If the system of equations , and has infinite solutions, then the value of is
- A.
- B.
- C.
- D.no real values
Show answer and solution
Correct option: A
Solution
For infinite solutions, the determinant of the coefficient matrix must be zero.
The matrix is:
|1 a 0|
|0 1 a|
|a 0 1|
det = 1(1·1 - a·0) - a(0·1 - a·a) + 0 = 1 - a(−a²) = 1 + a³
Setting 1 + a³ = 0 gives a³ = −1, so a = −1.
Verify: with a = −1, all three equations become x − y = 0, −z + y = 0, −x + z = 0, which are consistent and dependent (infinite solutions). Thus a = −1.
The matrix is:
|1 a 0|
|0 1 a|
|a 0 1|
det = 1(1·1 - a·0) - a(0·1 - a·a) + 0 = 1 - a(−a²) = 1 + a³
Setting 1 + a³ = 0 gives a³ = −1, so a = −1.
Verify: with a = −1, all three equations become x − y = 0, −z + y = 0, −x + z = 0, which are consistent and dependent (infinite solutions). Thus a = −1.
Previous-year question, JEE Main.