JEE MAIN - Mathematics (2020 - 8th January Morning Slot - No. 10)
For which of the following ordered pairs ($$\mu $$, $$\delta $$),
the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = $$\mu $$
4x + 4y + 4z = $$\delta $$
is inconsistent ?
x + 2y + 3z = 1
3x + 4y + 5z = $$\mu $$
4x + 4y + 4z = $$\delta $$
is inconsistent ?
(1, 0)
(4, 3)
(4, 6)
(3, 4)
Explanation
For inconsistent system we need
$$\Delta $$ = 0 and atleast one of $$\Delta $$x, $$\Delta $$y, $$\Delta $$z $$ \ne $$ 0
$$ \therefore $$ $$\Delta $$ = $$\left| {\matrix{ 1 & 2 & 3 \cr 3 & 4 & 5 \cr 4 & 4 & 4 \cr } } \right|$$ = 0
$$\Delta $$x = $$\left| {\matrix{ 1 & 2 & 3 \cr \mu & 4 & 5 \cr \delta & 4 & 4 \cr } } \right|$$
= (-4) - 2($$\mu $$ - 5$$\delta $$) + 3(4$$\mu $$ - 4$$\delta $$)
$$ \Rightarrow $$ 2$$\mu $$ $$ \ne $$ $$\delta $$ + 2 ....(1)
Only ($$\mu $$, $$\delta $$) = (4, 3) does satisfy the equation (1).
$$\Delta $$ = 0 and atleast one of $$\Delta $$x, $$\Delta $$y, $$\Delta $$z $$ \ne $$ 0
$$ \therefore $$ $$\Delta $$ = $$\left| {\matrix{ 1 & 2 & 3 \cr 3 & 4 & 5 \cr 4 & 4 & 4 \cr } } \right|$$ = 0
$$\Delta $$x = $$\left| {\matrix{ 1 & 2 & 3 \cr \mu & 4 & 5 \cr \delta & 4 & 4 \cr } } \right|$$
= (-4) - 2($$\mu $$ - 5$$\delta $$) + 3(4$$\mu $$ - 4$$\delta $$)
$$ \Rightarrow $$ 2$$\mu $$ $$ \ne $$ $$\delta $$ + 2 ....(1)
Only ($$\mu $$, $$\delta $$) = (4, 3) does satisfy the equation (1).
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