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CBSE Class 12 Math 2023 All Sets Solved Paper

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Question : 10 of 20
Marks: +1, -0
If (a, b), (c, d) and (e, f) are the vertices of △ABC and △ denotes the area of △ABC, then∣acebdf111∣2\begin{vmatrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{vmatrix}^2is equal to:
Solution:  
Area of the triangle with vertices (a, b), (c, d) and (e, f) is given by:
12∣ab1cd1ef1∣=△\frac{12}{\begin{vmatrix} a & b & 1 \\ c & d & 1 \\ e & f & 1 \end{vmatrix}} = \triangle
⇒∣ab1cd1ef1∣=2△\Rightarrow \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ e & f & 1 \end{vmatrix} = 2\triangle
Now, ∣ab1cd1ef1∣=∣acebdf111∣\begin{vmatrix} a & b & 1 \\ c & d & 1 \\ e & f & 1 \end{vmatrix} = \begin{vmatrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{vmatrix} (∵∣AT∣=∣A∣)( \because |A^T| = |A| )
∴∣acebdf111∣=4△2\therefore \begin{vmatrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{vmatrix} = 4\triangle^2
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