Show that: y+z&x&y +x&z&x +y&y&z =(x+y+z)(x-z)^3 - Vidyadip

Question

Show that:
$\begin{vmatrix}\text{y}+\text{z}&\text{x}&\text{y}\\\text{z}+\text{x}&\text{z}&\text{x}\\\text{x}+\text{y}&\text{y}&\text{z}\end{vmatrix}=(\text{x}+\text{y}+\text{z})(\text{x}-\text{z})^3$

✓

Answer

Let $\text{L.H.S}=\begin{vmatrix}\text{y}+\text{z}&\text{x}&\text{y}\\\text{z}+\text{x}&\text{z}&\text{x}\\\text{x}+\text{y}&\text{y}&\text{z}\end{vmatrix}$
$=\begin{vmatrix}2(\text{x}+\text{y}+\text{z})&\text{x}+\text{y}+\text{z}&\text{x}+\text{y}+\text{z}\\\text{z}+\text{x}&\text{z}&\text{x}\\\text{x}+\text{y}&\text{y}&\text{z}\end{vmatrix}$ [Applying $R_1 → R_1 + R_2 + R_3]$
=$(\text{x}+\text{y}+\text{z})\begin{vmatrix}2&1&1\\\text{z}+\text{x}&\text{z}&\text{x}\\\text{x}+\text{y}&\text{y}&\text{z}\end{vmatrix}$
$=(\text{x}+\text{y}+\text{z})\begin{vmatrix}0&1&1\\0&\text{z}&\text{x}\\\text{x}-\text{z}&\text{y}&\text{z}\end{vmatrix}$ [Applying $C_1 → C_1 - C_2 - C_3]$
$=(\text{x}+\text{y}+\text{z})\left\{(\text{x}-\text{z})\times\begin{vmatrix}1&1\\\text{z}&\text{x}\end{vmatrix}\right\}$ [Expanding along $C_1$]
$=(\text{x}+\text{y}+\text{z})(\text{x}-\text{z})^3$

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

Evaluate the following integrals:$\int\frac{\text{x}^2}{\text{x}^2+7\text{x}+10}\text{ dx}$

Evaluate the following integrals:
$\int\frac{(\text{x}^2+1)(\text{x}^2+2)}{(\text{x}^2+3)(\text{x}^2+4)}\ \text{dx}$

Using vector method, prove that the point is collinear:
A(-3, -2, -5), B(1, 2, 3) and C(3, 4, 7)

Find the points of local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\sin2\text{x},0\leq\text{x}\leq\pi$

Find the vector equation of the plane passing through the intersection of the planes $\vec{\text{r}}.\Big(2\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\Big)=7,\ \vec{\text{r}}.\Big(2\hat{\text{i}}+5\hat{\text{j}}+3\hat{\text{k}}\Big)=9$ and through the point (2, 1, 3).

Show that $\text{y}=\text{A}\cos2\text{x}+\text{B}\sin2\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$

Find the points on the curve $y = x^3 - 3x,$ where the tangent to the curve is parallel to the chord joining (1, -2) and (2, 2).

Solve the following systems of linear equations by cramer's rule:

For the matrix $A = \left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 1&2&{ - 3} \\ 2&{ - 1}&3 \end{array}} \right]$, show that $A^3 - 6A^2 + 5A + 11I = 0.$ Hence find $A^{-1}.$

If O is the origin and the coordinates of A are (a, b, c) Find the direction cosines of OA and the equation of the plane through A at right angles to OA.