JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Matrices

If the system of linear equation

x - 4 y + 7 z = g, 3 y - 5 z = h, - 2 x + 5 y - 9 z = k

is consistent, then:

(A) $g + h + k = 0$
(B) $2 g + h + k = 0$ ✓ Correct
(C) $g + h + 2 k = 0$
(D) $g + 2 h + k = 0$

Answer: (B)

Solution

For the system

x - 4 y + 7 z = g

3 y - 5 z = h

- 2 x + 5 y - 9 z = k

the coefficient determinant is zero, so consistency requires the constants to satisfy the same linear dependence the coefficient rows do. Searching for a combination

λ R_{1} + μ R_{2} + ν R_{3} = 0

of the coefficients gives

2 R_{1} + R_{2} + R_{3} = 0

. The constants must obey the identical combination, so

2 g + h + k = 0

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option

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