Standard +0.8 This requires implicit differentiation to find dy/dx, setting it to zero, then solving the resulting system of equations (the original curve equation plus dy/dx = 0) simultaneously. While implicit differentiation itself is standard, solving the coupled system algebraically to find exact coordinates of turning points requires careful manipulation and is more demanding than routine differentiation questions.
\((1-2x)^2 + x(1-2x) + x^2 - x = 1\) or \(y^2 + \dfrac{(1-y)y}{2} + \dfrac{(1-y)^2}{4} - \dfrac{1-y}{2} = 1\)
M1
AO 3.1a
\(3x^2 - 4x [= 0]\) or \(3y^2 + 2y - 5 [= 0]\)
A1
AO 1.1
\(x = 0,\ x = \dfrac{4}{3}\)
M1
AO 1.1
\(y = 1,\ y = -\dfrac{5}{3}\)
M1
AO 1.1
\((0, 1)\) and \(\left(\dfrac{4}{3}, -\dfrac{5}{3}\right)\) or \(x = 0,\ y = 1\) and \(x = \dfrac{4}{3},\ y = -\dfrac{5}{3}\)
A1
AO 3.2a
## Question 13:
$2y\dfrac{dy}{dx}$ | B1 | AO 1.1 | chain rule
$y + x\dfrac{dy}{dx}$ | B1 | AO 3.1a | product rule
(their previous terms) $+ 2x - 1 = 0$ | B1 | AO 1.1 | may award if "$= 0$" seen later, but not if RHS is $\dfrac{dy}{dx}$
their $y + 2x - 1 = 0$ | M1 | AO 2.1 | substitution of $\dfrac{dy}{dx} = 0$; dependent on award of at least one B mark; **NB** $\dfrac{dy}{dx} = \dfrac{1-2x-y}{2y+x}$
$(1-2x)^2 + x(1-2x) + x^2 - x = 1$ **or** $y^2 + \dfrac{(1-y)y}{2} + \dfrac{(1-y)^2}{4} - \dfrac{1-y}{2} = 1$ | M1 | AO 3.1a | elimination of $x$ or $y$ using expression or value obtained from use of $\dfrac{dy}{dx} = 0$; dependent on award of at least one B mark
$3x^2 - 4x [= 0]$ **or** $3y^2 + 2y - 5 [= 0]$ | A1 | AO 1.1 |
$x = 0,\ x = \dfrac{4}{3}$ | M1 | AO 1.1 | values of $x$ **or** $y$ found from their quadratic
$y = 1,\ y = -\dfrac{5}{3}$ | M1 | AO 1.1 | values of $y$ **or** $x$ found from substitution of both $x$ **or** both $y$ values; must see substitution unless values correct; **NB** may see extra points $y = -1$ or $\dfrac{1}{3}$ from substitution into original equation
$(0, 1)$ and $\left(\dfrac{4}{3}, -\dfrac{5}{3}\right)$ **or** $x = 0,\ y = 1$ and $x = \dfrac{4}{3},\ y = -\dfrac{5}{3}$ | A1 | AO 3.2a | **A0** if extra points in final answer; dependent on fully correct working throughout; if **M0M0** allow **SCB1** for 1 correct pair of coordinates and no others
13 Determine the coordinates of the turning points on the curve with equation
$$y ^ { 2 } + x y + x ^ { 2 } - x = 1 .$$
\hfill \mbox{\textit{OCR MEI Paper 2 2024 Q13 [9]}}