19N.1.SL.TZ0.T_14

pestleMathematicsAISLPaper 119N· sl-5-3-differentiating-polynomials-n-e-zsource ↗

The diagram shows the curve $y = \frac{x^{2}}{2} + \frac{2 a}{x}, x \neq 0$ .

The equation of the vertical asymptote of the curve is $x = k$ .

Write down the value of $k$ .

[1]

Find $\frac{d y}{d x}$ .

[3]

At the point where $x = 2$ , the gradient of the tangent to the curve is $0.5$ .

Find the value of $a$ .

[2]

Markscheme / solution

$(k =) 0$ (A1) (C1)

Note: Award (A1) for an answer of " $x = 0$ ".

[1 mark]

$x - \frac{2 a}{x^{2}}$ (A1)(A1)(A1) (C3)

Note: Award (A1) for $x$ , (A1) for $- 2 a$ , (A1) for $x^{- 2}$ or $\frac{1}{x^{2}}$ . Award at most (A1)(A1)(A0) if extra terms are seen.

[3 marks]

$0.5 = 2 - \frac{2 a}{2^{2}}$ (M1)

Note: Award (M1) for their correctly substituted derivative equated to $0.5$ .

$(a =) 3$ (A1)(ft) (C2)

Note: Follow through from part (b) providing their answer is not $a = 0$ as this value contradicts the graph.

[2 marks]

Examiners’ report

[N/A]