| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Parallel and perpendicular planes |
| Difficulty | Standard +0.3 This is a straightforward vectors question requiring standard techniques: (i) verify a line lies in a plane by checking the point satisfies the equation and the direction vector is perpendicular to the normal, (ii) find a plane equation using the cross product of two vectors to get the normal. Both parts are routine applications of well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms |
| Answer | Marks |
|---|---|
| (i) EITHER: State or imply general point of l has coordinates \((1, 1-2\lambda, 1+\lambda)\), or equivalent | B1 |
| Substitute in LHS of plane equation | M1 |
| Verify that the equation is satisfied | A1 |
| OR: State or imply the plane has equation \(\mathbf{r} \cdot (1 \ 2j \ -3k) = 5\), or equivalent | B1 |
| Substitute for r in LHS and expand the scalar product | M1 |
| Verify that the equation is satisfied | A1 |
| OR: Verify that a point of l lies on the plane | B1 |
| Find a second point on l and substitute its coordinates in the equation of p | M1 |
| Verify second point, e.g. (1, -1, 2) lies on the plane | A1 |
| OR: Verify that a point of l lies on the plane | B1 |
| Form scalar product of a direction vector of l with a vector normal to p | M1 |
| Verify scalar product is zero and l is parallel to p | A1 |
| (ii) EITHER: Use scalar product of relevant vectors to form an equation in \(a, b, c\), e.g. \(a - 2b + c = 0\) | M1* |
| State two correct equations in \(a, b, c\) | A1 |
| Solve simultaneous equations and find one ratio, e.g. \(a : b = c : 4 : 1 - 2c\), or equivalent | M1(dep*) |
| Obtain \(a : b : c = 4 : 1 : -2\), or equivalent | A1 |
| Substitute correctly in \(4x + y - 2z = d\) to find \(d\) | M1 |
| Obtain equation \(4x + y - 2z = 1\), or equivalent | A1 |
| OR: Attempt to add a 2-parameter equation for the plane with relevant vectors, e.g. \(\mathbf{r} = 2i + j + 4k + \lambda(1-2j \ + k) + \mu(i + 2j + 3k)\) | M2 |
| State a correct equation, e.g. \(\mathbf{r} = 2i + j + 4k + \lambda(1-2j \ + k) + \mu(i + 2j + 3k)\) | A1 |
| State 3 equations in \(x, y, z, \lambda, \mu\) | A1 |
| Eliminate \(\lambda\) and \(\mu\) | M1 |
| Obtain equation \(4x + y - 2z = 1\), or equivalent | A1 |
| 6 |
(i) EITHER: State or imply general point of l has coordinates $(1, 1-2\lambda, 1+\lambda)$, or equivalent | B1 |
Substitute in LHS of plane equation | M1 |
Verify that the equation is satisfied | A1 |
OR: State or imply the plane has equation $\mathbf{r} \cdot (1 \ 2j \ -3k) = 5$, or equivalent | B1 |
Substitute for r in LHS and expand the scalar product | M1 |
Verify that the equation is satisfied | A1 |
OR: Verify that a point of l lies on the plane | B1 |
Find a second point on l and substitute its coordinates in the equation of p | M1 |
Verify second point, e.g. (1, -1, 2) lies on the plane | A1 |
OR: Verify that a point of l lies on the plane | B1 |
Form scalar product of a direction vector of l with a vector normal to p | M1 |
Verify scalar product is zero and l is parallel to p | A1 |
(ii) EITHER: Use scalar product of relevant vectors to form an equation in $a, b, c$, e.g. $a - 2b + c = 0$ | M1* |
State two correct equations in $a, b, c$ | A1 |
Solve simultaneous equations and find one ratio, e.g. $a : b = c : 4 : 1 - 2c$, or equivalent | M1(dep*) |
Obtain $a : b : c = 4 : 1 : -2$, or equivalent | A1 |
Substitute correctly in $4x + y - 2z = d$ to find $d$ | M1 |
Obtain equation $4x + y - 2z = 1$, or equivalent | A1 |
OR: Attempt to add a 2-parameter equation for the plane with relevant vectors, e.g. $\mathbf{r} = 2i + j + 4k + \lambda(1-2j \ + k) + \mu(i + 2j + 3k)$ | M2 |
State a correct equation, e.g. $\mathbf{r} = 2i + j + 4k + \lambda(1-2j \ + k) + \mu(i + 2j + 3k)$ | A1 |
State 3 equations in $x, y, z, \lambda, \mu$ | A1 |
Eliminate $\lambda$ and $\mu$ | M1 |
Obtain equation $4x + y - 2z = 1$, or equivalent | A1 |
| | 6 |7 The line $l$ has equation $\mathbf { r } = \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$. The plane $p$ has equation $x + 2 y + 3 z = 5$.\\
(i) Show that the line $l$ lies in the plane $p$.\\
(ii) A second plane is perpendicular to the plane $p$, parallel to the line $l$ and contains the point with position vector $2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }$. Find the equation of this plane, giving your answer in the form $a x + b y + c z = d$.
\hfill \mbox{\textit{CAIE P3 2006 Q7 [9]}}