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Tim Gan Math

1999 A-Level H2 Mathematics

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Question 1Applications of Differentiation

The curve $C$ has equation $y=\frac{{{x}^{2}}+5}{\lambda x+1}$, where $\lambda $ is a non-zero constant.

  1. Obtain the equations of the asymptotes of $C$.
  2. Show that $C$ has exactly two points at which $\frac{\mathrm{d}y}{\mathrm{d}x}=0$.
  3. Draw a sketch of $C$ to illustrate the case $\lambda >0$, and a separate of $C$ to illustrate the case $\lambda <0$.
  4. Find the set of values of $\lambda $ for which the line $y=4x$ and $C$ has at least one point in common.
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Final Answer:

(i) OA: \(y = \frac{x}{\lambda} - \frac{1}{\lambda^2}\), VA: \(x = -\frac{1}{\lambda}\). (iv) \(\lambda \ge \frac{1}{20}\)