Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS4 Marks
Question
Using Cofactors of elements of third column, evaluate $\triangle=\begin{vmatrix}1&x&yz\\1&y&zx\\1&z&xy\end{vmatrix}.$
✓

Answer

The given determinant is $\begin{vmatrix}1&x&yz\\1&y&zx\\1&z&xy\end{vmatrix}.$
We have:
M13 = $\begin{vmatrix}1&y\\1&z\end{vmatrix}=z-y$
M23 = $\begin{vmatrix}1&x\\1&z\end{vmatrix}=z-x$
M33 = $\begin{vmatrix}1&x\\1&y\end{vmatrix}=y-x$
$\therefore$ A13 = cofactor of a13 = (-1)1+3 M13 = (z - y)
A23 = cofactors of a23 = (-1)2+3 M23 = - (z - x) = (x - z)
A33 = cofactors of a33 = (-1)3+3 M33 = (y - x)
We know that $\triangle$ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
$\therefore\triangle$ = a13A13 + a23A23 + a33A33
= yz(z - y) + zx (x - z) + xy (y - x)
= yz2 - y2z + x2z - xz2 + xy2 - x2y
=(x2z - y2z) + (yz2 - xz2) + (xy2 - x2y)
= z(x2 - y2) + z2(y - x) + xy(y - x)
=z(x - y)(x + y) + z2(y - x) + xy(y - x)
=(x- y)[zx + zy - z2 - xy]
=(x - y)[z(x - z) + y(z - x)]
=(x - y)(z - x)[-z + y]
=(x - y)(y - z)(z - x)

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

$\int\frac{\text{x}^2+3\text{x}-1}{(\text{x}+1)^2}\text{dx}$
→
If $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$, find $\left(\frac{d^2 y}{d x^2}\right)_{t=\frac{x}{4}}$.
→
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
→
Prove the following results:
$2\sin^{-1}\frac{3}{5}-\tan^{-1}\frac{17}{31}=\frac{\pi}{4}$
→
Evaluvate the following intregals:
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
→
Find the maximum and minimum values of the function $\text{f}(\text{x})=\frac{4}{\text{x}+2}+\text{x}$
→
Evaluate the following definite integrals:
$\int\limits_{0}^{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
→
Find the general solution $ x \frac { d y } { d x } + y - x + x y \cot x = 0 \ (x \neq 0)$.
→
Show that the line whose vector equation is $\vec{\text{r}}=2\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}+\lambda(\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})$ is parallel to the plane whose vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})=7$ Also, find the distance between thetm.
→
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
2x - 2y + 4z + 5 = 0 and 3x - 3y + 6z - 1 = 0
→