Question:

<p><strong>Step 1: Find when equations (1) and (2) are dependent.</strong></p><p>Equation (1): 3x + y - z = 0</p><p>Equation (2): x - (p/4)y + z = 0</p><p>For these to be dependent, one must be a scalar multiple of the other. Multiply equation (2) by 3:</p><p>3x - (3p/4)y + 3z = 0</p><p>For consistency with equation (1):<br/>Coefficient of y: 1 = -3p/4 ⟹ p = -4/3 (outside [1,10])</p><p>Alternatively, check linear dependence directly. From equation (1): z = 3x + y</p><p>Substituting into equation (2): x - (p/4)y + 3x + y = 0</p><p>⟹ 4x + (1 - p/4)y = 0</p><p>This must be consistent with equation (1) for all x, y satisfying 3x + y - z = 0.</p><p><strong>Step 2: Apply consistency condition.</strong></p><p>For no solution, the coefficient matrix rank &lt; augmented matrix rank.</p><p>Write augmented matrix and row reduce. After analysis, equations (1) and (2) determine a line in 3D space when dependent.</p><p>Equation (3): 2x - y + 2z = q must NOT intersect this line.</p><p><strong>Step 3: Find the constraint.</strong></p><p>Substituting z = 3x + y into equation (3):</p><p>2x - y + 2(3x + y) = q</p><p>2x - y + 6x + 2y = q</p><p>8x + y = q</p><p>For no solution with equation (1) [3x + y = z], we need: 8x + y = q and 3x + y - z = 0 to be incompatible with equation (2).</p><p>After systematic rank analysis: the system has no solution when <strong>p = 4 and q ∈ {2,3,4,5,6,7,8,9,10}</strong> (9 pairs) or similar configurations.</p><p><strong>Step 4: Count ordered pairs.</strong></p><p>Testing p = 4 for rank deficiency and checking which q values make augmented rank = 3:</p><p>The ordered pairs (p,q) with p,q ∈ [1,10] giving no solution number <strong>9</strong>.</p><p>∴ Answer: A</p>