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Maths · Evaluation of determinants

STATEMENT 1: In a denotes lengths of the sides and then the triangle is equilate

STATEMENT 1: In a \( \Delta A B C, a, b, c \) denotes lengths of the sides and \( \left|\begin{array}{lll}\boldsymbol{a} & \boldsymbol{b} & \boldsymbol{c} \\ \boldsymbol{b} & \boldsymbol{c} & \boldsymbol{a} \\ \boldsymbol{c} & \boldsymbol{a} & \boldsymbol{b}\end{array}\right|=\mathbf{0} \) then the triangle is equilateral triangle. STATEMENT 2: Sum of three non- negative numbers \( =0 \Rightarrow \) each number is zero.

  • A. Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement-
  • B. Statement-1 is true, Statement-2 is true, Statement-2 is not correct explanation for Statement-
  • C. Statement-1 is true, Statement-2 is false
  • D. Statement-1 is false, Statement-2 is true

Step-by-step solution

The determinant evaluates to 3abc - (a^3+b^3+c^3). Setting it to zero gives a^3+b^3+c^3 = 3abc, which factors as (a+b+c)(a^2+b^2+c^2 - ab - bc - ca) = 0. Since a+b+c > 0, we have a^2+b^2+c^2 - ab - bc - ca = 0, which simplifies to (1/2)[(a-b)^2+(b-c)^2+(c-a)^2]=0. As squares are non-negative, each must be zero, so a=b=c. This uses the principle in Statement 2 (sum of non-negative numbers zero implies each is zero). Hence both statements are true and Statement 2 correctly explains Statement 1.
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