2012 A-level H2 Mathematics (9740) Paper 1 Question 8 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Differentiating the given equation with respect to x,

1 - \frac{dy}{dx} = 2(x+y)(1 + \frac{dy}{dx})

1 = 2x + 2y + (2x+2y+1)\frac{dy}{dx}

2 = (2x+2y+1)(1 + \frac{dy}{dx})

1 + \frac{dy}{dx} = \frac{2}{2x+2y+1}

(ii)
\frac{d^2y}{dx^2} = - \frac{2}{(2x+2y+1)^2} (2 + 2\frac{dy}{dx})

= - (\frac{2}{2x+2y+1})^2 (1  + \frac{dy}{dx})

= - (1+ \frac{dy}{dx})^3

(iii)
When \frac{dy}{dx} = 0, \frac{d^2y}{dx^2} = -1 <0

Thus we conclude that the turning point is a maximum point.

KS Comments:

Some students can still get the implicit differentiation wrong. (iii) actually is very simple if students do not overthink and trust in what they understand.

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