Matrices & Determinants
Determinants
Grade Class 12
Question:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|
(A) λ = 2
(B) λ = 3
(C) for a = 2√<span style="text-decoration:overline">3</span>, √<span style="text-decoration:overline">5</span>/b + √<span style="text-decoration:overline">7</span>/c = 2
(D) for c = 2√<span style="text-decoration:overline">7</span>, √<span style="text-decoration:overline">3</span>/a + √<span style="text-decoration:overline">5</span>/b = 2
Step-by-Step Solution
Key Concept: The determinant equation represents a plane in 3D space. Given the condition, the expression simplifies to a constant value related to the intercepts of the plane.
The given determinant is |a sqrt(5) sqrt(7); sqrt(3) b sqrt(7); sqrt(3) sqrt(5) c| = 0. Expanding this, we get a(bc - sqrt(35)) - sqrt(5)(c*sqrt(3) - sqrt(21)) + sqrt(7)(sqrt(15) - b*sqrt(3)) = 0. This simplifies to abc - a*sqrt(35) - c*sqrt(15) + sqrt(105) + sqrt(105) - b*sqrt(21) = 0. Dividing by (a-sqrt(3))(b-sqrt(5))(c-sqrt(7)) leads to the result lambda = 2. Checking options, D is correct.
Correct Answer: D