Question 7:
--- 7(a) ---
7(a) | Uses an appropriate method for
finding the values of k ( for
example expanding appropriate
determinant) | AO1.1a | M1 | 1 –1 k
k –3 5 =0
1 –2 3
–3 5 k 5 k –3
+ +k =0
–2 3 1 3 1 –2
1 +3k – 5 + k (–2k + 3)=0
−2k2 +6k −4=0
k2 −3k+2=0
(k−2)(k−1)=0
k = 2 or 1
Obtains a quadratic equation in k | AO1.1a | M1
Obtains two correct values for k | AO1.1b | A1
(b) | Selects an appropriate method to
determine the appropriate
geometrical configuration and
substitutes ‘their’ first value of k | AO3.1a | M1 | when k =1
x− y + z =3
x−3y+5z = −1
x−2y+3z = −4
−2y+4z =−4
y−2z =7
y−2z =2; y−2z =7
Hence equations are inconsistent and the
three planes form a prism
when k = 2
x− y+2z =3
2x−3y+5z =−1
x−2y+3z =−4
R −2R :−y+z =−7
2 1
R −R :−y+z =−7
3 1
Hence equations are consistent and the
three planes form a sheaf – they meet in
line
Eliminates one variable or uses
row reduction | AO1.1a | M1
Obtains a contradiction and
makes correct deduction about
the geometric configuration
(must have correct value for k) | AO2.2a | R1
Substitutes ‘their’ 2nd value of k
into selected method to determine
the appropriate geometrical
configuration | AO1.1a | M1
Obtains a consistent set of
equations and makes correct
deduction about geometric
configuration (must have correct
value for k) | AO2.2a | R1
Q | Marking Instructions | AO | Marks | Typical Solution
(c) | Deduces that the planes must
meet in a line and hence that
k = 2 | AO2.2a | R1 | x− y+2z =3
2x−3y+5z =−1
x−2y+3z =−4
⇒−y+z =−7
Let
z =λ
Then
y =λ+7
and
x =3+ y−2z
=3+λ+7−2λ
=−λ+10
Selects method to find
solution: For example, sets
one variable = λ, substitutes
and attempts to find other
variables in terms of λ | AO1.1a | M1 | ALT
 1   1   1 
     
−1 × −2 = −1
     
     
 2   3  −1
x 10  1 
     
y = 7 +λ −1
     
     
z  0  −1 |  1   1   1 
     
−1 × −2 = −1
     
     
 2   3  −1
x 10  1 
     
y = 7 +λ −1
     
     
z  0  −1
Fully states correct solution
CAO | AO1.1b | A1
Total | 11
Q | Marking Instructions | AO | Marks | Typical Solution