AP EAMCET202419 May 2024Evening ShiftMathematicsDeterminantsActual
| array ccc 1 & 1 & 1 a^2 & b^2 & c^2 a^3 & b^3 & c^3 array |=
Options
- A(a-b)(b-c)(c-a)(a+b+c)
- B(a-b)(b-c)(c-a)
- C(a-b)(b-c)(a-c)(a b+b c+c a)
- D(a-b)(b-c)(c-a)(a b+b c+c a)
Correct answer
D. (a-b)(b-c)(c-a)(a b+b c+c a)
Step-by-step solution
Since, | array ccc 1 & 1 & 1 a^2 & b^2 & c^2 a^3 & b^3 & c^3 array |= | array ccc 0 & 1 & 0 a^2-b^2 & b^2 & c^2-b^2 a^3-b^3 & b^3 & c^3-b^3 array | C₁ C₁-C₂ C₃ C₃-C₂ =(a-b)(c-b) | array ccc 0 & 1 & 0 a+b & b^2 & b+c a^2+b^2+a b & b^3 & b^2+c^2+b c array |=(a-b)(c-b) | array ccc 0 & 1 & 0 a+b & b^2 & c-a a^2+b^2+a b & b^3 & c^2+b c-a^2-a b array | ( C₃ C₃-C₁ ) aligned & =(a-b)(c-b)(c-a) | array ccc 0 & 1 & 0 a+b & b^2 & 1 a^2+b^2+a b & b^3 & a+b+c array | & =-(a-b)(c-b)(c-a) (a+b)(a+b+c)- (a^2+b^2+a b ) & =(a-b)(b-c