JEE MAIN - Mathematics (2022 - 29th June Morning Shift - No. 6)
If the system of linear equations
2x + y $$-$$ z = 7
x $$-$$ 3y + 2z = 1
x + 4y + $$\delta$$z = k, where $$\delta$$, k $$\in$$ R has infinitely many solutions, then $$\delta$$ + k is equal to:
$$-$$3
3
6
9
Explanation
$$2x + y - z = 7$$
$$x - 3y + 2z = 1$$
$$x + 4y + \delta z = k$$
$$\Delta = \left| {\matrix{ 2 & 1 & { - 1} \cr 1 & { - 3} & 2 \cr 1 & 4 & \delta \cr } } \right| = - 7\delta - 21 = 0$$
$$\delta = - 3$$
$${\Delta _1} = \left| {\matrix{ 7 & 1 & { - 1} \cr 1 & { - 3} & 2 \cr k & 4 & { - 3} \cr } } \right|$$
$$ \Rightarrow 6 - k = 0 \Rightarrow k = 6$$
$$\delta + k = - 3 + 6 = 3$$
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