Download App

Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants5 Marks
Question
$\begin{vmatrix}\text{a}+\text{b}+\text{c}&-\text{c}&-\text{b}\\-\text{c}&\text{a}+\text{b}+\text{c}&-\text{a}\\-\text{b}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix}$
$=2(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a})$
✓

Answer

$\text{L.H.S}=\begin{vmatrix}\text{a}+\text{b}+\text{c}&-\text{c}&-\text{b}\\-\text{c}&\text{a}+\text{b}+\text{c}&-\text{a}\\-\text{b}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix}$
$=\begin{vmatrix}\text{a}&-\text{c}&-\text{b}\\\text{b}&\text{a}+\text{b}+\text{c}&-\text{a}\\\text{c}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix} [$Applying $C_1 \rightarrow C_1 + C_2 + C_3]$
$=\begin{vmatrix}\text{a}+\text{b}&\text{a}+\text{b}&-(\text{a}+\text{b})\\\text{b}+\text{c}&\text{b}+\text{c}&\text{b}+\text{c}\\\text{c}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix} [$Applying $R_1 \rightarrow R_1 + R_2$ and $R_2 \rightarrow R_2 + R_3]$
$=(\text{a}+\text{b})(\text{b}+\text{c})\begin{vmatrix}1&1&-1\\1&1&1\\\text{c}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix}$
$[$Taking out common factor from $R_1$ and $R_2]$
$=(\text{a}+\text{b})(\text{b}+\text{c})\begin{vmatrix}0&0&-2\\1&1&1\\\text{c}&-\text{a}&\text{a}+\text{b}+\text{c}\end{vmatrix} [$Applying $R_1 \rightarrow R_1- R_2]$
$=(\text{a}+\text{b})(\text{b}+\text{c})\{(-2)(-\text{a}-\text{c})\} [$Expanding along $R_1]$
$=2(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a})$
$=\text{R.H.S}$
Hence proved.

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the points where $f(x) = |\log_e x|$ is not differentiable.
→
For the matrix $\text{A}=\begin{bmatrix}1&1&1\\1&2&-3\\2&-1&3\end{bmatrix}$Show that $\ce{A^3 - 6A^2 + 5A + 11I = 0.}$ Hence, find $A^{-1}.$
 
→
A firm manufactures two types of products $A$ and $B$ and sells them at a profit of $Rs.2$ on type $A$ and $Rs.3$ on type $B.$ Each product is processed on two machines $M_1$ and $M_2.$ Type $A$ requires one minute of processing time on $M_1$ and two minutes of $M_2;$ type $B$ requires one minute on $M_1$ and one minute on $M_2.$ The machine $M_1$ is available for not more than $6$ hours $40$ minutes while machine $M_2$ is available for $10$ hours during any working day. Formulate the problem as a $LPP.$
→
If $\hat{\text{a}}$ and $\hat{\text{b}}$ are unit vectors inclined at an angle $\theta$, prove that$\tan\frac{\theta}{2}=\frac{\big|\hat{\text{a}}-\hat{\text{b}}\big|}{\big|\hat{\text{a}}+\hat{\text{b}}\big|}$
→
Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is $\frac{1}{4}$ and that to Tarun's rejection is $\frac{2}{3}$. Find the probability that at least one of them will be selected.
→
Solve the differential equation: $\frac{\text{dy}}{\text{dx}}-\frac{2\text{x}}{1+\text{x}^2}\text{y}=\text{x}^2+2$
→
Maximize $Z = 9x + 3y$ Subject to $2\text{x}+3\text{y}\leq13 , 3\text{x}+\text{y}\leq5 , \text{x},\text{y}\geq0$
→
$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$
→
Show that the following system of linear equation is inconsistent:
$3x − y − 2z = 2$
$2y − z = −1$
$3x − 5y = 3$
→
Five defective mangoes are acciedently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.
→