Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Prove that:
$\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$
✓

Answer

$\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$
$\text{L.H.S}=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
Apply: $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_1$
$=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&\text{b}^2+\text{ca}-\text{a}^2-\text{bc}&\text{b}^3-\text{a}^3\\0&\text{c}^2+\text{abb}+\text{ca}-\text{a}^2-\text{bc}&\text{c}^3-\text{a}^3 \end{vmatrix}$
$=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&(\text{b}^2-\text{a}^2)-\text{c}(\text{b}-\text{a})&\text{b}^3-\text{a}^3\\0&(\text{c}^2-\text{a}^2)-\text{b}(\text{c}-\text{a})&\text{c}^3-\text{a}^3 \end{vmatrix}$
$=\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&(\text{b}-\text{a})(\text{b}+\text{a}-\text{c})&\text{b}^3-\text{a}^3\\0&(\text{c}-\text{a})(\text{c}+\text{a}-\text{b})&\text{c}^3-\text{a}^3 \end{vmatrix}$
$=(\text{b}-\text{a})(\text{c}-\text{a})\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\0&(\text{b}+\text{a}-\text{c})&\text{b}^2+\text{a}^2+\text{ab}\\0&(\text{c}+\text{a}-\text{b})&\text{c}^2+\text{a}^2+\text{ac} \end{vmatrix}$
$=(\text{b}-\text{a})(\text{c}-\text{a})\big[(\text{b}+\text{a}-\text{c})(\text{c}^2+\text{a}^2+\text{ac})\\-(\text{b}^2+\text{a}^2+\text{ab})(\text{c}^2+\text{a}^2+\text{ac})\big]$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$
$=\text{R.H.S}$

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 foot of the perpendicular drawn from the point $\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}$ to the line $\vec{\text{r}}=\hat{\text{j}}+2\hat{\text{k}}+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the length of the perpendicylar
→
Differentiate the function given in Exercise:
$\text{x}^{\text{x}\cos\text{x}}+\frac{\text{x}^2+1}{\text{x}^2-1}$
→
A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximise his profit? Determine the maximum profit.
→
Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
→
Give that vectors $\overrightarrow{\text{a}}, \overrightarrow{\text{b}}, \overrightarrow{\text{c}}.$ form a triangle such that $\overrightarrow{\text{a}} = \overrightarrow{\text{b}} + \overrightarrow{\text{c}}.$ Find p, q, r, s such that area of triangle is $5\sqrt{6}$ where $\overrightarrow{\text{a}} = \text{p }\hat{i} + \text{q }\hat{j} + \text{r }\hat{k} = \overrightarrow{\text{b}} = \text{s }\hat{i} + \text{3 }\hat{j} + \text{4 }\hat{k} \text{ and} \overrightarrow{\text{c}} = \text{3 }\hat{i} + \hat{j} - \text{2}\hat{k}.$
→
Evaluate: $\int\frac{\text{x sin}^{-1}\text{x}}{{\sqrt{\text{x-1}^{2}}}}\text{dx}.$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2-1}}{\text{ax}}\Big),\text{x}\neq0$
→
Evaluate the following integrals:
$\int^\limits1_{-1}|2\text{x}+1|\text{dx}$
→
Using the method of interation, find the area of the region bounded by the following lines:
3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0.
→
Evaluate the following integrals as limit of sum:
$\int\limits^5_{0}(\text{x}+1)\text{dx}$
→