find $m$ in terms of $b$ and describe $T$.[3]

Solution:

Solution locked

680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs a course plan.

Create free account Sign in Get the H2 Math course · from S$39.98/mo H2 Math tuition

Create a free account to practise similar questions.

Create free account Sign in Get the H2 Math course · from S$39.98/mo H2 Math tuition

Answer:(a)(i) $k=2$ (ii) Maximum $(-4,-18)$, minimum $(8,30)$, $y$-intercept $(0,-34)$, no $x$-intercept; asymptotes $x=2$, $y=2x+2$. (iii) $0<r<20$ (b) Intended: $m=-b-1$, translation 1 unit left; not uniquely determined without a restriction on $T$.

Need help? Join our JC Math tuition classes.

Learn more