| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Classifying stationary points on surfaces |
| Difficulty | Challenging +1.2 This is a Further Maths multivariable calculus question requiring partial derivatives, the Hessian matrix for classifying stationary points, and finding a tangent plane equation. While these are Further Maths topics (making it harder than typical A-level), the techniques are standard and mechanical once learned—find where both partials equal zero, compute second derivatives for classification, and apply the tangent plane formula. No novel insight or complex problem-solving required. |
| Spec | 8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero8.05f Nature of stationary points: classify using Hessian matrix8.05g Tangent planes: equation at a given point on surface |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (i) | (a) |
| Answer | Marks |
|---|---|
| z(cid:32)34 | B1 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | The connection must be made |
| Answer | Marks |
|---|---|
| (b) | f (cid:32)4, f (cid:32)(cid:16)2, f (cid:32)f (cid:32)3 |
| Answer | Marks |
|---|---|
| H (cid:31)0(cid:159) saddle-point | M1 |
| Answer | Marks |
|---|---|
| [3] | 2.1 |
| Answer | Marks |
|---|---|
| 2.2a | e |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (ii) | p |
| Answer | Marks |
|---|---|
| or 10x(cid:14)16y(cid:16)z(cid:32)4 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | Quoting correct form, including |
Question 6:
6 | (i) | (a) | f (cid:32)4x(cid:14)3y and
x
f (cid:32)(cid:16)2y(cid:14)3x(cid:14)17
y
Both are zero when
4x(cid:14)3y(cid:32)0 and (cid:16)2y(cid:14)3x(cid:14)17(cid:32)0
x(cid:32)(cid:16)3, y(cid:32)4
z(cid:32)34 | B1
B1
M1
A1
A1
[5] | 1.1
1.1
2.1
1.1
1.1 | The connection must be made
explicitlny
BC
(b) | f (cid:32)4, f (cid:32)(cid:16)2, f (cid:32)f (cid:32)3
xx yy xy yx
f f 4 3
xx xy
H (cid:32) (cid:32) (cid:32)(cid:16)17
f f 3 (cid:16)2
yx yy
H (cid:31)0(cid:159) saddle-point | M1
A1
e
E1
[3] | 2.1
i
c
1.1
2.2a | e
mEvaluating all second partial
derivatives and attempt to use the
Hessian
FT from their above values
Correct conclusion from a negative
determinant value
6 | (ii) | p
z(cid:32)f(a,b)(cid:14)(x(cid:16)a)f (a,b)
x S
(cid:14)(y(cid:16)b)f (a,b)
y
i.e. z(cid:32)38(cid:14)10(cid:11)x(cid:16)1(cid:12)(cid:14)16(cid:11)y(cid:16)2(cid:12)
or 10x(cid:14)16y(cid:16)z(cid:32)4 | M1
A1
[2] | 1.1
1.1 | Quoting correct form, including
attempt to substitute in values
Any correct simplified form6 A surface $S$ has equation $z = \mathrm { f } ( x , y )$, where $\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y$. It is given that $S$ has a single stationary point, $P$.
\begin{enumerate}[label=(\roman*)]
\item (a) Determine the coordinates of $P$.\\
(b) Determine the nature of $P$.
\item Find the equation of the tangent plane to $S$ at the point $Q ( 1,2,38 )$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure Q6 [10]}}