| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Angle between line and plane |
| Difficulty | Standard +0.8 This is a multi-part vectors question requiring understanding of line-plane relationships, skew/intersecting lines, and angle calculations. Part (i) is standard (perpendicularity condition), part (ii) requires solving simultaneous equations for line intersection, and part (iii) involves inverse application of the angle formula with trigonometric manipulation. The combination of three distinct techniques and the non-routine angle condition in part (iii) places this above average difficulty. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Equate scalar product of direction vector of \(l\) and \(p\) to zero | M1 | |
| Solve for \(a\) and obtain \(a = -6\) | A1 | [2 marks] |
| (ii) Express general point of \(l\) correctly in parametric form, e.g. \(3i + 2j + k + \mu(2i + j + 2k)\) or \((1 - \mu)(3i + 2j + k) + \mu(i + j - k)\) | B1 | |
| Equate at least two pairs of corresponding components of \(l\) and the second line and solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain either \(\lambda = \frac{2}{3}\) or \(\mu = \frac{1}{3}\); or \(\lambda = \frac{2}{a-1}\) or \(\mu = \frac{1}{a-1}\); or reach \(\lambda(a - 4) = 0\) or \((1 + \mu)(a - 4) = 0\) | A1 | |
| Obtain \(a = 4\) having ensured (if necessary) that all three component equations are satisfied | A1 | [4 marks] |
| (iii) Using the correct process for the moduli, divide scalar product of direction vector if \(l\) and normal to \(p\) by the product of their moduli and equate to the sine of the given angle, or form an equivalent horizontal equation | M1* | |
| Use \(\frac{2}{\sqrt{5}}\) as sine of the angle | A1 | |
| State equation in any form, e.g. \(\frac{a + 6}{\sqrt{(a^2 + 4 + 1)\sqrt{l + 4 + 4}}} = \frac{2}{\sqrt{5}}\) | A1 | |
| Solve for \(a\) | M1 (dep*) | |
| Obtain answers for \(a = 0\) and \(a = \frac{60}{31}\), or equivalent | A1 | [5 marks total] |
**(i)** Equate scalar product of direction vector of $l$ and $p$ to zero | M1 |
Solve for $a$ and obtain $a = -6$ | A1 | [2 marks]
**(ii)** Express general point of $l$ correctly in parametric form, e.g. $3i + 2j + k + \mu(2i + j + 2k)$ or $(1 - \mu)(3i + 2j + k) + \mu(i + j - k)$ | B1 |
Equate at least two pairs of corresponding components of $l$ and the second line and solve for $\lambda$ or for $\mu$ | M1 |
Obtain either $\lambda = \frac{2}{3}$ or $\mu = \frac{1}{3}$; or $\lambda = \frac{2}{a-1}$ or $\mu = \frac{1}{a-1}$; or reach $\lambda(a - 4) = 0$ or $(1 + \mu)(a - 4) = 0$ | A1 |
Obtain $a = 4$ having ensured (if necessary) that all three component equations are satisfied | A1 | [4 marks]
**(iii)** Using the correct process for the moduli, divide scalar product of direction vector if $l$ and normal to $p$ by the product of their moduli and equate to the sine of the given angle, or form an equivalent horizontal equation | M1* |
Use $\frac{2}{\sqrt{5}}$ as sine of the angle | A1 |
State equation in any form, e.g. $\frac{a + 6}{\sqrt{(a^2 + 4 + 1)\sqrt{l + 4 + 4}}} = \frac{2}{\sqrt{5}}$ | A1 |
Solve for $a$ | M1 (dep*) |
Obtain answers for $a = 0$ and $a = \frac{60}{31}$, or equivalent | A1 | [5 marks total]
**Guidance:** Allow use of the cosine of the angle to score M1M1.10 The line $l$ has equation $\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$, where $a$ is a constant. The plane $p$ has equation $x + 2 y + 2 z = 6$. Find the value or values of $a$ in each of the following cases.\\
(i) The line $l$ is parallel to the plane $p$.\\
(ii) The line $l$ intersects the line passing through the points with position vectors $3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }$ and $\mathbf { i } + \mathbf { j } - \mathbf { k }$.\\
(iii) The acute angle between the line $l$ and the plane $p$ is $\tan ^ { - 1 } 2$.
\hfill \mbox{\textit{CAIE P3 2013 Q10 [11]}}