OCR Further Additional Pure AS 2018 June — Question 2 9 marks

Exam BoardOCR
ModuleFurther Additional Pure AS (Further Additional Pure AS)
Year2018
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStationary points and optimisation
TypeStationary points of surface (multivariable)
DifficultyStandard +0.8 This is a multivariable calculus problem requiring partial differentiation and solving a system of nonlinear equations. While the verification at the origin is straightforward, finding the second stationary point involves solving coupled equations (18x² + 2xy = 0 and 2y/9 + x² = 0), which is more sophisticated than typical A-level single-variable calculus and represents Further Maths content that goes beyond standard A-level.
Spec8.05e Stationary points: where partial derivatives are zero
2 The surface with equation \(z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y\) has two stationary points.
  1. Verify that one of these stationary points is at the origin.
  2. Find the coordinates of the second stationary point.
Question 2:
AnswerMarks Guidance
2(i) z
18x2 2xy
x
z
 2 y x2
y 9
z z
When x = y = 0, both and are zero
AnswerMarks
x yM1
A1
A1
AnswerMarks
B12.1
1.1
1.1
AnswerMarks
2.2aGood attempt at 1st partial derivative
1st correct
2nd correct
Verified, checked or noted.
FT provided both p.d.s have both x
AnswerMarks
and y termsCondone lack of visible
c h eck that z = 0 also
[4]
AnswerMarks
(ii)z
For x, y  0,  0  y = –9x
x
z
and 0  y 9 x2
y 2
–9x = 9 x2  (x  0) x = 2
2
AnswerMarks
(x, y, z) = (2, –18, 12)M1 *
A1
M1 dep.
A1
AnswerMarks
A11.1a
2.2a
2.1
1.1
AnswerMarks
1.1Setting both first p.d.s equal to 0
FT Both preliminary statements
correct provided both p.d.s have both
x and y terms
Eliminating and solving for x
FT x correct
cao
[5]
Question 2:
2 | (i) | z
18x2 2xy
x
z
 2 y x2
y 9
z z
When x = y = 0, both and are zero
x y | M1
A1
A1
B1 | 2.1
1.1
1.1
2.2a | Good attempt at 1st partial derivative
1st correct
2nd correct
Verified, checked or noted.
FT provided both p.d.s have both x
and y terms | Condone lack of visible
c h eck that z = 0 also
[4]
(ii) | z
For x, y  0,  0  y = –9x
x
z
and 0  y 9 x2
y 2
–9x = 9 x2  (x  0) x = 2
2
(x, y, z) = (2, –18, 12) | M1 *
A1
M1 dep.
A1
A1 | 1.1a
2.2a
2.1
1.1
1.1 | Setting both first p.d.s equal to 0
FT Both preliminary statements
correct provided both p.d.s have both
x and y terms
Eliminating and solving for x
FT x correct
cao
[5]
2 The surface with equation $z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y$ has two stationary points.\\
(i) Verify that one of these stationary points is at the origin.\\
(ii) Find the coordinates of the second stationary point.

\hfill \mbox{\textit{OCR Further Additional Pure AS 2018 Q2 [9]}}
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Q1 8 Q2 9 Q3 3 Q4 11 Q5 8 Q6 8 Q7 13