Challenging +1.8 This is a Further Maths question requiring determinant calculation of a 3×3 matrix with algebraic entries, setting it equal to zero (singular matrix condition), and solving the resulting equation for x. While the determinant expansion is algebraically intensive and requires careful manipulation to factor/simplify, it follows a standard procedure without requiring novel insight. The 7-mark allocation reflects the computational length rather than exceptional conceptual difficulty.
S is a singular matrix such that
\(\det \mathbf{S} = \begin{vmatrix} a & a & x \\ x-b & a-b & x+1 \\ x^2 & a^2 & ax \end{vmatrix}\)
Express the possible values of \(x\) in terms of \(a\) and \(b\).
[7 marks]
Question 13:
13 | Explains that detM=0 when M is
singular
(Seen anywhere) | AO2.4 | R1 | a a x
S is singular xb ab x10
x2 a2 ax
0 a x
det S xa ab x1
x2 a2 a2 ax
0 a x
(xa) 1 ab x1
xa a2 ax
0 a x
det S(xa) 1 ab x1
xa 0 0
a x
(xa)(xa)
ab x1
(xa)(xa)(abx)
(xa)(xa)(abx)0
a
xa, a,
b
Seeks factor by combining rows or
columns to find a first linear factor for
example C 'C C
1 1 2 | AO3.1a | M1
Extracts first factor correctly | AO1.1b | A1
Combines rows or columns to find a
second linear factor R 'R aR
3 3 1 | AO1.1a | M1
Extracts second factor correctly | AO1.1b | A1
Completes expansion and obtains final
factor | AO1.1b | A1
Deduces correct values of x
FT ‘their’ factors | AO2.2a | A1F
Total | 7
Q | Marking Instructions | AO | Marks | Typical Solution
S is a singular matrix such that
$\det \mathbf{S} = \begin{vmatrix} a & a & x \\ x-b & a-b & x+1 \\ x^2 & a^2 & ax \end{vmatrix}$
Express the possible values of $x$ in terms of $a$ and $b$.
[7 marks]
\hfill \mbox{\textit{AQA Further Paper 2 Q13 [7]}}