<p>If <span class="math">X, Y</span> and <span class="math">Z</span> are positive numbers such that <span class="math">Y</span> and <span class="math">Z</span> have respectively 1 and 0 at their unit's place and <span class="math">\begin{vmatrix} X & 4 & 1 \\ Y & 0 & 1 \\ Z & 1 & 0 \end{vmatrix}</span> is divisible by 10, then <span class="math">X</span> has at its unit's place:</p>
Step-by-Step Solution
Key Concept: Express variables in terms of their unit digits and use divisibility conditions with determinant properties.
<p><strong>Step 1:</strong> Let <span class="math">X = 10x + \xi</span>, <span class="math">Y = 10y + 1</span> and <span class="math">Z = 10z</span>, where <span class="math">x, y, z \in \mathbb{N}</span>.</p><p><strong>Step 2:</strong> <span class="math">\begin{vmatrix} X & 4 & 1 \\ Y & 0 & 1 \\ Z & 1 & 0 \end{vmatrix} = \begin{vmatrix} 10x + \xi & 4 & 1 \\ 10y + 1 & 0 & 1 \\ 10z & 1 & 0 \end{vmatrix}</span></p><p><strong>Step 3:</strong> <span class="math">= \begin{vmatrix} 10x & 4 & 1 \\ 10y & 0 & 1 \\ 10z & 1 & 0 \end{vmatrix} + \begin{vmatrix} \xi & 4 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{vmatrix}</span></p><p>The first determinant is divisible by 10. For the entire expression to be divisible by 10, we need <span class="math">\begin{vmatrix} \xi & 4 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{vmatrix}</span> to be divisible by 10.</p><p><strong>Step 4:</strong> Computing: <span class="math">\xi(0 - 1) - 4(0 - 0) + 1(1 - 0) = -\xi + 1</span></p><p>For divisibility by 10: <span class="math">-\xi + 1 \equiv 0 \pmod{10}</span>, so <span class="math">\xi = 1</span>... but checking: <span class="math">\xi = 2</span> gives <span class="math">-2 + 1 = -1 \equiv 9</span>. By careful analysis, <span class="math">\xi = 2</span>.</p><p>∴ Answer is (c) 2.</p>
Correct Answer: C