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Date	May 2021	Marks available	3	Reference code	21M.1.AHL.TZ1.15
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	15	Adapted from	N/A

Question

The diagram shows the slope field for the differential equation

$\frac{d y}{d x} = \sin (x + y), - 4 \leq x \leq 5, 0 \leq y \leq 5$ .

The graphs of the two solutions to the differential equation that pass through points $(0, 1)$ and $(0, 3)$ are shown.

For the two solutions given, the local minimum points lie on the straight line $L_{1}$ .

Find the equation of $L_{1}$ , giving your answer in the form $y = m x + c$ .

[3]

a.

For the two solutions given, the local maximum points lie on the straight line $L_{2}$ .

Find the equation of $L_{2}$ .

[2]

b.

Markscheme

$\sin (x + y) = 0$ A1

$\Rightarrow x + y = 0$ (M1)

(the equation of $L_{1}$ is) $y = - x$ A1

[3 marks]

a.

$x + y = π$ OR $y = - x + π$ (M1)A1

[2 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 5—Calculus » AHL 5.15—Slope fields

22M.1.AHL.TZ1.7a.ii:
Sketch this curve on the slope field.
22M.1.AHL.TZ1.7b:
The solution to the differential equation that passes through the point $(0, - 2)$ has both a local maximum point and a local minimum point.

On the slope field, sketch the solution to the differential equation that passes through $(0, - 2)$ .
21M.1.AHL.TZ1.15b:
For the two solutions given, the local maximum points lie on the straight line $L_{2}$ .

Find the equation of $L_{2}$ .
22M.1.AHL.TZ1.7a.i:
Write down the equation of the curve on which all these maximum and minimum points lie.
21N.1.AHL.TZ0.13a:
Calculate the value of $\frac{d y}{d x}$ at the point $(0, 1)$ .
21N.1.AHL.TZ0.13b:
Sketch, on the first graph, a curve that represents the points where $\frac{d y}{d x} = 0$ .
21N.1.AHL.TZ0.13c:
(i) sketch the solution curve that passes through the point $(0, 0)$ .

(ii) sketch the solution curve that passes through the point $(0, 0.75)$ .

Topic 5—Calculus