Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Using properties of determinants, prove the following:
$\begin{vmatrix} 1 & \text{x} & \text{x}^{2} \\ \text{x}^{2} & 1 & \text{x} \\ \text{x} & \text{x}^{2} & 1 \end{vmatrix} = (1 - \text{x}^{3})^{2}.$
✓

Answer

Let| A| $ = \begin{vmatrix} 1 & \text{x} & \text{x}^{2} \\ \text{x}^{2} & 1 & \text{x} \\ \text{x} & \text{x}^{2} & 1 \end{vmatrix} $
Apply $C_1 \rightarrow C_1 + C_2 + C_3$
$ |\text{A}|= \begin{vmatrix} 1 +& \text{x} + \text{x}^{2} &\text{x} &\text{x}^{2}\\ 1 +& \text{x } + \text{x}^{2} & 1 & \text{x} \\ 1 +& \text{x} + \text{x}^{2} & \text{x}^{2}& 1 \end{vmatrix} $
$\Rightarrow |\text{A}|=(1 + \text{x} + \text{x}^{2}) \begin{vmatrix} 1 & \text{x} & \text{x}^{2} \\ 1 & 1 & \text{x} \\ 1 & \text{x}^{2} &1 \end{vmatrix} $
$\text{Apply R}_{2}\rightarrow\text{R}_{2}-\text{ R}_{1},\text{ R}_{3}\rightarrow\text{R}_{3}- \text{R}_{1}$
$\Rightarrow |\text{A}|=(1 + \text{x} + \text{x}^{2}) \begin{vmatrix} 1 & \text{x} & \text{x}^{2} \\ 0 & 1 - \text{x} & \text{x} - \text{x}^{2} \\ 0 & \text{x}^{2} - \text{x} &1 - \text{x}^{2} \end{vmatrix} $
Take (1 - x) common from $R_2$ and $R_3$
$ |\text{A}|=(1 + \text{x} + \text{x}^{2})(1 - \text{x})^{2} \begin{vmatrix} 1 & \text{x} & \text{x}^{2} \\ 0 & 1 & \text{x} \\ 0 & - \text{x} &1 + \text{x} \end{vmatrix} $
Expanding along $C_1$
$|A|= (1 + x + x^2 )(1 - x)^2 (1 + x + x^2)$
$= (1 - x^3)^2$
[$\because 1 - x^3 = (1 - x)(1 + x + x^2)$].

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

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}^2+3\text{y}+\text{y}^2=5\ \text{at}\ (1,1)$
→
Prove that the curves $xy = 4$ and $x^2 + y^2 = 8$ touch each other.
→
Solve the following systems of linear equations by cramer's rule:
2x - 3z + w = 1,
x - y + 2w = 1,
-3y + z + w = 1,
x + y + z = 1
→
Differentiate the following w.r.t. x:
$\tan^{-1}\bigg(\frac{\sqrt{1+\text{x}^2}+\sqrt{1-\text{x}^2}}{\sqrt{1+\text{x}^2}-\sqrt{1-\text{x}^2}}\bigg),-1<\text{x}<1,\text{ x}\neq0$
→
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is $\frac{8}{27}$ of the volume of the sphere.
→
By computing the shortest distance determine whether the following pairs of lines intersect or not:3
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{-5}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-7}{1}=\frac{\text{z}-5}{3}$
→
If the points $(x, -2), (5, 2), (8, 8)$ are collinear, find $x$ using determinants.
→
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\frac{{2R}}{{\sqrt 3 }}$ Also find the maximum volume.
→
Using the method of integration, find the area of the region bounded by the lines
$\text{2x + y = 4, 3x - 2y = 6 and x -3y + 5 = 0}$
→
Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1-\cos\theta)\ \text{at}\ \theta=\frac{-\pi}{2}$
→