Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
Evaluate:
$\begin{vmatrix}\text{a}&\text{b}+\text{c}&\text{a}^2\\\text{b}&\text{c}+\text{a}&\text{b}^2\\\text{c}&\text{a}+\text{b}&\text{c}^2\end{vmatrix}$
✓

Answer

$\begin{vmatrix}\text{a}&\text{b}+\text{c}&\text{a}^2\\\text{b}&\text{c}+\text{a}&\text{b}^2\\\text{c}&\text{a}+\text{b}&\text{c}^2\end{vmatrix}$
Apply: $C_2 → C_2 + C_1$
$=\begin{vmatrix}\text{a}&\text{b}+\text{c}+\text{a}&\text{a}^2\\\text{b}&\text{c}+\text{a}+\text{b}&\text{b}^2\\\text{c}&\text{a}+\text{b}+\text{c}&\text{c}^2\end{vmatrix}$
Take $(a + b + c)$ common from $C_2$​​​​​​​
$=(\text{b}+\text{c}+\text{a})\begin{vmatrix}\text{a}&1&\text{a}^2\\\text{b}&1&\text{b}^2\\\text{c}&1&\text{c}^2\end{vmatrix}$
Apply: $R_2 → R_2 - R_1, R_3 → R_3 - R_1$​​​​​​​
$=(\text{b}+\text{c}+\text{a})\begin{vmatrix}\text{a}&1&\text{a}^2\\\text{b}-\text{a}&0&\text{b}^2-\text{a}^2\\\text{c}-\text{a}&0&\text{c}^2-\text{a}^2\end{vmatrix}$
$=(\text{b}+\text{c}+\text{a})(\text{b}-\text{a})(\text{c}-\text{a})\begin{vmatrix}\text{a}&1&\text{a}^2\\1&0&\text{b}+\text{a}\\1&0&\text{c}+\text{a}\end{vmatrix}$
$=(\text{b}+\text{c}+\text{a})(\text{b}-\text{a})(\text{c}-\text{a})(\text{b}-\text{c})$

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 area lying above the x-axis and under the paraola $y^2 = 4x - x^2$.
→
Find the approximate change in the surface area of a cube of side $x$ metres caused by decreasing the side by $1\%.$
→
Integrate the function in Exercise:
$\frac{1}{\sqrt{(\text{x}-\text{a})(\text{x}-\text{b})}}$a
→
In Figure ABCD is a regular hexagon, which vectors are:
  1. Collinear.
  2. Equal.
  3. Co-initial.
  4. Collinear but not equal.
→
Evaluate the following integrals:
$\int\frac{\text{dx}}{(\text{x}^2+1)(\text{x}^2+4)}$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{bmatrix}$
→
Find the shortest distance between the following pairs of lines whose cartesian equation are:
$\frac{\text{x}-1}{-1}=\frac{\text{y}+2}{1}=\frac{\text{z}-3}{-2}$ and $\frac{\text{x}-1}{1}=\frac{\text{y}+1}{2}=\frac{\text{z}+1}{-2}$
→
Solve the following differential equation:
$\text{x dy}=(2\text{y}+2\text{x}^4+\text{x}^2)\text{dx}$
→
Solve the following systems of homogeneous linear equations by matrix method:
3x + y - 2z = 0
x + y + z = 0
x - 2y + z = 0
→
If $\text{y}=\text{x}\sin^{-1}\text{x}+\sqrt{1-\text{x}^2},$ prove that $\frac{\text{dy}}{\text{dx}}=\sin^{-1}\text{x}$
→