2024 RI Promo Q8
2024 RI Promo Q8
11 marks
- The diagram shows the curve \(y = \mathrm{f}(x)\) with a maximum point at \(C(4,3)\). The curve crosses the axes at the points \(A(0,2)\) and \(B(3,0)\). The lines \(x = 2\) and \(y = 0\) are the asymptotes of the curve.
Sketch the graph of \(y = \mathrm{f}'(x)\), clearly stating the equations of the asymptotes and the coordinates of the points corresponding to \(A\), \(B\) and \(C\) where appropriate.[3] - The curve \(C_1\) has equation \(y = \frac{ax^2 + bx - 8}{x - 2}\), where \(a\) and \(b\) are constants. It is given that \(C_1\) has an asymptote \(y = 3 - 2x\).
- State the value of \(a\) and show that \(b = 7\).[3]
- Sketch \(C_1\), clearly stating the equations of any asymptotes, the coordinates of any stationary points and of any points where \(C_1\) crosses the axes.[3]
- The curve \(C_1\) is transformed by a translation of \(2\) units in the negative \(x\)-direction, followed by a stretch with scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis, to form the curve \(C_2\). Find the equation of \(C_2\).[2]
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