1&a&a^2 ^2&1&a &a^2&1 =(a^3-1)^2 - Vidyadip

Question

$\begin{vmatrix}1&\text{a}&\text{a}^2\\\text{a}^2&1&\text{a}\\\text{a}&\text{a}^2&1\end{vmatrix}=(\text{a}^3-1)^2$

✓

Answer

Let us consider the L.H.S of the given equation.
Let $\begin{vmatrix}1&\text{a}&\text{a}^2\\\text{a}^2&1&\text{a}\\\text{a}&\text{a}^2&1\end{vmatrix}$
Applying C₁ → C₁ + C₂ + C₃, we have,
$=\begin{vmatrix}1+\text{a}+\text{a}^2&\text{a}&\text{a}^2\\1+\text{a}+\text{a}^2&1&\text{a}\\1+\text{a}+\text{a}^2&\text{a}^2&1\end{vmatrix}$
Taking term (1 + a + a₂) common, we have,
$=(1+\text{a}+\text{a}^2)\begin{vmatrix}1&\text{a}&\text{a}^2\\1&1&\text{a}\\1&\text{a}^2&1\end{vmatrix}$
Applying R₂ → R₂ - R₁ and R₃ → R₃ - R₁, we have,
$=(1+\text{a}+\text{a}^2)\begin{vmatrix}1&\text{a}&\text{a}^2\\0&1-\text{a}&\text{a}(1-\text{a})\\0&-\text{a}(1-\text{a})&(1-\text{a})(1+\text{a})\end{vmatrix}$
Taking the term (1 - a) common from R₂ and R₃, we have,
$=(1+\text{a}+\text{a}^2)(1-\text{a}^2)\begin{vmatrix}1&\text{a}&\text{a}^2\\0&1&\text{a}\\0&-\text{a}&(1+\text{a})\end{vmatrix}$
$=(1+\text{a}+\text{a}^2)(1-\text{a}^2)(1+\text{a}+\text{a}^2)$
$=(1+\text{a}+\text{a}^2)(1-\text{a}^2)$
$=\big[(1+\text{a}+\text{a}^2)(1-\text{a}^2)\big]^2$
$=\big[(\text{a}^3-1)\big]^2$
$=\text{R.H.S}$

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→

Similar questions

Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}-3)\sqrt{\text{x}+1}}\text{ dx}$

Solve the following systems of homogeneous linear equations by matrix method:

3x + 2y - z = 0

x + 4y + 3z = 0

If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix},$ show that A² - 5A + 7I₂ = 0.

A manufacturer makes two types A and B of tea-cups. Three machines are needed for the manufacture and the time in minutes required for each cup on the machines is given below:

	Machines
	I	II	III
A	12	18	6
B	6	0	9

Each machine is available for a maximum of 6 hours per day. If the profit on each cup A is 75 paise and that on each cup B is 50 paise, show that 15 tea-cups of type A and 30 of type B should be manufactured in a day to get the maximum profit.

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i} + 3\hat{j} + 4\hat{k}$ to the plane $\overrightarrow{\text{r}}. (2\hat{i} + \hat{j} + 3\hat{k}) - 26 = 0.$ Also find image of P in the plane.

A bag contains 4 white, 7 black and 5 red balls. Three balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.

Evaluate the following integrals:
$\int\sin^7\text{x}\text{ dx}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin^2\text{A}&\cot\text{A}&1\\\sin^2\text{B}&\cot\text{B}&1\\\sin^2\text{C}&\cot\text{C}&1\end{vmatrix}$

Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+2\text{y}=0,\text{y}(0)=1,\text{y}(0)=3$
Function $\text{y}=\text{e}^\text{x}+\text{e}^{2\text{x}}$

If $\begin{bmatrix}\text{x}&4&1\end{bmatrix}\begin{bmatrix}2&1&2\\1&0&2\\0&2&-4\end{bmatrix}\begin{bmatrix}\text{x}\\4\\-1\end{bmatrix}=0,$ find x.