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

By KS Teng

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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