Standard +0.3 Part (i) is a routine 3×3 system solved by Gaussian elimination or matrices—standard Further Maths fare. Part (ii) requires understanding that infinite solutions occur when the third equation is a linear combination of the first two, which is a conceptual step up but still a well-practiced technique in FP1. Overall slightly easier than average due to the straightforward setup and clear path to solution.
Find the solution to the following simultaneous equations.
$$\begin{array} { r r r }
x + y + & z = & 3 \\
2 x + 4 y + 5 z = & 9 \\
7 x + 11 y + 12 z = & 20
\end{array}$$
Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations.
$$\begin{array} { r r r l }
x + & y + & z = & 3 \\
2 x + & 4 y + & 5 z = & 9 \\
7 x + & 11 y + & p z = & k
\end{array}$$
(cid:22)(cid:68)(cid:3)(cid:14)(cid:3)(cid:28)(cid:69)(cid:3)(cid:32)(cid:3)k (cid:159) k = 27
M1
M1
eA1
E1
M1
A1
Answer
Marks
[6]
1.1
i
2.1
c
1.1
2.2a
3.1a
Answer
Marks
1.1
n
calculating det M
e
m
setting their 2p – 26 equal to 0 and
solving
conclude that p = 13
Deducing R = 3R + 2R
3 1 2
Answer
Marks
27 from correct reasoning
OR
M1 state that singular matrix
requires each row to be a
combination of the other two
M1 state that R = 3R + 2R
3 1 2
or equivalent
A1 conclude that p = 13
8(i)
n
8(ii)
e
m
i
c
e
p
S
Question 8:
8 | (i) | (cid:16)1
(cid:167)x(cid:183) (cid:167)1 1 1 (cid:183) (cid:167) 3 (cid:183)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
y (cid:32) 2 4 5 9
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
z 7 11 12 20
(cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185)
x = 5, y = –9, z = 7 | M1
A1
[2] | 1.2
1.1 | BC or in vector form | OR elimination method
M1 reduce to two equations in
two unknowns
8 | (ii) | (cid:167)1 1 1(cid:183)
(cid:168) (cid:184)
M(cid:32) 2 4 5
(cid:168) (cid:184)
(cid:168) (cid:184)
(cid:169) 7 11 p(cid:185)
Det M(cid:32)4p(cid:14)35(cid:14)22(cid:16)28(cid:16)2p(cid:16)55
M is singular, so 2p – 26 = 0
(cid:159) p = 13
The third equation must be a linear
combination of the first two p
(cid:68)(cid:3)(cid:14)(cid:3)(cid:21)(cid:69)(cid:3)(cid:32)(cid:3)(cid:26)(cid:15)(cid:3)(cid:68)(cid:3)(cid:14)(cid:3)(cid:23)(cid:69)(cid:3)(cid:32)(cid:3)(cid:20)(cid:20)(cid:15)(cid:3)(cid:68)(cid:3)(cid:14)(cid:3)(cid:24)(cid:69)(cid:3)(cid:32)(cid:3)(cid:20)(cid:22)(cid:3)
(cid:159)(cid:3)(cid:68)(cid:3)(cid:32)(cid:3)(cid:22)(cid:15)(cid:3)(cid:69)(cid:3)(cid:32)(cid:3)(cid:21)(cid:3)
(cid:22)(cid:68)(cid:3)(cid:14)(cid:3)(cid:28)(cid:69)(cid:3)(cid:32)(cid:3)k (cid:159) k = 27 | M1
M1
eA1
E1
M1
A1
[6] | 1.1
i
2.1
c
1.1
2.2a
3.1a
1.1 | n
calculating det M
e
m
setting their 2p – 26 equal to 0 and
solving
conclude that p = 13
Deducing R = 3R + 2R
3 1 2
27 from correct reasoning | OR
M1 state that singular matrix
requires each row to be a
combination of the other two
M1 state that R = 3R + 2R
3 1 2
or equivalent
A1 conclude that p = 13
--- 8(i) ---
8(i)
n
--- 8(ii) ---
8(ii)
e
m
i
c
e
p
S
8 (i) Find the solution to the following simultaneous equations.
$$\begin{array} { r r r }
x + y + & z = & 3 \\
2 x + 4 y + 5 z = & 9 \\
7 x + 11 y + 12 z = & 20
\end{array}$$
(ii) Determine the values of $p$ and $k$ for which there are an infinity of solutions to the following simultaneous equations.
$$\begin{array} { r r r l }
x + & y + & z = & 3 \\
2 x + & 4 y + & 5 z = & 9 \\
7 x + & 11 y + & p z = & k
\end{array}$$
\hfill \mbox{\textit{OCR Further Pure Core 1 Q8 [8]}}