Question

The number of real values of 'a', for which the system of equations $2x+3y+az = 0$, $x+ay-2z=0$ and $3x+y+3z = 0$ has nontrivial solutions is

• A. 2
• B. 1
• C. 0
• D. Infinity

Hint:

For homogeneous systems, always check $\det(A)=0$ for non-trivial solutions. If the resulting equation has no real roots, then no such real parameter exists.

Answer 1

The Correct Option is C

Step 1: Condition for non-trivial solutions:
A homogeneous system of linear equations admits non-trivial solutions if and only if the determinant of its coefficient matrix is zero.

Step 2: Form the coefficient matrix:
The given system has the coefficient matrix \[ M = \begin{pmatrix} 2 & 3 & a \\ 1 & a & -2 \\ 3 & 1 & 3 \end{pmatrix}. \] Thus, for non-trivial solutions, we require \[ \det(M) = 0. \]
Step 3: Evaluate the determinant:
Expanding along the first row, \[ \det(M) = 2 \begin{vmatrix} a & -2 \\ 1 & 3 \end{vmatrix} - 3 \begin{vmatrix} 1 & -2 \\ 3 & 3 \end{vmatrix} + a \begin{vmatrix} 1 & a \\ 3 & 1 \end{vmatrix}. \] Simplifying each term, \[ 2(3a + 2) - 3(3 + 6) + a(1 - 3a) = 0, \] \[ 6a + 4 - 27 + a - 3a^2 = 0, \] \[ -3a^2 + 7a - 23 = 0. \] Rewriting, \[ 3a^2 - 7a + 23 = 0. \]
Step 4: Check for real solutions:
To determine the number of real values of $a$, compute the discriminant: \[ \Delta = (-7)^2 - 4(3)(23) = 49 - 276 = -227. \] Since $\Delta<0$, the quadratic equation has no real roots.

Step 5: Conclusion:
Hence, there are no real values of $a$ for which the system has non-trivial solutions.
\[ \boxed{0} \]