Standard +0.8 This requires implicit differentiation of a product of three terms, setting dy/dx = 0 for horizontal tangent, then solving the resulting system of equations to find unique stationary point coordinates. The algebraic manipulation is non-trivial but follows standard techniques for implicit differentiation problems.
6 The equation of a curve is \(x y ( x + y ) = 2 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.
EITHER State \(x^2\frac{dy}{dx} + 2xy\), or equivalent, as derivative of \(x^2y\)
B1
State \(y^2 + 2xy\frac{dy}{dx}\), or equivalent, as derivative of \(xy^2\)
B1
OR
Answer
Marks
Guidance
State \(xy(1 + \frac{dy}{dx})\), or equivalent, as a term in an attempt to apply the product rule
B1
State \((y + x\frac{dy}{dx})(x+y)\), or equivalent, in an attempt to apply the product rule
B1
Equate attempted derivative of LHS to zero and set \(\frac{dy}{dx}\) equal to zero
M1
Obtain a horizontal equation, e.g. \(y^2 = -2xy\), or \(y = -2x\), or equivalent
A1\(\sqrt{\ }\)
Explicitly reject \(y = 0\) as a possibility
A1
Obtain an equation in \(x\) (or in \(y\))
M1
Obtain \(x = a\)
A1
Obtain \(y = -2a\) only
A1
[8 marks]
**EITHER** State $x^2\frac{dy}{dx} + 2xy$, or equivalent, as derivative of $x^2y$ | B1 |
State $y^2 + 2xy\frac{dy}{dx}$, or equivalent, as derivative of $xy^2$ | B1 |
**OR**
State $xy(1 + \frac{dy}{dx})$, or equivalent, as a term in an attempt to apply the product rule | B1 |
State $(y + x\frac{dy}{dx})(x+y)$, or equivalent, in an attempt to apply the product rule | B1 |
Equate attempted derivative of LHS to zero and set $\frac{dy}{dx}$ equal to zero | M1 |
Obtain a horizontal equation, e.g. $y^2 = -2xy$, or $y = -2x$, or equivalent | A1$\sqrt{\ }$ |
Explicitly reject $y = 0$ as a possibility | A1 |
Obtain an equation in $x$ (or in $y$) | M1 |
Obtain $x = a$ | A1 |
Obtain $y = -2a$ only | A1 | [8 marks] | [The first M1 is dependent on at least one B mark having been earned.] [SR: for an attempt using $(x+y) = 2a^2/xy$, the B marks are given for the correct derivatives of the two sides of the equation, and the M1 for setting $\frac{dy}{dx}$ equal to zero.] [SR: for an attempt which begins by expressing $y$ in terms of $x$, give M1A1 for a reasonable attempt at differentiation, M1A1$\sqrt{\ }$ for setting $\frac{dy}{dx}$ equal to zero and obtaining an equation free of surds, A1 for solving and obtaining $x = a$; then M1 for obtaining an equation for $y$, A1 for $y = -2a$ and A1 for finding and rejecting $y = a$ as a possibility.]
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6 The equation of a curve is $x y ( x + y ) = 2 a ^ { 3 }$, where $a$ is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the $x$-axis, and find the coordinates of this point.
\hfill \mbox{\textit{CAIE P3 2008 Q6 [8]}}