Moderate -0.3 This is a standard C4 vectors question requiring systematic checking of three cases (parallel/intersect/skew) using direction vectors and solving simultaneous equations. While it involves multiple steps, the procedure is routine and well-practiced, making it slightly easier than average for A-level.
Set up 3 equations: \(1+2\lambda=\mu-1\); \(-\lambda=5-\mu\); \(3+5\lambda=2-5\mu\)
M1
Minor errors only accepted; must be consistent
Attempt to find \(\lambda\) or \(\mu\) from 2 equations then find \(\mu\) or \(\lambda\) from any of the 3
M1
Or find \(\lambda\) from (i)(ii) then from (ii)(iii); inconsistency dep*A1 also awarded
\((\lambda,\mu) = (3,8)\) or \((-2\frac{3}{5}, 2\frac{2}{5})\) or \((-\frac{11}{15}, \frac{8}{15})\) etc.
A1
Accept equivalent proper/improper fractional or decimal values
Substitute correct values into a correct equation (not the immediate last one used) to demonstrate inconsistency
M1
e.g. after \((3,8)\), sub in (iii): \(3+5\times3 \neq 2-5\times8\)
State "skew"
A1
Dependent on \(3 \times M1 + A1\)
Identify direction vectors; state not identical/multiples; evaluate \(\cos(\text{angle})\) and state \(\neq \pm1\); state "not parallel"
B1
dvs must be identified: \(\begin{pmatrix}2\\-1\\5\end{pmatrix}\) and \(\begin{pmatrix}1\\-1\\-5\end{pmatrix}\)
# Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Set up 3 equations: $1+2\lambda=\mu-1$; $-\lambda=5-\mu$; $3+5\lambda=2-5\mu$ | M1 | Minor errors only accepted; must be consistent |
| Attempt to find $\lambda$ or $\mu$ from 2 equations then find $\mu$ or $\lambda$ from any of the 3 | M1 | Or find $\lambda$ from (i)(ii) then from (ii)(iii); inconsistency dep*A1 also awarded |
| $(\lambda,\mu) = (3,8)$ or $(-2\frac{3}{5}, 2\frac{2}{5})$ or $(-\frac{11}{15}, \frac{8}{15})$ etc. | A1 | Accept equivalent proper/improper fractional or decimal values |
| Substitute correct values into a correct equation (not the immediate last one used) to demonstrate inconsistency | M1 | e.g. after $(3,8)$, sub in (iii): $3+5\times3 \neq 2-5\times8$ |
| State "skew" | A1 | Dependent on $3 \times M1 + A1$ |
| Identify direction vectors; state not identical/multiples; evaluate $\cos(\text{angle})$ and state $\neq \pm1$; state "not parallel" | B1 | dvs must be identified: $\begin{pmatrix}2\\-1\\5\end{pmatrix}$ and $\begin{pmatrix}1\\-1\\-5\end{pmatrix}$ |
---