| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard three-part question on line intersection and planes requiring routine techniques: equating parametric equations to find intersection, using dot product for angle between direction vectors, and finding plane equation via cross product of direction vectors. All methods are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Express general point of \(l\) or \(m\) in component form, e.g. \((1+s,\ 1-s,\ 1+2s)\) or \((4+2t,\ 6+2t,\ 1+t)\) | B1 | |
| Equate at least two corresponding pairs of components and solve for \(s\) or \(t\) | M1 | |
| Obtain \(s = -1\) or \(t = -2\) | A1 | |
| Verify that all three component equations are satisfied | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out correct process for evaluating the scalar product of the direction vectors of \(l\) and \(m\) | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer \(74.2°\) (or 1.30 radians) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use scalar product to obtain \(a - b + 2c = 0\) and \(2a + 2b + c = 0\) | B1 | EITHER path |
| Solve and obtain one ratio, e.g. \(a:b\) | M1 | |
| Obtain \(a:b:c = 5:-3:-4\), or equivalent | A1 | |
| Substitute coordinates of a relevant point and values for \(a\), \(b\) and \(c\) in general equation of plane and evaluate \(d\) | M1 | |
| Obtain answer \(5x - 3y - 4z = -2\), or equivalent | A1 | [5] |
| Using two points on \(l\) and one on \(m\), or vice versa, state three equations in \(a\), \(b\), \(c\) and \(d\) | B1 | OR 1 |
| Solve and obtain one ratio, e.g. \(a:b\) | M1 | |
| Obtain a ratio of three of the unknowns, e.g. \(a:b:c = -5:3:4\) | A1 | |
| Use coordinates of a relevant point and found ratio to find the fourth unknown, e.g. \(d\) | M1 | |
| Obtain answer \(-5x + 3y + 4z = 2\), or equivalent | A1 | [5] |
| Form a correct 2-parameter equation for the plane, e.g. \(\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - \mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) | B1 | OR 2 |
| State three equations in \(x\), \(y\), \(z\), \(\lambda\) and \(\mu\) | M1 | |
| State three correct equations | A1 | |
| Eliminate \(\lambda\) and \(\mu\) | M1 | |
| Obtain answer \(5x - 3y - 4z = -2\), or equivalent | A1 | [5] |
| Attempt to calculate vector product of direction vectors of \(l\) and \(m\) | M1 | OR 3 |
| Obtain two correct components of the product | A1 | |
| Obtain correct product, e.g. \(-5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\) | A1 | |
| Form a plane equation and use coordinates of a relevant point to calculate \(d\) | M1 | |
| Obtain answer \(-5x + 3y + 4z = 2\), or equivalent | A1 | [5] |
## Question 10:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express general point of $l$ or $m$ in component form, e.g. $(1+s,\ 1-s,\ 1+2s)$ or $(4+2t,\ 6+2t,\ 1+t)$ | B1 | |
| Equate at least two corresponding pairs of components and solve for $s$ or $t$ | M1 | |
| Obtain $s = -1$ or $t = -2$ | A1 | |
| Verify that all three component equations are satisfied | A1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out correct process for evaluating the scalar product of the direction vectors of $l$ and $m$ | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer $74.2°$ (or 1.30 radians) | A1 | [3] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use scalar product to obtain $a - b + 2c = 0$ and $2a + 2b + c = 0$ | B1 | EITHER path |
| Solve and obtain one ratio, e.g. $a:b$ | M1 | |
| Obtain $a:b:c = 5:-3:-4$, or equivalent | A1 | |
| Substitute coordinates of a relevant point and values for $a$, $b$ and $c$ in general equation of plane and evaluate $d$ | M1 | |
| Obtain answer $5x - 3y - 4z = -2$, or equivalent | A1 | [5] |
| Using two points on $l$ and one on $m$, or vice versa, state three equations in $a$, $b$, $c$ and $d$ | B1 | OR 1 |
| Solve and obtain one ratio, e.g. $a:b$ | M1 | |
| Obtain a ratio of three of the unknowns, e.g. $a:b:c = -5:3:4$ | A1 | |
| Use coordinates of a relevant point and found ratio to find the fourth unknown, e.g. $d$ | M1 | |
| Obtain answer $-5x + 3y + 4z = 2$, or equivalent | A1 | [5] |
| Form a correct 2-parameter equation for the plane, e.g. $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - \mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ | B1 | OR 2 |
| State three equations in $x$, $y$, $z$, $\lambda$ and $\mu$ | M1 | |
| State three correct equations | A1 | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain answer $5x - 3y - 4z = -2$, or equivalent | A1 | [5] |
| Attempt to calculate vector product of direction vectors of $l$ and $m$ | M1 | OR 3 |
| Obtain two correct components of the product | A1 | |
| Obtain correct product, e.g. $-5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ | A1 | |
| Form a plane equation and use coordinates of a relevant point to calculate $d$ | M1 | |
| Obtain answer $-5x + 3y + 4z = 2$, or equivalent | A1 | [5] |10 The lines $l$ and $m$ have vector equations
$$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$
respectively.\\
(i) Show that $l$ and $m$ intersect.\\
(ii) Calculate the acute angle between the lines.\\
(iii) Find the equation of the plane containing $l$ and $m$, giving your answer in the form $a x + b y + c z = d$.
\hfill \mbox{\textit{CAIE P3 2010 Q10 [12]}}