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2006 Paper 3 Q1
D: 1700.0 B: 1500.0
curve sketching asymptotes rational function tangent at origin intersection of curves number of real roots oblique asymptote vertical asymptote

Sketch the curve with cartesian equation \[ y = \frac{2x(x^2-5)}{x^2-4} \] and give the equations of the asymptotes and of the tangent to the curve at the origin. Hence determine the number of real roots of the following equations:

  1. \(3x(x^2-5)= (x^2-4)(x+3)\,\);
  2. \(4x(x^2-5)= (x^2-4)(5x-2)\,\);
  3. \(4x^2(x^2-5)^2= (x^2-4)^2(x^2+1)\,\).

Solution: \begin{align*} && y &= \frac{2x(x^2-5)}{x^2-4} \\ &&&= 2x(x^2-5)(-\tfrac14)(1-\tfrac14x^2)^{-1} \\ &&&= \tfrac52x + \cdots \\ &&&= \frac{2x(x^2-4)-2x}{x^2-4} \\ &&&= 2x - \frac{2x}{x^2-4} \end{align*}

TikZ diagram
  1. We are looking for the intersections of \(y = \frac23(x+3)\) and \(y = f(x)\)
    TikZ diagram
    Therefore 3 real roots.
  2. We are looking for intersections of \(y = \frac12(5x-2)\) and \(y = f(x)\)
    TikZ diagram
    so one solution.
  3. We are looking for intersections of \(y = f(x)^2\) and \(y = x^2+1\), or \(y = \sqrt{x^2+1}\) and \(y = f(x)\) where \(f(x) \geq 0\)
    TikZ diagram
    So \(3\) solutions.

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1989 Paper 3 Q4
D: 1700.0 B: 1500.0
curve sketching asymptotes rational function tangent at origin roots of polynomials intersection of curves substitution

Sketch the curve whose cartesian equation is \[ y=\frac{2x(x^{2}-5)}{x^{2}-4}, \] and give the equations of the asymptotes and of the tangent to the curve at the origin. Hence, or otherwise, determine (giving reasons) the number of real roots of the following equations:

  1. \(4x^{2}(x^{2}-5)=(5x-2)(x^{2}-4)\);
  2. \(4x^{2}(x^{2}-5)^{2}=(x^{2}-4)^{2}(x^{2}+1)\);
  3. \(4z^{2}(z-5)^{2}=(z-4)^{2}(z+1)\).

Solution:

TikZ diagram
The gradient at the origin is \(\frac{5}{2}\) which we can observe by looking at the taylor series
  1. \begin{align*} && 4x^{2}(x^{2}-5)&=(5x-2)(x^{2}-4) \\ && \frac{2x(x^2-5)}{x^2-4} &= \frac{5x-2}{2x} = \frac52 -\frac1x \end{align*} Therefore it will have two roots as the hyperbola will intersect our graph in two places. (In the upper left and lower right quadrants).
  2. \begin{align*} && 4x^{2}(x^{2}-5)^{2}&=(x^{2}-4)^{2}(x^{2}+1) \\ && \frac{2x(x^2-5)}{x^2-4} &= \frac{(x^2-4)(x^2+1)}{2x} \end{align*}
    TikZ diagram
    No solutions \begin{align*} && 4z^{2}(z-5)^{2}&=(z-4)^{2}(z+1) \\ && \frac{2z(z^2-5)}{z^2-4} &= \frac{(z+5)(z-4)(z+1)}{(z-5)(z+4)} \end{align*}
    TikZ diagram
    5 solutions

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