JEE MATHEMATICS MCQ: MATRICES

Solution approach

1. Correct Answer + Core Reason: 0, 3. The values are obtained by simplifying the determinant and solving for y.
   
   Step-by-Step Solution:
   
   Step 1: Apply the column operation $C_1 \rightarrow C_1+C_2+C_3$ to the determinant.
   
   Step 2: The determinant becomes (3-y)|\beginlll1 & 1-y & 1-y 1 & 1+y & 1-y 1 & 1-y & 1+y\endlll|$(3-y)\left|\begin{{lll}}1 & 1-y & 1-y \\1 & 1+y & 1-y \\1 & 1-y & 1+y\end{{lll}}\right|$
   
   Step 3: Apply the row operations $R_3 \rightarrow R_3-R_2$ and $R_2 \rightarrow R_2-R_1$.
   
   Step 4: The determinant becomes (3-y)|\beginccc1 & 1-y & 1-y 0 & 2 y & 0 0 & -2 y & 2 y\endccc|$(3-y)\left|\begin{{ccc}}1 & 1-y & 1-y \\0 & 2 y & 0 \\0 & -2 y & 2 y\end{{ccc}}\right|$
   
   Step 5: Expand the determinant: $(3-y)(4y^2)$.
   
   Step 6: Set the determinant equal to zero and solve for y: $(3-y)(4y^2) = 0$.
   
   Step 7: The solutions are $y = 0$ and $y = 3$.
   
   Why other options are wrong:
   - Option B: This is incorrect because the determinant equation does not yield these values.
   - Option C: This is incorrect because the determinant equation does not yield these values.
   - Option D: This is incorrect because the determinant equation does not yield these values.
   
   Key Concepts:
   - Determinants
   - Properties of determinants
   - Solving equations
   
   Memory Aids:
   - Remember to use row and column operations to simplify determinants.
   
   Quick Tips:
   - Look for opportunities to simplify determinants using row and column operations.