A-Level Mathematics May/June 2025 Q1: A curve has equation y = 2x + 12/x². Find the equation of the tangent to the curve at t…

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  The equation of the curve is given by $y = 2x + \frac{12}{x^2}$.

  Step 1: Find the derivative of the function. First, rewrite the equation using negative indices to make differentiation easier: $y = 2x + 12x^{-2}$

  Now, differentiate with respect to $x$ to find the gradient function, $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{d}{dx}(2x) + \frac{d}{dx}(12x^{-2})$ $\frac{dy}{dx} = 2 + 12(-2)x^{-2-1}$ $\frac{dy}{dx} = 2 - 24x^{-3}$

  Step 2: Find the gradient of the tangent at the given point. The tangent is at the point $(-2, -1)$. We substitute $x = -2$ into the derivative to find the gradient, $m$. $m = 2 - 24(-2)^{-3}$ $m = 2 - \frac{24}{(-2)^3}$ $m = 2 - \frac{24}{-8}$ $m = 2 - (-3)$ $m = 5$

  Step 3: Find the equation of the tangent. We use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, with the gradient $m=5$ and the point $(x_1, y_1) = (-2, -1)$. $y - (-1) = 5(x - (-2))$ $y + 1 = 5(x + 2)$ $y + 1 = 5x + 10$

  Finally, rearrange the equation into the form $y = mx + c$: $y = 5x + 10 - 1$ $y = 5x + 9$

  The equation of the tangent is $y = 5x + 9$.

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  How the marks are awarded

  • B1 — Correctly differentiating the curve's equation. This involves differentiating $2x$ to get $2$ and, crucially, differentiating $12x^{-2}$ to get $-24x^{-3}$.
  • M1 — Substituting the value $x = -2$ from the given point into the derived gradient function, $\frac{dy}{dx}$, which must contain a term of the form $ax^{-3}$.
  • DM1 — Using a valid method to find the equation of a straight line, by combining the calculated gradient (5) and the given point (–2, –1). This mark is dependent on the previous M1 mark.
  • A1 — Stating the final, fully correct equation of the tangent in the required form, which is $y = 5x + 9$.

  Common mistakes

  • Incorrect differentiation of $12/x^2$. A common error is to get $12(-2)x^{-1}$ or $+24x^{-3}$, which loses the first mark and all subsequent accuracy marks.
  • Arithmetic errors with negative numbers. For example, calculating $(-2)^3$ as $-6$ or $8$ instead of $-8$, leading to an incorrect gradient.
  • Errors in finding the line equation. For instance, making a sign error in the point-slope formula, like $y - 1 = 5(x - 2)$, or incorrectly calculating the y-intercept 'c'.
  • Finding the equation of the normal instead of the tangent by using the negative reciprocal of the gradient (e.g., using $m = -1/5$).

  Examiner tip: Always rewrite terms like $k/x^n$ as $kx^{-n}$ before differentiating to apply the power rule correctly and avoid common errors.

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