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3 and 4 .Determinants and Matrices
normal
Consider system of equations $ x + y -az = 1$ ; $2x + ay + z = 1$ ; $ax + y -z = 2$
A
for $a \ne 1$ system has unique solution.
B
if system has no solution then $'a'$ must be $1$ .
C
for $a \in \left\{ {1,\frac{{ - 1 \pm \sqrt 5 }}{2}} \right\}$ , system has no solution.
D
for $a = \frac{{ - 1 \pm \sqrt 5 }}{2}$ , system has infinite number of solutions.
Solution
$\Delta=\left|\begin{array}{ccc}{1} & {1} & {-a} \\ {2} & {a} & {1} \\ {a} & {1} & {-1}\end{array}\right|=1(-a-1)-1(-2-a)-a\left(2-a^{2}\right)$
$=a^{3}-2 a+1=(a-1)\left(a^{2}+a-1\right)$
$=(a-1)\left(a-\frac{-1-\sqrt{5}}{2}\right)\left(a-\frac{-1+\sqrt{5}}{2}\right)$
For $\Delta=0$
$a=1, \frac{-1 \pm \sqrt{5}}{2}$
for each value system has no solution.
Standard 12
Mathematics
