A-Level Mathematics May/June 2024 Q6(a): A curve passes through the point (3/5, -3) and is such that dy/dx = -20/(5x-3)². Find t…

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  To find the equation of the curve, we must integrate the expression for the gradient, $\frac{dy}{dx}$.

  Given: $\frac{dy}{dx} = -\frac{20}{(5x-3)^2}$

  First, rewrite the expression using a negative index: $\frac{dy}{dx} = -20(5x-3)^{-2}$

  Now, integrate with respect to $x$ to find $y$: $y = \int -20(5x-3)^{-2} \, dx$

  Using the reverse chain rule, $\int (ax+b)^n \, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c$: $y = \frac{-20(5x-3)^{-1}}{5 \times (-1)} + c$ $y = \frac{-20(5x-3)^{-1}}{-5} + c$ $y = 4(5x-3)^{-1} + c$

  This can also be written as: $y = \frac{4}{5x-3} + c$

  We are given that the curve passes through the point $(\frac{4}{5}, -3)$. Note: The point must be $(4/5, -3)$ for the function to be defined. Substituting $x = \frac{4}{5}$ and $y = -3$ to find the constant of integration, $c$: $-3 = \frac{4}{5(\frac{4}{5}) - 3} + c$ $-3 = \frac{4}{4 - 3} + c$ $-3 = \frac{4}{1} + c$ $-3 = 4 + c$ $c = -3 - 4$ $c = -7$

  Therefore, the equation of the curve is: $y = \frac{4}{5x-3} - 7$

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  How the marks are awarded

  • M1 — Awarded for attempting to integrate the expression for dy/dx, achieving the form k(5x-3)⁻¹. This is shown in the working line y = [-20(5x-3)⁻¹]/[5 x (-1)] + c.
  • A1 — Awarded for the correct integrated expression 4(5x-3)⁻¹. The constant of integration + c is not required for this mark.
  • DM1 — Awarded for substituting the coordinates of the given point, x = 4/5 and y = -3, into an integrated equation containing a constant c. This mark is dependent on the first M1 mark being awarded.
  • A1 — Awarded for the correct and final equation of the curve, y = 4/(5x-3) - 7 or an equivalent form. The value c = -7 alone is not sufficient unless the full equation is stated.

  Common mistakes

  • Forgetting to divide by the derivative of the inner function (5) during integration, leading to an incorrect coefficient. For example, getting y = 20(5x-3)⁻¹ + c.
  • Making a sign error during integration or simplification, for example ending up with y = -4(5x-3)⁻¹ + c.
  • Making an arithmetic error when solving for the constant c. For example, from -3 = 4 + c, incorrectly calculating c = -1 or c = 7.
  • Forgetting to include the constant of integration + c after integrating, which makes it impossible to find the specific curve equation.

  Examiner tip: Always remember to include the constant of integration, + c, after indefinite integration, and then use the given coordinates to solve for its value.

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