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Date November 2020 Marks available 5 Reference code 20N.1.AHL.TZ0.H_11
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Find and Hence Question number H_11 Adapted from N/A

Question

Consider the curve C defined by y2=sin(xy) , y0.

Show that dydx=ycos(xy)2y-xcos(xy).

[5]
a.

Prove that, when dydx=0 , y=±1.

[5]
b.

Hence find the coordinates of all points on C, for 0<x<4π, where dydx=0.

[5]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt at implicit differentiation       M1

2ydydx=cos(xy)xdydx+y       A1M1A1


Note: Award A1 for LHS, M1 for attempt at chain rule, A1 for RHS.


2ydydx=xdydxcos(xy)+ycos(xy)

2ydydx-xdydxcos(xy)=ycos(xy)

dydx(2y-xcos(xy))=ycos(xy)       M1


Note: Award M1 for collecting derivatives and factorising.


dydx=ycos(xy)2y-xcos(xy)       AG


[5 marks]

a.

setting dydx=0

ycos(xy)=0       (M1)

(y0)cos(xy)=0       A1

sin(xy)(=±1-cos2(xy)=±1-0)=±1  OR  xy=(2n+1)π2(n)  OR  xy=π2, 3π2,       A1


Note: If they offer values for xy, award A1 for at least two correct values in two different ‘quadrants’ and no incorrect values.


y2(=sin(xy))>0       R1

y2=1       A1

y=±1       AG


[5 marks]

b.

y=±11=sin(±x)sinx=±1  OR  y=±10=cos(±x)cosx=0       (M1)

(sinx=1)(π2,1),(5π2,1)       A1A1

(sinx=-1)(3π2,-1),(7π2,-1)       A1A1


Note:
Allow ‘coordinates’ expressed as x=π2, y=1 for example.
Note: Each of the A marks may be awarded independently and are not dependent on (M1) being awarded.

Note: Mark only the candidate’s first two attempts for each case of sinx.

[5 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 5 —Calculus » SL 5.4—Tangents and normal
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Topic 5 —Calculus » AHL 5.14—Implicit functions, related rates, optimisation
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