AP EAMCET · Maths · Determinants

If \([x]\) is the greatest integer less than or equal to \(x\) and \(|x|\) is the modulus of \(x\), then the system of three equations
\(\begin{aligned} & 2 x+3|y|+5[z]=0, x+|y|-2[z]=4, \\ & x+|y|+[z]=1 \text { has } \end{aligned}\)

A a unique solution
B finitely many solutions
C infinitely many solutions
D no solution

Verified Solution

Answer & Solution

Correct Answer

(C) infinitely many solutions

Step-by-step Solution

Detailed explanation

Given system of three equations \(\begin{aligned} 2 x+3|y|+5[z] & =0 \\ x+|y|-2[z] & =4 \\ x+|y|+[z] & =1 \end{aligned}\) and According to Cramer's rule, \(x=\frac{\Delta_1}{\Delta},|y|=\frac{\Delta_2}{\Delta} \text { and }[z]=\frac{\Delta_3}{\Delta}\) where,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If \([x]\) is the greatest integer less than or equal to \(x\) and \(|x|\) is the modulus of \(x\), then the system of three equations\(\begin{aligned} & 2 x+3|y|+5[z]=0, x+|y|-2[z]=4, \\ & x+|y|+[z]=1 \text { has } \end{aligned}\)

Step-by-step Solution

If \([x]\) is the greatest integer less than or equal to \(x\) and \(|x|\) is the modulus of \(x\), then the system of three equations
\(\begin{aligned} & 2 x+3|y|+5[z]=0, x+|y|-2[z]=4, \\ & x+|y|+[z]=1 \text { has } \end{aligned}\)