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Date November 2017 Marks available 8 Reference code 17N.1.AHL.TZ0.H_7
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Determine Question number H_7 Adapted from N/A

Question

The folium of Descartes is a curve defined by the equation x 3 + y 3 3 x y = 0 , shown in the following diagram.

N17/5/MATHL/HP1/ENG/TZ0/07

Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the y -axis.

Markscheme

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

x 3 + y 3 3 x y = 0

3 x 2 + 3 y 2 d y d x 3 x d y d x 3 y = 0     M1A1

 

Note:     Differentiation wrt y is also acceptable.

 

d y d x = 3 y 3 x 2 3 y 2 3 x   ( = y x 2 y 2 x )     (A1)

 

Note:     All following marks may be awarded if the denominator is correct, but the numerator incorrect.

 

y 2 x = 0     M1

EITHER

x = y 2

y 6 + y 3 3 y 3 = 0     M1A1

y 6 2 y 3 = 0

y 3 ( y 3 2 ) = 0

( y 0 ) y = 2 3     A1

x = ( 2 3 ) 2   ( = 4 3 )     A1

OR

x 3 + x y 3 x y = 0     M1

x ( x 2 2 y ) = 0

x 0 y = x 2 2     A1

y 2 = x 4 4

x = x 4 4

x ( x 3 4 ) = 0

( x 0 ) x = 4 3     A1

y = ( 4 3 ) 2 2 = 2 3     A1

[8 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » SL 5.3—Differentiating polynomials, n E Z
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