Using properties of determinants, prove the following: & & ^ 2 & … - Vidyadip

Question

Using properties of determinants, prove the following:$\begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{vmatrix} = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)( \alpha + \beta + \gamma) $

✓

Answer

$\text{LHS:} \triangle = \begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{vmatrix}$$\text{R}_{3} \rightarrow \text{R}_{3} + \text{R}_{1} = \triangle = \begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \alpha +\beta+ \gamma &\alpha +\beta+ \gamma & \alpha +\beta+ \gamma \end{vmatrix}$
$= (\alpha + \beta + r) \begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ 1 & 1 & 1 \end{vmatrix} $
$\text{Applying C}_{1} \rightarrow \text{C}_{1} -\text{C}_{2}$ $\text{and C}_{2} \rightarrow \text{C}_{2} - \text{C}_{3}$
$\triangle = ( \alpha + \beta + \gamma) \begin{vmatrix} \alpha- \beta & \beta - \gamma & \gamma \\ \alpha^{2} - \beta^{2} & \beta^{2} -\gamma^{2} & \gamma^{2} \\ 0 & 0 & 1 \end{vmatrix} $
Expanding by last row to get
$\triangle = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)( \alpha + \beta + \gamma) = \text{RHS} $

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 "5 Marks Questions" from DETERMINANTS→DETERMINANTS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is black,
B = the card drawn is a king.

Two biased dice are thrown together. For the first die $\text{P}(6)=\frac{1}{2},$ the other scores being equally likely while for the second die, $\text{P}(1)=\frac{2}{5}$ and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.

Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$

Solve the following system of equations by matrix method:
$3x + 7y = 4$
$x + 2y = -1$

Solve the following differential equation
$\sin^4\text{x}\frac{\text{dy}}{\text{dx}}=\cos\text{x}$

Evaluate the following integrals:
$\int\limits^{5}_0\frac{\sqrt[4]{\text{x}+4}}{\sqrt[4]{\text{x}+4}+\sqrt[9]{9-\text{x}}}\text{ dx}$

If $\text{A}=\begin{bmatrix}3&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}7&3\end{bmatrix},$ then find a non-zero matrix C such that AC = BC.

A manufacturer has three machines installed in his factory. machines $I$ and $II$ are capable of being operated for at most $12$ hours whereas Machine $III$ must operate at least for $5$ hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

Item	Number of hours required by the machine
	$I$	$II$	$III$
$A$	$1$	$2$	$1$
$B$	$2$	$1$	$\frac{5}{4}$

He makes a profit of Rs. $6.00$ on item A and Rs. $4.00$ on item $B$. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.

Using differentials, find the approximate values of the following:
$\Big(\frac{17}{81}\Big)^{\frac{1}{4}}$

Express the matrix $\text{A}=\begin{bmatrix}4&2&-1 \\3 & 5&7\\1&-2&1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.