| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line of intersection of planes |
| Difficulty | Standard +0.3 This is a standard two-part question on planes requiring routine application of formulas: (i) finding angle between planes using dot product of normal vectors, and (ii) finding line of intersection by solving simultaneous equations and expressing parametrically. Both are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.04d Angles: between planes and between line and plane4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Find scalar product of the normals to the planes | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and find \(\cos^{-1}\) of the result | M1 | |
| Obtain \(67.8°\) (or \(1.18\) radians) | A1 | [3] |
| (ii) EITHER Carry out complete method for finding point on line | M1 | |
| Obtain one such point, e.g. \((2, -3, 0)\) or \(\left(-\frac{17}{7}, 0, \frac{6}{7}\right)\) or \((0, -17, -4)\) or \(\ldots\) | A1... | |
| Either State \(3a - b + 2c = 0\) and \(a + b - 4c = 0\) or equivalent | B1 | |
| Attempt to solve for one ratio, e.g. \(a : b\) | M1 | |
| Obtain \(a : b : c = 1 : 7 : 2\) or equivalent | A1 | |
| State a correct final answer, e.g. \(\mathbf{r} = [2, -3, 0] + \lambda[1, 7, 2]\) | A1✱ | |
| Or 1 Obtain a second point on the line | A1 | |
| Subtract position vectors to obtain direction vector | M1 | |
| Obtain \([1, 7, 2]\) or equivalent | A1 | |
| State a correct final answer, e.g. \(\mathbf{r} = [2, -3, 0] + \lambda[1, 7, 2]\) | A1✱ | |
| Or 2 Use correct method to calculate vector product of two normals | M1 | |
| Obtain two correct components | A1 | |
| Obtain \([2, 14, 4]\) or equivalent | A1 | |
| State a correct final answer, e.g. \(\mathbf{r} = [2, -3, 0] + \lambda[1, 7, 2]\) | A1✱ | |
| [✱ is dependent on both M marks in all three cases] | ||
| OR 3 Express one variable in terms of a second variable | M1 | |
| Obtain a correct simplified expression, e.g. \(x = \frac{1}{2}(4+z)\) | A1 | |
| Express the first variable in terms of third variable | M1 | |
| Obtain a correct simplified expression, e.g. \(x = \frac{1}{7}(17+y)\) | A1 | |
| Form a vector equation for the line | M1 | |
| State a correct final answer, e.g. \(\mathbf{r} = [0, -17, -4] + \lambda[1, 7, 2]\) | A1 | |
| OR 4 Express one variable in terms of a second variable | M1 | |
| Obtain a correct simplified expression, e.g. \(z = 2x - 4\) | A1 | |
| Express third variable in terms of the second variable | M1 | |
| Obtain a correct simplified expression, e.g. \(y = 7x - 17\) | A1 | |
| Form a vector equation for the line | M1 | |
| State a correct final answer, e.g. \(\mathbf{r} = [0, -17, -4] + \lambda[1, 7, 2]\) | A1 | [6] |
**(i)** Find scalar product of the normals to the planes | M1 |
Using the correct process for the moduli, divide the scalar product by the product of the moduli and find $\cos^{-1}$ of the result | M1 |
Obtain $67.8°$ (or $1.18$ radians) | A1 | [3]
**(ii)** **EITHER** Carry out complete method for finding point on line | M1 |
Obtain one such point, e.g. $(2, -3, 0)$ or $\left(-\frac{17}{7}, 0, \frac{6}{7}\right)$ or $(0, -17, -4)$ or $\ldots$ | A1... |
**Either** State $3a - b + 2c = 0$ and $a + b - 4c = 0$ or equivalent | B1 |
Attempt to solve for one ratio, e.g. $a : b$ | M1 |
Obtain $a : b : c = 1 : 7 : 2$ or equivalent | A1 |
State a correct final answer, e.g. $\mathbf{r} = [2, -3, 0] + \lambda[1, 7, 2]$ | A1✱ |
**Or 1** Obtain a second point on the line | A1 |
Subtract position vectors to obtain direction vector | M1 |
Obtain $[1, 7, 2]$ or equivalent | A1 |
State a correct final answer, e.g. $\mathbf{r} = [2, -3, 0] + \lambda[1, 7, 2]$ | A1✱ |
**Or 2** Use correct method to calculate vector product of two normals | M1 |
Obtain two correct components | A1 |
Obtain $[2, 14, 4]$ or equivalent | A1 |
State a correct final answer, e.g. $\mathbf{r} = [2, -3, 0] + \lambda[1, 7, 2]$ | A1✱ |
[✱ is dependent on both M marks in all three cases] |
**OR 3** Express one variable in terms of a second variable | M1 |
Obtain a correct simplified expression, e.g. $x = \frac{1}{2}(4+z)$ | A1 |
Express the first variable in terms of third variable | M1 |
Obtain a correct simplified expression, e.g. $x = \frac{1}{7}(17+y)$ | A1 |
Form a vector equation for the line | M1 |
State a correct final answer, e.g. $\mathbf{r} = [0, -17, -4] + \lambda[1, 7, 2]$ | A1 |
**OR 4** Express one variable in terms of a second variable | M1 |
Obtain a correct simplified expression, e.g. $z = 2x - 4$ | A1 |
Express third variable in terms of the second variable | M1 |
Obtain a correct simplified expression, e.g. $y = 7x - 17$ | A1 |
Form a vector equation for the line | M1 |
State a correct final answer, e.g. $\mathbf{r} = [0, -17, -4] + \lambda[1, 7, 2]$ | A1 | [6]6 Two planes have equations $3 x - y + 2 z = 9$ and $x + y - 4 z = - 1$.\\
(i) Find the acute angle between the planes.\\
(ii) Find a vector equation of the line of intersection of the planes.
\hfill \mbox{\textit{CAIE P3 2013 Q6 [9]}}