Standard +0.3 This is a straightforward Further Maths question on solving simultaneous equations in three variables with a parameter. Part (i) requires systematic elimination (standard technique), yielding a solution in terms of 'a'. Part (ii) is a simple check of whether x=y=0 is possible. While it's Further Maths content, the algebraic manipulation is routine and methodical with no conceptual subtlety, making it slightly easier than average overall.
Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\).
$$x - 2y - z = 6$$
$$3x + y + 5z = -4$$
$$-4x + 2y - 3z = a$$
[4]
Determine whether the intersection of the three planes could be on the \(z\)-axis. [2]
So coordinates (46(cid:14)9a,38(cid:14)8a,(cid:16)36(cid:16)7a)
M1
M1
A1
I
A1
C
Answer
Marks
[4]
1.1a
1.1
1.1
M
Answer
Marks
2.5
Implied used of matrices
N
Attempt to use inverse matrix
BC
E
All three elements correct.
Answer written as coordinates,
with at least one element
Answer
Marks
correct.
Alternate Method
M1 attempt at elimination of
one variable
M1 attempt at elimination of
second variable
Answer
Marks
Guidance
4
(ii)
E
46
46(cid:14)9a(cid:32)0 so a(cid:32)(cid:16)
9 P
(cid:167) 46(cid:183)
38(cid:14)8(cid:117) (cid:168) (cid:16) (cid:184) (cid:122)0 so thSere is no intersection on the
(cid:169) 9 (cid:185)
z-axis as both x and y coordinates would need to be
Answer
Marks
zero
M1
A1
Answer
Marks
[2]
3.1a
2.1
Find value of a for x or
y-coordinate to be zero
Complete clear argument with
correct conclusion (allow FT
Answer
Marks
from their (i))
Alternate method
M1 x(cid:32) y(cid:32)0 so z(cid:32)(cid:16)6 on first
plane
A1 z(cid:32)(cid:16)0.8 on second plane so
no intersection on z-axis.
Alternate Method
M1 attempt at elimination of
one variable
M1 attempt at elimination of
second variable
Question 4:
4 | (i) | (cid:170) 1 (cid:16)2 (cid:16)1(cid:186)(cid:170)x(cid:186) (cid:170) 6 (cid:186)
(cid:171) (cid:187)(cid:171) (cid:187) (cid:171) (cid:187)
3 1 5 y (cid:32) (cid:16)4
(cid:171) (cid:187)(cid:171) (cid:187) (cid:171) (cid:187)
(cid:171) (cid:172)(cid:16)4 2 (cid:16)3(cid:187) (cid:188) (cid:171) (cid:172)z(cid:187) (cid:188) (cid:171) (cid:172) a (cid:187) (cid:188)
(cid:170)x(cid:186) (cid:170) 13 8 9 (cid:186)(cid:170) 6 (cid:186)
(cid:171) (cid:187) (cid:171) (cid:187)(cid:171) (cid:187)
(cid:159) y (cid:32) 11 7 8 (cid:16)4
(cid:171) (cid:187) (cid:171) (cid:187)(cid:171) (cid:187)
(cid:171) (cid:172)z(cid:187) (cid:188) (cid:171) (cid:172)(cid:16)10 (cid:16)6 (cid:16)7(cid:187) (cid:188) (cid:171) (cid:172) a (cid:187) (cid:188)
(cid:170) 46(cid:14)9a (cid:186)
(cid:171) (cid:187)
= 38(cid:14)8a
(cid:171) (cid:187)
(cid:171) (cid:172)(cid:16)36(cid:16)7a(cid:187)
(cid:188)
So coordinates (46(cid:14)9a,38(cid:14)8a,(cid:16)36(cid:16)7a) | M1
M1
A1
I
A1
C
[4] | 1.1a
1.1
1.1
M
2.5 | Implied used of matrices
N
Attempt to use inverse matrix
BC
E
All three elements correct.
Answer written as coordinates,
with at least one element
correct. | Alternate Method
M1 attempt at elimination of
one variable
M1 attempt at elimination of
second variable
4 | (ii) | E
46
46(cid:14)9a(cid:32)0 so a(cid:32)(cid:16)
9 P
(cid:167) 46(cid:183)
38(cid:14)8(cid:117) (cid:168) (cid:16) (cid:184) (cid:122)0 so thSere is no intersection on the
(cid:169) 9 (cid:185)
z-axis as both x and y coordinates would need to be
zero | M1
A1
[2] | 3.1a
2.1 | Find value of a for x or
y-coordinate to be zero
Complete clear argument with
correct conclusion (allow FT
from their (i)) | Alternate method
M1 x(cid:32) y(cid:32)0 so z(cid:32)(cid:16)6 on first
plane
A1 z(cid:32)(cid:16)0.8 on second plane so
no intersection on z-axis.
Alternate Method
M1 attempt at elimination of
one variable
M1 attempt at elimination of
second variable
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates of the point where the following three planes intersect. Give your answers in terms of $a$.
$$x - 2y - z = 6$$
$$3x + y + 5z = -4$$
$$-4x + 2y - 3z = a$$
[4]
\item Determine whether the intersection of the three planes could be on the $z$-axis. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS Q4 [6]}}