Standard +0.8 This question requires implicit differentiation with a product rule and a fractional power, then solving a cubic equation to find x-values where y=1, and finally substituting to find gradients at multiple points. The combination of algebraic manipulation, implicit differentiation technique, and solving the resulting cubic makes this moderately harder than a standard implicit differentiation question.
15 You must show detailed reasoning in this question.
The equation of a curve is
$$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$
Find the gradient of the curve at each of the points where \(y = 1\).
\(x - 4\sqrt{x} + 3 = 0\) or \(4\sqrt{x} = x + 3\); \(x = 1\) or \(9\)
A1
AO 2.1
\(3y^2\frac{dy}{dx}\)
A1
AO 1.1
\(-x\times\frac{dy}{dx} - y\) or \(x\times\frac{dy}{dx} + y\)
B1
AO 3.1a
\(3y^2\frac{dy}{dx} - x\frac{dy}{dx} - y + \frac{2}{\sqrt{x}} [= 0]\); substitution of \(y=1\) and their \(x=1\) or their \(x=9\)
M1
AO 2.1
\(m = -\frac{1}{2}\) [at \((1,1)\)]
A1
AO 1.1
\(m = -\frac{1}{18}\) [at \((1,9)\)]
M1, A1, A1 [9]
AO 1.1
## Question 15:
substitution of $y = 1$ | M1 | AO 1.1a |
$x - 4\sqrt{x} + 3 = 0$ or $4\sqrt{x} = x + 3$; $x = 1$ or $9$ | A1 | AO 2.1 |
$3y^2\frac{dy}{dx}$ | A1 | AO 1.1 |
$-x\times\frac{dy}{dx} - y$ or $x\times\frac{dy}{dx} + y$ | B1 | AO 3.1a |
$3y^2\frac{dy}{dx} - x\frac{dy}{dx} - y + \frac{2}{\sqrt{x}} [= 0]$; substitution of $y=1$ and their $x=1$ or their $x=9$ | M1 | AO 2.1 | allow one sign error
$m = -\frac{1}{2}$ [at $(1,1)$] | A1 | AO 1.1 | dependent on at least two terms correct on LHS following differentiation; allow following wrong rearrangement after differentiating
$m = -\frac{1}{18}$ [at $(1,9)$] | M1, A1, A1 [9] | AO 1.1 | allow $-0.05555$...to 2 sf or better
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15 You must show detailed reasoning in this question.
The equation of a curve is
$$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$
Find the gradient of the curve at each of the points where $y = 1$.
\hfill \mbox{\textit{OCR MEI Paper 2 2018 Q15 [9]}}