Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants4 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: C2 → C2 + C1
$=\begin{vmatrix}\text{a}&\text{b}+\text{c}+\text{a}&\text{a}^2\\\text{b}&\text{c}+\text{a}+\text{b}&\text{b}^2\\\text{c}&\text{a}+\text{b}+\text{c}&\text{c}^2\end{vmatrix}$
Take (a + b + c) common from C2
$=(\text{b}+\text{c}+\text{a})\begin{vmatrix}\text{a}&1&\text{a}^2\\\text{b}&1&\text{b}^2\\\text{c}&1&\text{c}^2\end{vmatrix}$
Apply: R2 → R2 - R1, R3 → R3 - R1
$=(\text{b}+\text{c}+\text{a})\begin{vmatrix}\text{a}&1&\text{a}^2\\\text{b}-\text{a}&0&\text{b}^2-\text{a}^2\\\text{c}-\text{a}&0&\text{c}^2-\text{a}^2\end{vmatrix}$
$=(\text{b}+\text{c}+\text{a})(\text{b}-\text{a})(\text{c}-\text{a})\begin{vmatrix}\text{a}&1&\text{a}^2\\1&0&\text{b}+\text{a}\\1&0&\text{c}+\text{a}\end{vmatrix}$
$=(\text{b}+\text{c}+\text{a})(\text{b}-\text{a})(\text{c}-\text{a})(\text{b}-\text{c})$

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 "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

Write the value of $\tan^{-1}\text{x}+\tan^{-1}\Big(\frac{1}{\text{x}}\Big)$ x < 0.
→
Find the cartesian form of the equations of the following planes.
$\vec{\text{r}}=(1+\text{s}+\text{t})\hat{\text{i}}+(2-\text{s}+\text{t})\hat{\text{j}}+(3-2\text{s}+2\text{t})\hat{\text{k}}$ 
→
A merchant plans to sell two types of personal computers - a desktop model and a portable, model that will costRs. 25,000 andRs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make an L.P.P. and solve it graphically.
→
Find a particular solution of the differential equation $(\text{x}-\text{y})(\text{dx}+\text{dy})=\text{dx}-\text{dy},\ \text{given that y}=-1,$ $\text{when x}=0. \ (\text{Hint: put x}-\text{y}=\text{t})$ 
→
The two adjacent sides of a parallelogram are $2\hat{i} - 4\hat{j} - 5\hat{k} \text{ and } 2\hat{i} + 2\hat{j} + 3\hat{k}.$ Find the two unit vectors parallel its diagonals. Using the diagonal vectors, find the area of the parallelogram.
→
If $\text{A}=\frac{1}{9}\begin{bmatrix}-8 & 1 & 4\\4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix},$ prove that A-1 = A3.
→
If $\text{A}=\begin{bmatrix}1&2&0\\3&-4&5\\0&-1&3\end{bmatrix},$ compute A2 - 4A + 3I3.
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\text{ dx}\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\ \text{dx}$
→
$\text{Evaluate:}\int\frac{\text{x + 2}}{\sqrt{\text{x}^{2}+\text{2x}+}\text{3}}\text{dx}$
→
Prove that the line of section of the planes 5x + 2y - 4z + 2 = 0 and 2x + 8y + 2z - 1 = 0 is parallel to the plane 4x - 2y - 5z - 2 = 0.
→