JAMIA 2022 — Mathematics PYQ

1. Observe the Row/Column Sums:

All three determinants represent circulant or symmetric matrices where each row (or column) contains the same elements.

• For the first determinant:

  $$\left|\begin{array}{ccc}x& amp;3& amp;6\\ 3& amp;6& amp;x\\ 6& amp;x& amp;3\end{array}\right|=0$$

  The sum of each row is $x+3+6=x+9$.

• For the second determinant:

  $$\left|\begin{array}{ccc}2& amp;x& amp;7\\ x& amp;7& amp;2\\ 7& amp;2& amp;x\end{array}\right|=0$$

  The sum of each row is $2+x+7=x+9$.

• For the third determinant:

  $$\left|\begin{array}{ccc}4& amp;5& amp;x\\ 5& amp;x& amp;4\\ x& amp;4& amp;5\end{array}\right|=0$$

  The sum of each row is $4+5+x=x+9$.

2. Using Property of Determinants:

A key property of such matrices is that if the sum of all elements in each row (or column) is constant, then that sum is a factor of the determinant.

By performing the column operation $C_1\to C_1+C_2+C_3$ on any of these determinants, the first column becomes:

For the determinant to be zero, one possible condition is:

3. Verification:

Let's check if $x=-9$ satisfies the first equation:

Adding all columns to the first column:

Similarly, for the other two determinants, when $x=-9$, the sum of each row becomes $0$, making the determinant $0$.

Final Answer:

The value of $x$ is $-9$.