| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2021 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Moderate -0.8 This question tests basic understanding of 3D surfaces through sections and contours, requiring only substitution of constant values (y=k for sections, z=k for contours) and sketching a simple parabola. The concepts are straightforward with minimal calculation, making it easier than average for Further Maths students who should be comfortable with 3D coordinate geometry. |
| Spec | 8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives8.05c Sections and contours: sketch and relate to surface |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | (i) |
| [1] | 1.2 | z = x2 + 4ax for any a, 0 < |
| Answer | Marks |
|---|---|
| (ii) | (Part of) a ∪-shaped parabola in the x-z plane |
| Thro’ (0,0), with Min ≈ at (−2a, −4a2) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.1 | (BC) FT Details correct; must exist only for x: −6 ≤ x ≤ 6 | |
| (b) | E.g. when z = 1, 1 = x2 + 4xy | B1 |
| [1] | 1.2 | b = x2 + 4xy for any b, −16 ≤ b ≤ 84, b ≠ 0 |
Question 2:
2 | (a) | (i) | E.g. when y = 1, z = x2 + 4x | B1
[1] | 1.2 | z = x2 + 4ax for any a, 0 < | a | ≤ 2
Only one chosen case is required. If the general case is offered,
the suitable range of values of the chosen parameter must be
noted (though condone the inclusion of “a” = 0).
(ii) | (Part of) a ∪-shaped parabola in the x-z plane
Thro’ (0,0), with Min ≈ at (−2a, −4a2) | M1
A1
[2] | 1.1
1.1 | (BC) FT Details correct; must exist only for x: −6 ≤ x ≤ 6
(b) | E.g. when z = 1, 1 = x2 + 4xy | B1
[1] | 1.2 | b = x2 + 4xy for any b, −16 ≤ b ≤ 84, b ≠ 0
Only one chosen case is required. If the general case is offered,
it must only be noted that “b” ≠ 0.2 The surface $S$ is given by $z = x ^ { 2 } + 4 x y$ for $- 6 \leqslant x \leqslant 6$ and $- 2 \leqslant y \leqslant 2$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the equation of any one section of $S$ which is parallel to the $x$-z plane
\item Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
\end{enumerate}\item Write down the equation of any one contour of $S$ which does not include the origin.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2021 Q2 [4]}}