| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Standard +0.3 This is a standard C4 vectors question with routine procedures: (i) angle between lines using dot product formula (bookwork), (ii) showing lines are skew by solving simultaneous equations (standard technique), (iii) finding parameter value for intersection (straightforward algebra). All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Work with correct pair of direction vectors | M1 | Of any two 3x3 vectors rel to question |
| Demonstrate correct method for finding scalar product | M1 | Of any vector relevant to question |
| Demonstrate correct method for finding modulus | M1 | Of any vector relevant to question |
| 24, 24.0 (24.006363...) (degrees) 0.419 (0.418999...) (rad) | A1 4 | Mark earliest value, allow trunc/rounding |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to set up 3 equations | M1 | Of type \(3 + 2s = 5, 3x = 3 + t, -2 = 4x = 2 - 2t\) |
| Find correct values of \((s,t) = (1,0)\) or \((1,4)\) or \((5,12)\) | A1 | Or 2 diff values of \(s\) (or of \(t\)) |
| Substitute their \((s,t)\) into equation not used | M1 | and make a relevant deduction |
| Correctly demonstrate failure | A1 4 | dep on all 3 prev marks |
| Answer | Marks | Guidance |
|---|---|---|
| Subst their \((s,t)\) from first 2 eqns into new 3rd eqn | M1 | New 3rd eqn of type \(a - 4x = 2 - 2t\) |
| \(a = 6\) | A1 2 |
**(i)**
Work with correct pair of direction vectors | M1 | Of any two 3x3 vectors rel to question
Demonstrate correct method for finding scalar product | M1 | Of any vector relevant to question
Demonstrate correct method for finding modulus | M1 | Of any vector relevant to question
24, 24.0 (24.006363...) (degrees) 0.419 (0.418999...) (rad) | A1 4 | Mark earliest value, allow trunc/rounding
**(ii)**
Attempt to set up 3 equations | M1 | Of type $3 + 2s = 5, 3x = 3 + t, -2 = 4x = 2 - 2t$
Find correct values of $(s,t) = (1,0)$ or $(1,4)$ or $(5,12)$ | A1 | Or 2 diff values of $s$ (or of $t$)
Substitute their $(s,t)$ into equation not used | M1 | and make a relevant deduction
Correctly demonstrate failure | A1 4 | dep on all 3 prev marks
**(iii)**
Subst their $(s,t)$ from first 2 eqns into new 3rd eqn | M1 | New 3rd eqn of type $a - 4x = 2 - 2t$
$a = 6$ | A1 2 |
---6 The line $l _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 3 \\ 0 \\ - 2 \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)$. The line $l _ { 2 }$ has equation $\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right)$.\\
(i) Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$.\\
(ii) Show that $l _ { 1 }$ and $l _ { 2 }$ are skew.\\
(iii) One of the numbers in the equation of line $l _ { 1 }$ is changed so that the equation becomes $\mathbf { r } = \left( \begin{array} { l } 3 \\ 0 \\ a \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)$. Given that $l _ { 1 }$ and $l _ { 2 }$ now intersect, find $a$.
\hfill \mbox{\textit{OCR C4 2011 Q6 [10]}}