NDA2018MathematicsDeterminantsActual
If A, B and C are the angles of a triangle and | array ccc 1 & 1 & 1 1+ A & 1+ B & 1+ C ~A + ^2 ~A & ~B + ^2 ~B & C + ^2 C array |=0 then which one of the following is correct?
Options
- AThe triangle ABC is isosceles
- BThe triangle ABC is equilateral
- CThe triangle ABC is scalene
- DNo conclusion can be drawn with regard to the nature of the triangle
Correct answer
A. The triangle ABC is isosceles
Step-by-step solution
| array ccc 1 & 1 & 1 1+ A & 1+ B & 1+ C A + ^2 ~A & ~B + ^2 ~B & C + ^2 C array | aligned & | array ccc 1 & 0 & 0 1+ A & ~A - B & ~A - C ~A + ^2 ~A & ^2 ~A - ^2 ~B & ^2 ~A - ^2 C array |+0=0 & Applying C ₂ C ₁- C ₂ and C ₃ C ₁- C ₃ . & =( A - B ) ( A - C ) & | array ccc 1 & 0 & 0 1+ A & 1 & 1 ~A + ^2 ~A & ( ~A + B ) & ( A + C ) array |=0 & ( A- B)( A- C)( B- C)=0 & ~A = B or A = C or B = C either A = & B or B = C or A = C . aligned