Solution:
1. Apply Row Operations:
To simplify the determinant, we can use row operations. Let's perform:
The determinant becomes:
pa−pa−pamp;bamp;q−bamp;0amp;camp;0amp;r−c=0
2. Expand the Determinant:
Expanding along the first row (R1):
p[(q−b)(r−c)−0]−b[(a−p)(r−c)−0]+c[0−(a−p)(q−b)]=0
p(q−b)(r−c)−b(a−p)(r−c)−c(a−p)(q−b)=0
3. Adjust the Signs:
Since we need terms like p−a, let's replace (a−p) with −(p−a):
p(q−b)(r−c)−b[−(p−a)](r−c)−c[−(p−a)](q−b)=0
p(q−b)(r−c)+b(p−a)(r−c)+c(p−a)(q−b)=0
4. Divide by (p−a)(q−b)(r−c):
To reach the required form, divide the entire equation by (p−a)(q−b)(r−c):
(p−a)(q−b)(r−c)p(q−b)(r−c)+(p−a)(q−b)(r−c)b(p−a)(r−c)+(p−a)(q−b)(r−c)c(p−a)(q−b)=0
5. Convert b and c terms to q and r:
We need the numerators to be q and r. We can rewrite the fractions:
Substitute these back into the equation:
p−ap+(q−bq−1)+(r−cr−1)=0
Correct Option: (d) 2
