If $(a, b),(c, d)$ and $(e, f)$ are the vertices of $\Delta ABC$ and $\Delta$ denotes the area of $\Delta ABC$, then $\left|\begin{array}{lll}a & c & e \\ b & d & f \\ 1 & 1 & 1\end{array}\right|^{2}$ is equal to:

CBSE Class 12 · 2023 · DeterminantsMultiple Choice

(a, b), (c, d)

and

(e, f)

are the vertices of

Δ A B C

and

Δ

denotes the area of

Δ A B C

, then

a b 1 c d 1 e f 1^{2}

is equal to:

Answer: (B)

The area satisfies

Δ = \frac{1}{2} a c e b d f 111

, so that determinant equals

2Δ

. Since a determinant equals its transpose's determinant,

a b 1 c d 1 e f 1 = 2Δ

as well. Squaring gives

(2Δ)^{2} = 4 Δ^{2}

.
Correct option: (B)

4 Δ^{2}

CBSE Class 12 Board · 2023 · Single correct option