| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2018 |
| Session | December |
| Marks | 11 |
| Topic | Vector Product and Surfaces |
| Type | Finding stationary points on surfaces |
| Difficulty | Challenging +1.2 This is a standard Further Maths multivariable calculus question requiring partial derivatives, setting them to zero, and using the second derivative test (Hessian determinant). Part (a) is routine verification; part (b) requires analyzing how the discriminant changes with parameter t, which adds modest complexity but follows a well-established method taught in FM courses. |
| Spec | 8.05e Stationary points: where partial derivatives are zero8.05f Nature of stationary points: classify using Hessian matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = tix^2 + y^2 + 3xy - y\) | ||
| \(\frac{\partial z}{\partial x} = 2tx + 3y\), \(\frac{\partial z}{\partial y} = 2y + 3x - 1\) | B1, B1 | 3.1a, 1.1 |
| When \(t = \frac{2}{3}\), if there is a stationary point then | ||
| \(\frac{\partial z}{\partial x} = \frac{2}{3}x + 3y = 0\) and \(\frac{\partial z}{\partial y} = 2y + 3x - 1 = 0\) | M1 | 3.1a |
| \(2\frac{\partial z}{\partial x} = 9x + 6y = 0\) | ||
| \(3\frac{\partial z}{\partial y} = 6y + 9x - 3 = 0\) which is a contradiction, so there is no stationary point on \(S\) when \(t = \frac{2}{3}\). | E1 | 3.2a |
| Total: [4] | Finding the coordinates of the stationary points is not required |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\partial^2 z}{\partial x^2} = 2t\), \(\frac{\partial^2 z}{\partial y^2} = 2\), \(\frac{\partial^2 z}{\partial x\partial y} = \frac{\partial^2 z}{\partial y\partial x} = 3\) | B1, B1 | 1.1, 1.1 |
| All 4 (NB: \(z_{xy} = z_{yx}\) may be soi) | ||
| \(H = \begin{pmatrix} 2t & 3 \\ 3 & 2 \end{pmatrix} \Rightarrow \det(H) = 4t - 9\) | M1, A1 | 3.1a, 1.1 |
| If \(t > \frac{9}{4}\), \(S\) has a (local) minimum since \(\det(H) > 0\) and \(z_{xx} > 0\) | M1, A1 | 3.1a, 2.4 |
| If \(t < \frac{9}{4}\), \(S\) has a saddle-point since \(\det(H) < 0\) | A1 | 2.4 |
| Total: [7] | Two second partial derivatives correct; Attempt to find Hessian in term of \(t\); Correct; considering value of \(\det(H)\) either side of \(t = \frac{9}{4}\); correct nature of stationary point with justification; correct nature of stationary point with justification |
**Part (a)**
| $z = tix^2 + y^2 + 3xy - y$ | | |
| $\frac{\partial z}{\partial x} = 2tx + 3y$, $\frac{\partial z}{\partial y} = 2y + 3x - 1$ | B1, B1 | 3.1a, 1.1 |
| When $t = \frac{2}{3}$, if there is a stationary point then | | |
| $\frac{\partial z}{\partial x} = \frac{2}{3}x + 3y = 0$ and $\frac{\partial z}{\partial y} = 2y + 3x - 1 = 0$ | M1 | 3.1a |
| $2\frac{\partial z}{\partial x} = 9x + 6y = 0$ | | |
| $3\frac{\partial z}{\partial y} = 6y + 9x - 3 = 0$ which is a contradiction, so there is no stationary point on $S$ when $t = \frac{2}{3}$. | E1 | 3.2a |
| Total: [4] | Finding the coordinates of the stationary points is not required | |
**Part (b)**
| $\frac{\partial^2 z}{\partial x^2} = 2t$, $\frac{\partial^2 z}{\partial y^2} = 2$, $\frac{\partial^2 z}{\partial x\partial y} = \frac{\partial^2 z}{\partial y\partial x} = 3$ | B1, B1 | 1.1, 1.1 |
| All 4 (NB: $z_{xy} = z_{yx}$ may be soi) | | |
| $H = \begin{pmatrix} 2t & 3 \\ 3 & 2 \end{pmatrix} \Rightarrow \det(H) = 4t - 9$ | M1, A1 | 3.1a, 1.1 |
| If $t > \frac{9}{4}$, $S$ has a (local) minimum since $\det(H) > 0$ and $z_{xx} > 0$ | M1, A1 | 3.1a, 2.4 |
| If $t < \frac{9}{4}$, $S$ has a saddle-point since $\det(H) < 0$ | A1 | 2.4 |
| Total: [7] | Two second partial derivatives correct; Attempt to find Hessian in term of $t$; Correct; considering value of $\det(H)$ either side of $t = \frac{9}{4}$; correct nature of stationary point with justification; correct nature of stationary point with justification | |7 For each value of $t$, the surface $S _ { t }$ has equation $z = t x ^ { 2 } + y ^ { 2 } + 3 x y - y$.
\begin{enumerate}[label=(\alph*)]
\item Verify that there are no stationary points on $S _ { t }$ when $t = \frac { 9 } { 4 }$.
\item Determine, as $t$ varies, the nature of any stationary point(s) of $S _ { t }$.\\
(You do not have to find the coordinates of the stationary points.)
\section*{OCR}
Oxford Cambridge and RSA
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2018 Q7 [11]}}