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NDA 2019 Maths · Mathematics · Determinants · medium

The homogeneous system x+y+z=0, x+2y+3z=0, 2x+3y+kz=0 has a non-trivial solution for k equal to

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The homogeneous system x+y+z=0x+y+z=0, x+2y+3z=0x+2y+3z=0, 2x+3y+kz=02x+3y+kz=0 has a non-trivial solution for kk equal to
  1. A. 55
  2. B. 44
  3. C. 33
  4. D. 66
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Answer: (B) 44

A homogeneous system has non-trivial solutions iff the determinant of the coefficients is zero. Expanding, ∣11112323k∣=1(2k−9)−1(k−6)+1(3−4)=k−4\begin{vmatrix}1&1&1\\1&2&3\\2&3&k\end{vmatrix}=1(2k-9)-1(k-6)+1(3-4)=k-4. So k=4k=4. This also follows by noticing that the third equation is the sum of the first two when k=4k=4.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake