If `|{:(a,b,a),(b,c,1),(c,a,1):}|=2010` and if `|{:(c-a,c-b,ab),(a-b,a-c,bc),(b-c,b-a,ca):}|-|{:(c-a,c-b,c^(2)),(a-b,a-c,a^(2)),(b-c,b-a,b^(2)):}|=p`, then the number of positive divisors of `p` is

Options

A. (a) `36`
B. (b) `49`
C. (c) `64`
D. (d) `81`

Explanation

To solve the given problem, we need to evaluate two determinants and find the difference between them. Let's break it down step by step.

### Step 1: Evaluate the first determinant
We have the determinant:
\[ D_1 = \begin{vmatrix} a & b & a \\ b & c & 1 \\ c & a & 1 \end{vmatrix} \]
Using the determinant formula for a 3x3 matrix, we can expand this determinant:
\[ D_1 = a \begin{vmatrix} c & 1 \\ a & 1 \end{vmatrix} - b \begin{vmatrix} b & 1 \\ c & 1 \end{vmatrix} + a \begin{vmatrix} b & c \\ c & a \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \(\begin{vmatrix} c & 1 \\ a & 1 \end{vmatrix} = c - a\)
2. \(\begin{vmatrix} b & 1 \\ c & 1 \end{vmatrix} = b - c\)
3. \(\begin{vmatrix} b & c \\ c & a \end{vmatrix} = ab - c^2\)

Substituting these back into \(D_1\):
\[ D_1 = a(c - a) - b(b - c) + a(ab - c^2) \]
\[ = ac - a^2 - b^2 + bc + a^2b - ac^2 \]
\[ = a^2b - ac^2 - b^2 + bc \]

### Step 2: Evaluate the second determinant
Now we evaluate:
\[ D_2 = \begin{vmatrix} c-a & c-b & ab \\ a-b & a-c & bc \\ b-c & b-a & ca \end{vmatrix} \]

Using similar expansion techniques:
\[ D_2 = (c-a) \begin{vmatrix} a-c & bc \\ b-a & ca \end{vmatrix} - (c-b) \begin{vmatrix} a-b & bc \\ b-c & ca \end{vmatrix} + ab \begin{vmatrix} a-b & a-c \\ b-c & b-a \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \(\begin{vmatrix} a-c & bc \\ b-a & ca \end{vmatrix} = (a-c)ca - (b-a)bc\)
2. \(\begin{vmatrix} a-b & bc \\ b-c & ca \end{vmatrix} = (a-b)ca - (b-c)bc\)
3. \(\begin{vmatrix} a-b & a-c \\ b-c & b-a \end{vmatrix} = (a-b)(b-a) - (a-c)(b-c)\)

### Step 3: Calculate the difference
Now we need to find \(D_2 - D_1\) and set it equal to \(p\):
\[ p = D_2 - D_1 \]

### Step 4: Set the equation
From the problem, we know:
\[ |D_1| = 2010 \]
Thus, we can substitute this value into our equation to find \(p\).

### Step 5: Find the number of positive divisors of \(p\)
To find the number of positive divisors of \(p\), we need to factor \(p\) into its prime factors and use the formula for counting divisors.