Question

The values of a and b, for which the system of equations $2x + 3y + 6z = 8$ $x + 2y + az = 5$ $3x + 5y + 9z = b$ has no solution, are :

• A. $a \neq 3, b = 3$
• B. $a = 3, b \neq 13$
• C. $a = 3, b = 13$
• D. $a \neq 3, b \neq 13$

Hint:

Always check if one row of the matrix is a linear combination of others. Here, Row 1 + Row 2 = Row 3 for the coefficients. This immediately tells you that $\Delta = 0$ and the solution depends entirely on whether the constants follow the same rule ($8 + 5 = b$).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A system of linear equations has no solution if the lines/planes are parallel but distinct, or if the determinant of coefficients (\(\Delta\)) is zero while at least one coordinate determinant (\(\Delta_x, \Delta_y, \Delta_z\)) is non-zero.
Step 3: Detailed Explanation:
The coefficient determinant is:
\[ \Delta = \begin{vmatrix} 2 & 3 & 6 \\ 1 & 2 & a \\ 3 & 5 & 9 \end{vmatrix} \]
Expand along the first row:
\(\Delta = 2(18 - 5a) - 3(9 - 3a) + 6(5 - 6)\)
\(\Delta = 36 - 10a - 27 + 9a - 6 = 3 - a\).
For the system to have either "no solution" or "infinite solutions", \(\Delta = 0 \Rightarrow a = 3\).
Now, let's examine the equations with \(a = 3\):
Eq 1: \(2x + 3y + 6z = 8\)
Eq 2: \(x + 2y + 3z = 5\)
Eq 3: \(3x + 5y + 9z = b\)
Note that if we add Eq 1 and Eq 2:
\((2x + x) + (3y + 2y) + (6z + 3z) = 8 + 5\)
\(3x + 5y + 9z = 13\).
For consistency (infinite solutions), the third equation must be identical: \(b = 13\).
For no solution, the third equation must represent a parallel plane that does not coincide: \(b \neq 13\).
Step 4: Final Answer:
The condition for no solution is \(a = 3\) and \(b \neq 13\).