Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Using the proprties of determinants in Exercise:
$\begin{vmatrix}\text{y}^2\text{z}^2&\text{yz}&\text{y}+\text{z}\\\text{z}^2\text{x}^2&\text{zx}&\text{z}+\text{x}\\\text{x}^2\text{y}^2&\text{xy}&\text{x}+\text{y}\end{vmatrix}=0$
✓

Answer

We have, $\begin{vmatrix}\text{y}^2\text{z}^2&\text{yz}&\text{y}+\text{z}\\\text{z}^2\text{x}^2&\text{zx}&\text{z}+\text{x}\\\text{x}^2\text{y}^2&\text{xy}&\text{x}+\text{y}\end{vmatrix}$
$[\text{Multiplying R}_1, \text{R}_2, \text{R}_3\text{ by x, y, z respectively}]$
$=\frac{1}{\text{xyz}}\begin{vmatrix}\text{xy}^2\text{z}^2&\text{xyz}&\text{xy}+\text{xz}\\\text{x}^2\text{yz}^2&\text{xyz}&\text{yz}+\text{xy}\\\text{x}^2\text{y}^2\text{z}&\text{xyz}&\text{xz}+\text{yz}\end{vmatrix}$
$\big[\text{Taking (xyz) common from C}_1\text{ and C}_2\big]$
$=\frac{1}{\text{xyz}}(\text{xyz})^2\begin{vmatrix}\text{yz}&1&\text{xy}+\text{xz}\\\text{xz}&1&\text{yz}+\text{xy}\\\text{xy}&1&\text{xz}+\text{yz}\end{vmatrix}$
$[\text{Applying C}_3\rightarrow\text{C}_3+\text{C}_1]$
$=\text{xyz}\begin{bmatrix}\text{yz}&1&\text{xy}+\text{yz}+\text{zx}\\\text{xz}&1&\text{xy}+\text{yz}+\text{zx}\\\text{xy}&1&\text{xy}+\text{yz}+\text{zx}\end{bmatrix}$
$\big[\text{Taking (xy}+\text{yz}+\text{zx})\text{ common from }\text{C}_3\big]$
$=\text{xyz (yz}+\text{yz}+\text{zx})\begin{vmatrix}\text{yz}&1&1\\\text{xz}&1&1\\\text{xy}&1&1\end{vmatrix}$
$=0$
$\big[\because\text{C}_2\text{ and C}_3\text{ are identical}\big]$

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

Differentiate the function $x^{x \cos x}+\frac{x^{2}+1}{x^{2}-1}$ w.r.t. x.
→
In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true'; if it falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly.
→
Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio $\text{1 : 2 : 4. }$The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.
→
If $\text{A}=\begin{bmatrix}2&-3&5\\ 3&2&-4\\ 1&1&-2\end{bmatrix}$, find $A^{-1}$ and hence solve the system of linear equations:
2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y + 2z = -3
→
If $\vec{\text{a}} \times \vec{\text{b}} = \vec{\text{c}} \times \vec{\text{d}}$ and $\vec{\text{a}} \times \vec{\text{c}} = \vec{\text{b}} \times \vec{\text{d}},$ show that $\vec{\text{a}} - \vec{\text{d}}$ is parallel to $\vec{\text{b}} - \vec{\text{c}},$ where $\vec{\text{a}} \neq \vec{\text{d}}$ and $\vec{\text{b}} \neq \vec{\text{c}}.$
→
Differentiate the following functions with respect to x:
$\text{e}^{3\text{x}}\cos2\text{x}$
→
Find the values of a and b such that the function f defined by $\text{f(x)}=\begin{cases}\frac{\text{x}-4}{|\text{x}-4|}+\text{a},&\text{if x}<4\\\text{a+}\text{b},&\text{if x}=4\\\frac{\text{x}-4}{|\text{x}-4|}+\text{b},&\text{if x}>4\end{cases}$ is a continuous function at x = 4.
→
For each of the differential equations given in find the general solution:
$(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=1$
→
Prove that the curves $xy = 4$ and $x^2 + y^2 = 8$ touch each other.
→
Differentiate the functions given in Exercise:
$(\log\text{x})^\text{x}+\text{x}^{\log\text{x}}$
→