JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
Consider the system of linear equations \(-x+y+2 z=0\) \(3 x-a y+5 z=1\) \(2 x-2 y-a z=7\) Let \(S_{1}\) be the set of all \(\mathrm{a} \in {R}\) for which the system is inconsistent and \(S_{2}\) be the set of all \(a \in {R}\) for which the system has infinitely many solutions. If \(n\left(S_{1}\right)\) and \(n\left(S_{2}\right)\) denote the number of elements in \(S_{1}\) and \(\mathrm{S}_{2}\) respectively, then
- A \(\mathrm{n}\left(\mathrm{S}_{1}\right)=2, \mathrm{n}\left(\mathrm{S}_{2}\right)=2\)
- B \(\mathrm{n}\left(\mathrm{S}_{1}\right)=1, \mathrm{n}\left(\mathrm{S}_{2}\right)=0\)
- C \(\mathrm{n}\left(\mathrm{S}_{1}\right)=2, \mathrm{n}\left(\mathrm{S}_{2}\right)=0\)
- D \(\mathrm{n}\left(\mathrm{S}_{1}\right)=0, \mathrm{n}\left(\mathrm{S}_{2}\right)=2\)
Answer & Solution
Correct Answer
(C) \(\mathrm{n}\left(\mathrm{S}_{1}\right)=2, \mathrm{n}\left(\mathrm{S}_{2}\right)=0\)
Step-by-step Solution
Detailed explanation
\(\Delta=\left|\begin{array}{ccc}-1 & 1 & 2 \\ 3 & -a & 5 \\ 2 & -2 & -a\end{array}\right|\) \(=-1\left(a^{2}+10\right)-1(-3 a-10)+2(-6+2 a)\) \(=-a^{2}-10+3 a+10-12+4 a\) \(\Delta=-a^{2}+7 a-12\) \(\Delta=-\left[a^{2}-7 a+12\right]\) \(\Delta=-[(a-3)(a-4)]\)…
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