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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 408
Question 408 of 472
Medium
MCQ
The ordered pair (a, b), for which the system of linear equations 3x - 2y + z = b 5x - 8y + 9z = 3 2x + y + az = -1 has no solution, is :
A
\[\left( {3,{1 \over 3}} \right)\]
B
\[\left( { - 3,{1 \over 3}} \right)\]
C
\[\left( { - 3, - {1 \over 3}} \right)\]
D
\[\left( {3, - {1 \over 3}} \right)\]
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Official solution
\[\left| {
(
3
−
2
1
5
−
8
9
2
1
a
)
\begin{pmatrix} 3 & { - 2} & 1 \\ 5 & { - 8} & 9 \\ 2 & 1 & a \\ \end{pmatrix}
3
5
2
−
2
−
8
1
1
9
a
} \right| = 0 \Rightarrow - 14a - 42 = 0 \Rightarrow a = - 3\] Now 3 (equation (1)) - (equation (2)) - 2 (equation (3)) is \[3(3x - 2y + z - b) - (5x - 8y + 9z - 3) - 2(2x + y + az + 1) = 0\] \[ \Rightarrow - 3b + 3 - 2 = 0 \Rightarrow b = {1 \over 3}\] So for no solution
a
=
−
3
a = - 3
a
=
−
3
and \[b \ne {1 \over 3}\]
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