Graphs & Transformations: Hyperbola, circle & periodic function — JPJC 2025 H2 Math Prelim Paper 1
What this question tests
Question
(a) The curves \(C_{1}\) and \(C_{2}\) have equations \[ \frac{x^{2}}{9}-\frac{y^{2}}{4}=1 \qquad\text{and}\qquad y^{2}+x^{2}=k^{2} \] respectively, where \(k\) is a positive constant.
- Sketch \(C_{1}\) and \(C_{2}\) on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes.
- State the range of values of \(k\) for \(C_{1}\) and \(C_{2}\) to intersect.
- State the equations of the common lines of symmetry for both \(C_{1}\) and \(C_{2}\).
(b) The function \(\mathrm{f}\), with domain the set of all real values, is given by \[ \mathrm{f}(x)=\begin{cases} -2x+6 & \text{for } 0<x\leq 3,\\ 3x-9 & \text{for } 3<x\leq 5,\end{cases} \] and that \(\mathrm{f}(x)=\mathrm{f}(x+5)\).
- Find \(\mathrm{f}(46)\).
- Sketch the graph of \(y=\mathrm{f}(x)\) for \(-5\leq x\leq 5\).
- Hence, state the roots of \(\mathrm{f}(-x)=0\) for \(-5\leq x\leq 5\).
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(a)(i) [3 marks]
(a)(ii) [1 mark]
For \(C_{1}\) and \(C_{2}\) to intersect, \(k \geq 3\) (since \(k > 0\)).
(a)(iii) [1 mark]
Common lines of symmetry for both \(C_{1}\) and \(C_{2}\) are \(x = 0\) and \(y = 0\).
(b)(i) [1 mark]
Since \(\mathrm{f}\) is periodic with period 5: \[ \mathrm{f}(46) = \mathrm{f}(41) = \mathrm{f}(36) = \cdots = \mathrm{f}(1) = -2(1)+6 = 4. \]
(b)(ii) [2 marks]
(b)(iii) [1 mark]
For \(-5 \leq x \leq 5\), the roots of \(\mathrm{f}(-x) = 0\) are \(-3\) and \(2\).