Project ForgeThe system of linear equations in real (x,y) given by (x y)[[2, 5-2α],[α, 1]] = (0 0) involves a real parameter α and has infinitely many non-trivial solutions for special value(s) of α. Which one or more among the following options is/are non-trivial solution(s) of (x,y) for such special value(s) of α?
- x=2, y=−2
- x=−1, y=4
- Cx=1, y=1
- Dx=4, y=−2
Explanation
Writing out the two equations (2x+αy=0 and (5-2α)x+y=0), infinitely many non-trivial solutions require the determinant 2(1)-(5-2α)α = 2α²-5α+2 = 0, giving α=2 or α=0.5. For α=2: 2x+2y=0 → x=-y, so (x,y)=(2,-2) works — option (A). For α=0.5: 2x+0.5y=0 → y=-4x, so (x,y)=(-1,4) works — option (B). Neither (C) x=1,y=1 nor (D) x=4,y=-2 satisfies either relation.
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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