Question:
The set of values of \(k\) for which the system of simultaneous equations
\[
x+y+kz = 1, \quad 2x+2y-3, \quad x+2y+2kz = k
\]
has no real solution is:
The set of values of \(k\) for which the system of simultaneous equations
\[
x+y+kz = 1, \quad 2x+2y-3, \quad x+2y+2kz = k
\]
has no real solution is:
Show Hint
For simultaneous equations, identify parameters which eliminate variables causing inconsistency; those values correspond to no real solution.
Updated On: Jul 18, 2026
- \(\{0\}\)
- \(\mathbb{R} - \{0\}\)
- \(\{2\}\)
- \(\{-1,0,1\}\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Analyze the system.
For the system to have no real solution, the determinant of the coefficient matrix (if 3x3) or consistency condition must fail.
Here, checking the coefficient of \(z\) in the equations:
- If \(k=0\), \(z\) disappears and the remaining equations may become inconsistent.
Step 2: Check \(k=0\).
- Substituting \(k=0\), system reduces to two equations in \(x\) and \(y\) that cannot satisfy all simultaneously.
Step 3: Conclusion.
Hence, the set of \(k\) for which the system has no real solution is:
\[ \boxed{\{0\}} \]
For the system to have no real solution, the determinant of the coefficient matrix (if 3x3) or consistency condition must fail.
Here, checking the coefficient of \(z\) in the equations:
- If \(k=0\), \(z\) disappears and the remaining equations may become inconsistent.
Step 2: Check \(k=0\).
- Substituting \(k=0\), system reduces to two equations in \(x\) and \(y\) that cannot satisfy all simultaneously.
Step 3: Conclusion.
Hence, the set of \(k\) for which the system has no real solution is:
\[ \boxed{\{0\}} \]
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