Challenging +1.2 This is a standard implicit differentiation problem requiring finding dy/dx, setting it to zero for stationary points, and solving for x. While it involves algebraic manipulation of a cubic implicit equation with a parameter, the technique is routine for P3 level and follows a predictable method. The 6 marks reflect moderate algebraic complexity rather than conceptual difficulty.
\includegraphics{figure_5}
The diagram shows the curve with equation
$$x^3 + xy^2 + ay^2 - 3ax^2 = 0,$$
where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\). [6]
Solve for \(x\) and obtain answer \(x = \sqrt{3a}\)
A1
OR1: Rearrange equation in the form \(y^2 = \frac{3ax^2 - x^3}{x + a}\) and attempt differentiation of one side
B1
Use correct quotient or product rule to differentiate RHS
M1
Obtain correct derivative of RHS in any form
A1
Set \(\frac{dy}{dx}\) equal to zero and obtain an equation in \(x\)
M1
Obtain a correct horizontal equation free of surds
A1
Solve for \(x\) and obtain answer \(x = \sqrt{3a}\)
A1
OR2: Rearrange equation in the form \(y = \left(\frac{3ax^2 - x^3}{x + a}\right)^{\frac{1}{2}}\) and differentiation of RHS
B1
Use correct quotient or product rule and chain rule
M1
Obtain correct derivative in any form
A1
Equate derivative to zero and obtain an equation in \(x\)
M1
Obtain a correct horizontal equation free of surds
A1
Solve for \(x\) and obtain answer \(x = \sqrt{3a}\)
A1
[6]
**EITHER:** State $2ay\frac{dy}{dx}$ as derivative of $ay^2$ | B1 |
State $y^2 + 2xy\frac{dy}{dx}$ as derivative of $xy^2$ | B1 |
Equate derivative of LHS to zero and set $\frac{dy}{dx}$ equal to zero | M1 |
Obtain $3x^2 + y^2 - 6ax = 0$, or horizontal equivalent | A1 |
Eliminate $y$ and obtain an equation in $x$ | M1 |
Solve for $x$ and obtain answer $x = \sqrt{3a}$ | A1 |
**OR1:** Rearrange equation in the form $y^2 = \frac{3ax^2 - x^3}{x + a}$ and attempt differentiation of one side | B1 |
Use correct quotient or product rule to differentiate RHS | M1 |
Obtain correct derivative of RHS in any form | A1 |
Set $\frac{dy}{dx}$ equal to zero and obtain an equation in $x$ | M1 |
Obtain a correct horizontal equation free of surds | A1 |
Solve for $x$ and obtain answer $x = \sqrt{3a}$ | A1 |
**OR2:** Rearrange equation in the form $y = \left(\frac{3ax^2 - x^3}{x + a}\right)^{\frac{1}{2}}$ and differentiation of RHS | B1 |
Use correct quotient or product rule and chain rule | M1 |
Obtain correct derivative in any form | A1 |
Equate derivative to zero and obtain an equation in $x$ | M1 |
Obtain a correct horizontal equation free of surds | A1 |
Solve for $x$ and obtain answer $x = \sqrt{3a}$ | A1 | [6]
\includegraphics{figure_5}
The diagram shows the curve with equation
$$x^3 + xy^2 + ay^2 - 3ax^2 = 0,$$
where $a$ is a positive constant. The maximum point on the curve is $M$. Find the $x$-coordinate of $M$ in terms of $a$. [6]
\hfill \mbox{\textit{CAIE P3 2013 Q5 [6]}}