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Question

Using properties of deteminants, prove that ∣ ∣ ∣ ∣ ∣ ∣ ∣(a+b)2ccca(b+c)2aabb(c+a)2b∣ ∣ ∣ ∣ ∣ ∣ ∣=2(a+b+c)3.

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Solution

LHs=∣ ∣ ∣ ∣ ∣ ∣ ∣(a+b)2ccca(b+c)2aabb(c+a)2b∣ ∣ ∣ ∣ ∣ ∣ ∣

=1abc∣ ∣ ∣(a+b)2c2c2a2(b+c)2a2b2b2(c+a)2∣ ∣ ∣

On applying C2C2C1andC3C3C1, we get-

=1abc∣ ∣ ∣(a+b)2c2(a+b)2c2(a+b)2a2(b+c)2a20b20(c+a)2b2∣ ∣ ∣

=1abc∣ ∣ ∣(a+b)2(a+b+c)(cab)(a+b+c)(cab)a2(a+b+c)(b+ca)0b20(a+b+c)(c+ab)∣ ∣ ∣

Taking (a+b+c) common from C2andC3,we get-

(a+b+c)2abc∣ ∣ ∣(a+b)2cabcaba2b+ca0b20c+ab∣ ∣ ∣

=(a+b+c)2abc∣ ∣ ∣(a+b)2cabcaba2b+ca0b20c+ab∣ ∣ ∣

On applying R1R1(R2+R3), we get-

=(a+b+c)2abc∣ ∣ ∣2ab2b2aa2b+ca0b20c+ab∣ ∣ ∣

On applying C3C3+1bC1, we get-

(a+b+c)2abc∣ ∣ ∣ ∣2ab2b0a2b+caa2bb20c+a∣ ∣ ∣ ∣

Expanding the determinant now-

(a+b+c)2abc[2ab(bc+ab+c2+acaca2)+2b(a2c+a3a2b)]
On simplifying, we get-

(a+b+c)2abc×2abc(a+b+c)

=2(a+b+c)3

Hence,proved.





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