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Question

The system of equations x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 3y - 4z = 0 possess a non-trivial solution over the set of rationals, then 2k is an integral element of the interval
  1. (20, 30)
  2. (40, 50)
  3. [30, 40]
  4. [10, 20]


Solution

The correct option is C [30, 40]
For the given system to have a non-trival solution, we must have
∣ ∣1k33k2234∣ ∣=0∣ ∣1k302k11032k10∣ ∣=0
[Applying R2R23R1,R3R32R1]
20k +11 (3-2k) = 0 k=332

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