Question
If p a, q b, r c and the system of equations aligned & p x+a y+a z=0 \\ & b x+q y+b z=0 \\ & c x+c y+r z=0 aligned has a non-trivial solution, then the value of p p-a + q q-b + r r-c is
BITSAT
Mathematics
Determinants
2022
BITSAT 2022
BITSAT Mathematics Question (2022) — Solution
Options
- A. 1
- B. 2
- C. 1 2
- D. 0
Answer
B. 2
Step-by-step solution
As the given system of equations has a non-trivial solution. = | array lll p & a & a \\ b & q & b \\ c & c & r array |=0 Applying C_2 C_2-C_1 and C_3 C_3-C_1 = | array ccc p & a-p & a-p \\ b & q-b & 0 \\ c & 0 & r-c array |=0 Expanding along C_3, we get (a-p) | array cc b & q-b \\ c & 0 array |+(r-c) | array cc p & a-p \\ b & q-b array |=0 aligned & (a-p)(-c)(q-b)+(r-c) \\ & \ p(q-b)-b(a-p)\ =0 \\ & \\ & (p-a)(q-b) c+p(r-c)(q-b)+b(r-c)(p-a)=0 aligned Dividing by (p-a)(q-b)(r-c), we get c r-c + p p-a + b q-b =0 p p-a + q q-b + r r-c = q-b q-b + r-c r-c =2
Related: Mathematics — Determinants · All PYQ Banks