Question
(a) Find the set of values of k for which the following system of equations has
no solution.
x + 2y − 3z = k
3x + y + 2z = 4
5x + 7z = 5
(b) Describe the geometrical relationship of the three planes represented by this system of equations.
▶️Answer/Explanation
Markscheme
(a)
\(\left( {\begin{array}{*{20}{c}}
1&2&{ – 3}&k \\
3&1&2&4 \\
5&0&7&5
\end{array}} \right)\) M1
\({R_1} – 2{R_2}\)
\(\left( {\begin{array}{*{20}{c}}
{ – 5}&0&{ – 7}&{k – 8} \\
3&1&2&4 \\
5&0&7&5
\end{array}} \right)\) (A1)
\({R_1} + {R_3}\)
\(\left( {\begin{array}{*{20}{c}}
0&0&0&{k – 3} \\
3&1&2&4 \\
5&0&7&5
\end{array}} \right)\) (A1)
Hence no solutions if \(k \in \mathbb{R}\), \(k \ne 3\) A1
(b) Two planes meet in a line and the third plane is parallel to that line.
[5 marks]
Examiners report
Most candidates realised that some form of row operations was appropriate here but arithmetic errors were fairly common. Many candidates whose arithmetic was correct gave their answer as k = 3 instead of \(k \ne 3\) . Very few candidates gave a correct answer to (b) with most failing to realise that stating that there was no common point was not enough to answer the question.
Question
The system of equations
\[2x – y + 3z = 2\]
\[3x + y + 2z = – 2\]
\[ – x + 2y + az = b\]
is known to have more than one solution. Find the value of a and the value of b.
▶️Answer/Explanation
Markscheme
EITHER
using row reduction (or attempting to eliminate a variable) M1
\(\left( {\begin{array}{*{20}{ccc|c}}
2&{ – 1}&3&2 \\
3&1&2&{ – 2} \\
{ – 1}&2&a&b
\end{array}} \right)\begin{array}{*{20}{l}}
{} \\
{ \to 2R2 – 3R1} \\
{ \to 2R3 + R1}
\end{array}\)
\(\left( {\begin{array}{*{20}{ccc|c}}
2&{ – 1}&3&2 \\
0&5&{ – 5}&{ – 10} \\
0&3&{2a + 3}&{2b + 2}
\end{array}} \right)\begin{array}{*{20}{l}}
{} \\
{ \to R2/5} \\
{}
\end{array}\) A1
Note: For an algebraic solution award A1 for two correct equations in two variables.
\(\left( {\begin{array}{*{20}{ccc|c}}
2&{ – 1}&3&2 \\
0&1&{ – 1}&{ – 2} \\
0&3&{2a + 3}&{2b + 2}
\end{array}} \right)\begin{array}{*{20}{l}}
{} \\
{} \\
{ \to R3 – 3R2}
\end{array}\)
\(\left( {\begin{array}{*{20}{ccc|c}}
2&{ – 1}&3&2 \\
0&1&{ – 1}&{ – 2} \\
0&0&{2a + 6}&{2b + 8}
\end{array}} \right)\)
Note: Accept alternative correct row reductions.
recognition of the need for 4 zeroes M1
so for multiple solutions a = – 3 and b = – 4 A1A1
[5 marks]
OR
\(\left| {\begin{array}{*{20}{c}}
2&{ – 1}&3 \\
3&1&2 \\
{ – 1}&2&a
\end{array}} \right| = 0\) M1
\( \Rightarrow 2(a – 4) + (3a + 2) + 3(6 + 1) = 0\)
\( \Rightarrow 5a + 15 = 0\)
\( \Rightarrow a = – 3\) A1
\(\left| {\begin{array}{*{20}{c}}
2&{ – 1}&2 \\
3&1&{ – 2} \\
{ – 1}&2&b
\end{array}} \right| = 0\) M1
\( \Rightarrow 2(b + 4) + (3b – 2) + 2(6 + 1) = 0\) A1
\( \Rightarrow 5b + 20 = 0\)
\( \Rightarrow b = – 4\) A1
[5 marks]
Examiners report
Many candidates attempted an algebraic approach that used excessive time but still allowed few to arrive at a solution. Of those that recognised the question should be done by matrices, some were unaware that for more than one solution a complete line of zeros is necessary.
Question
The three planes
\(2x – 2y – z = 3\)
\(4x + 5y – 2z = – 3\)
\(3x + 4y – 3z = – 7\)
intersect at the point with coordinates (a, b, c).
a.Find the value of each of a, b and c.[2]
The equations of three planes are
\(2x – 4y – 3z = 4\)
\( – x + 3y + 5z = – 2\)
\(3x – 5y – z = 6\).
Find a vector equation of the line of intersection of these three planes.[4]
▶️Answer/Explanation
Markscheme
(a) use GDC or manual method to find a, b and c (M1)
obtain \(a = 2,{\text{ }}b = – 1,{\text{ }}c = 3\) (in any identifiable form) A1
[2 marks]
use GDC or manual method to solve second set of equations (M1)
obtain \(x = \frac{{4 – 11t}}{2};{\text{ }}y = \frac{{ – 7t}}{2};{\text{ }}z = t\) (or equivalent) (A1)
\(r = \left( {\begin{array}{*{20}{c}}
2 \\
0 \\
0
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
{ – 5.5} \\
{ – 3.5} \\
1
\end{array}} \right)\) (accept equivalent vector forms) M1A1
Note: Final A1 requires r = or equivalent.
[4 marks]
Examiners report
Generally well done.
Moderate success here. Some forgot that an equation must have an = sign.