1.

The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)], [(m + 1)(m + 2), (m + 2)] and [(m + 2)(m + 3), (m + 3)]. Then which one among the following is correct?(a) The area of the triangle is dependent on m(b) The area of the triangle is independent on m(c) Answer cannot be predicted(d) Data inadequateThe question was asked by my school principal while I was bunking the class.This interesting question is from Application of Determinants in chapter Determinants of Mathematics – Class 12

Answer»

The CORRECT answer is (b) The area of the triangle is independent on m

Explanation: The area o the triangle with the given POINT as vertices is,

1/2 \(\begin{vmatrix}m(m+1) & (m+1) & 1 \\(m+1)(m+2) & (m+2) & 1 \\(m+2)(m+3) & (m+3) & 1 \end {vmatrix}\)

 = 1/2 \(\begin{vmatrix}m^2 + m & (m+1) & 1 \\m^2 + 3m + 2 & (m+2) & 1 \\m^2 + 5m + 6 & (m+3) & 1 \end {vmatrix}\)

Now, by performing the row operation R2 = R2 – R1 and R3 = R3 – R2

= 1/2 \(\begin{vmatrix}m^2 + m & (m+1) & 1 \\2m + 2 & 1 & 0 \\2m + 4 & 1 & 0 \end {vmatrix}\)

Now, breaking the DETERMINANT we get,

= 1/2 (2m + 2 – 2m – 4)

= -1

Thus, it is independent of m.


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