Prove that: 1&b+c&b^2+c^2 1&c+a&c^2+a^2 1&a+b&a^2+b^2 =(a+b)(b-c)(c-a)

Let L.H.S= 1&b+c&b^2+c^2 1&c+a&c^2+a^2 1&a+b&a^2+b^2 = 0&(b+c)-(c+a)&(b^2+c^2)-(c^2+a^2) 0&(c+a)-(a+b)&(c^2+a^2)-(a^2+b^2) 1&a+b&a^2+b^2 [Applying R1 → R1 - R2 and R2 → R2 - R3] = 0&b-a&b^2-a^2 0&c-b&c^2-b^2 1&a+b&a^2+b^2 =(-1)^2 0&a-b&a^2-b^2 0&b-c&b^2-c^2 1&a+b&a^2+b^2 [Taking out (-1) common from R1 and R2] =(a-b)(b-c) 0&1&a+b 0&1&b+c 1&a+b&a^2+b^2 =(a+b)(b-c) 1 1 & a+b 1 & b+c =(a+b)(b-c)(c-a) =R.H.S

Prove that: 1&b+c&b^2+c^2 1&c+a&c^2+a^2 1&a+b&a^2+b^2 =(a+b)(b-c)…

Download App

Question

Prove that:
$\begin{vmatrix}1&\text{b}+\text{c}&\text{b}^2+\text{c}^2\\1&\text{c}+\text{a}&\text{c}^2+\text{a}^2\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2 \end{vmatrix}=(\text{a}+\text{b})(\text{b}-\text{c})(\text{c}-\text{a})$

✓

Answer

Let $\text{L.H.S}=\begin{vmatrix}1&\text{b}+\text{c}&\text{b}^2+\text{c}^2\\1&\text{c}+\text{a}&\text{c}^2+\text{a}^2\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2 \end{vmatrix}$
$=\begin{vmatrix}0&(\text{b}+\text{c})-(\text{c}+\text{a})&(\text{b}^2+\text{c}^2)-(\text{c}^2+\text{a}^2)\\0&(\text{c}+\text{a})-(\text{a}+\text{b})&(\text{c}^2+\text{a}^2)-(\text{a}^2+\text{b}^2)\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2\end{vmatrix}$ [Applying R₁ → R₁ - R₂ and R₂ → R₂ - R₃]
$=\begin{vmatrix}0&\text{b}-\text{a}&\text{b}^2-\text{a}^2\\0&\text{c}-\text{b}&\text{c}^2-\text{b}^2\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2\end{vmatrix}$
$=(-1)^2\begin{vmatrix}0&\text{a}-\text{b}&\text{a}^2-\text{b}^2\\0&\text{b}-\text{c}&\text{b}^2-\text{c}^2\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2\end{vmatrix}$ [Taking out (-1) common from R₁ and R₂]
$=(\text{a}-\text{b})(\text{b}-\text{c})\begin{vmatrix}0&1&\text{a}+\text{b}\\0&1&\text{b}+\text{c}\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2\end{vmatrix}$
$=(\text{a}+\text{b})(\text{b}-\text{c})\left\{1\times\begin{vmatrix} 1 & \text{a}+\text{b} \\ 1 & \text{b}+\text{c} \end{vmatrix}\right\}$
$=(\text{a}+\text{b})(\text{b}-\text{c})(\text{c}-\text{a})$
$=\text{R.H.S}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Determinants→Determinants — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Determinants — Maths STD 12 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions