NDA 2 2025 Maths, question 19

Topic: Algebra

Q19

If $a,b,c$ are the sides of a triangle $ABC$ and $p$ is the perimeter of the triangle, then what is

$\left| \begin{matrix} p+c & a & b \\ c & p+a & b \\ c & a & p+b \end{matrix} \right|$

equal to?

(a) $p^{3}$
(b) $2p^{3}$ Correct answer
(c) $3p^{3}$
(d) $4p^{3}$

Solution.

Answer: (b) $2p^{3}$

Subtract row 3 from rows 1 and 2 to obtain (p,0,−p) and (0,p,−p).
Factor p from each of these rows, then add columns 1 and 2 to column 3.
Its only nonzero entry becomes c+a+p+b=2p.
Hence the determinant is p²×2p=2p³.