By using properties of determinants, show that: a-b-c&2a&2a 2b&b-… - Vidyadip

Question

By using properties of determinants, show that:
$\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}=(a+b+c)^3$

✓

Answer

$\text{L.H.S.}=\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}$$=\begin{vmatrix}a+b+c&a+b+c&a+b+c\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}\ \left[\text{R}_1\rightarrow\text{R}_1+\text{R}_2+\text{R}_3\right]$
$=(a+b+c)\begin{vmatrix}1&1&1\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}$
$=(a+b+c)\begin{vmatrix}1&0&0\\2b&-b-c-a&0\\2c&0&-c-a-b\end{vmatrix}\ \left[\text{C}_2\rightarrow\text{C}_2-\text{C}_1\ \text{and C}_3\rightarrow\text{C}_3-\text{C}_1\right]$

$=(a+b+c)1[(a+b+c)(a+b+c)-0]$
$=(a+b+c)(a+b+c)^2$
$=(a+b+c)^3=\text{R.H.S.}$

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 "3 Marks Question" from DETERMINANTS→DETERMINANTS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write a value of $\int\text{e}^{2\text{x}^2+\ln\text{x}}\text{ dx}$

An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities:

P(A fails|B has failed)
P(A fails alone)

Let $A = [-1, 1].$ Then, discuss whether the following functions from $A$ to itself are one-one, onto or bijective:$h(x) = x^2$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases: $xy = c^2$

Let $A = R_0 \times R,$ where $R_0$ denote the set of all non$-$zero real numbers. A binary operation $'⊙\ ' $is defined on $A$ as follows:
$(a, b) ⊙ (c, d) = (ac, bc + d)$ for all $(a, b), (c, d) \in R_0 \times R.$
Show that $'⊙\ '$ is commutative and associative on $A.$

A pair of dice is thrown 6 times. If getting a total of 9 is considered a success, what is the probability of at least 5 successes?

Integrate the following integrals:
$\int\cos3\text{x}\cos4\text{x dx}$

If $f : R \rightarrow R$ be the function defined by $f(x) = 4x^3 + 7,$ show that $f$ is a bijection.

$\text{If} \ \vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\ \ \vec{b}=-$ $\hat{i}+2\hat{j}+\hat{k}\ \text{and}\ \vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c},$ then find the value of $\lambda.$

For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?