Standard +0.8 This is a Further Maths question requiring understanding that a sheaf means all three planes pass through a common line. Students must find the line of intersection of two planes, then ensure the third plane contains this line by substituting parametric equations and solving for k and l. This involves multiple conceptual steps beyond routine plane intersection problems, but follows a systematic approach once the sheaf condition is understood.
15 The equations of three planes are
$$\begin{aligned}
- 4 x + k y + 7 z & = 4 \\
x - 2 y + 5 z & = 1 \\
2 x + 3 y + z & = 2
\end{aligned}$$
Given that the planes form a sheaf, determine the values of \(k\) and \(l\).
Question 15:
15 | − 4 k 7
1 − 2 5 = − 4 ( − 1 7 ) − k ( − 9 ) + 7 7
2 3 1
=117+9k
so det = 0 when k = −13
− 4 xxx −−+ 1 323 yyy +++ 7 z === 4l2 (1( )))
5 z 2
2 z ( 3
(1)+2(3): −7y+9z=8
(3)−2(2): 7y−9z=2−2l
so 2l−2=8l=5 | M1
M1
A1
M1
M1
A1
[6] | 3.1a
1.1
1.1
1.1
1.1
1.1 | finding det of matrix of coeffs
setting det = 0
k = −13
finding eqn in 2 variables
finding 2nd eqn in 2 variables
l = 5 | or B2 for use of linear
dependency to find k
or
M2 for 2(2) − 3(3)
Alternative method
15 The equations of three planes are
$$\begin{aligned}
- 4 x + k y + 7 z & = 4 \\
x - 2 y + 5 z & = 1 \\
2 x + 3 y + z & = 2
\end{aligned}$$
Given that the planes form a sheaf, determine the values of $k$ and $l$.
\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q15 [6]}}