A-Level Mathematics October/November 2024 Q2: The curve y = x² - a/x has a stationary point at (-3, b). Find the values of the consta…

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  To find the values of the constants a and b, we need to use the information that the curve has a stationary point at (-3, b).

  The equation of the curve is $y = x^2 - \frac{a}{x}$. First, we rewrite the equation in a form that is easier to differentiate: $y = x^2 - ax^{-1}$

  At a stationary point, the gradient of the curve, $\frac{dy}{dx}$, is equal to zero. We differentiate y with respect to x: $\frac{dy}{dx} = 2x - a(-1)x^{-2}$ $\frac{dy}{dx} = 2x + ax^{-2}$

  We are given that the stationary point occurs at $x = -3$. We set $\frac{dy}{dx} = 0$ and substitute $x = -3$: $2(-3) + a(-3)^{-2} = 0$ $-6 + \frac{a}{(-3)^2} = 0$ $-6 + \frac{a}{9} = 0$

  Now, we solve for a: $\frac{a}{9} = 6$ $a = 6 \times 9$ $a = 54$

  Now that we have the value of a, the equation of the curve is $y = x^2 - \frac{54}{x}$. The point (-3, b) lies on this curve. Therefore, we can substitute $x = -3$ and $y = b$ into the equation to find b: $b = (-3)^2 - \frac{54}{-3}$ $b = 9 - (-18)$ $b = 9 + 18$ $b = 27$

  Thus, the values of the constants are $a = 54$ and $b = 27$.

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  How the marks are awarded

  • B1 — The first mark is awarded for correctly differentiating the expression for y. Rewriting $y = x^2 - a/x$ as $y = x^2 - ax^{-1}$ is a helpful first step, leading to the derivative $\frac{dy}{dx} = 2x + ax^{-2}$.
  • M1 — The method mark is for using the information about the stationary point. This involves setting the derivative $\frac{dy}{dx}$ equal to zero and substituting the given x-coordinate, $x = -3$, to form an equation in terms of a, as shown in the step $2(-3) + a(-3)^{-2} = 0$.
  • A1 — The third mark is an accuracy mark for correctly solving the equation from the previous step to find the value of a. The calculation $-6 + a/9 = 0$ leads to $a = 54$.
  • A1 — The final accuracy mark is for finding the value of b. This is achieved by substituting the coordinates of the stationary point $(x=-3, y=b)$ and the value of $a=54$ into the original equation for the curve, leading to $b = 27$.

  Common mistakes

  • Incorrectly differentiating the -a/x term. A common error is to get a sign wrong, resulting in $\frac{dy}{dx} = 2x - ax^{-2}$, which leads to an incorrect value for a.
  • Making an error when substituting $x=-3$ into the derivative, for example calculating (-3)^2 as -9 instead of 9. This would lead to $-6 - a/9 = 0$ and $a=-54$.
  • Finding the value of a correctly but then making a sign error when calculating b. For instance, calculating $b = (-3)^2 - \frac{54}{-3}$ as 9 - 18 = -9 instead of the correct 9 - (-18) = 27.
  • Confusing the conditions for the two unknowns, for example, by substituting $y=b$ into the derivative equation, or setting the original equation for y to zero.

  Examiner tip: Remember that a given stationary point (x, y) provides two key pieces of information: the coordinates satisfy the original equation $y=f(x)$, and the x-coordinate makes the derivative f'(x) equal to zero.

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