Download App

Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants5 Marks
Question
Evaluate: $\begin{vmatrix}\text{x}+\lambda&\text{x}&\text{x}\\\text{x}&\text{x}+\lambda&\text{x}\\\text{x}&\text{x}&\text{x}+\lambda\end{vmatrix}$
✓

Answer

$\triangle=\begin{vmatrix}\text{x}+\lambda&\text{x}&\text{x}\\\text{x}&\text{x}+\lambda&\text{x}\\\text{x}&\text{x}&\text{x}+\lambda\end{vmatrix}$
$=\begin{vmatrix}\lambda&0&\text{x}\\-\lambda&\lambda&\text{x}\\0&-\lambda&\text{x}+\lambda\end{vmatrix} [$Applying $C_1 \rightarrow C_1 - C_2, C_2 \rightarrow C_2 - C_3]$
$=\begin{vmatrix}\lambda&0&\text{x}\\-\lambda&0&2\text{x}+\lambda\\0&-\lambda&\text{x}+\lambda\end{vmatrix} [$Applying $R_1 \rightarrow R_2 + R_3]$
$=\lambda\begin{vmatrix}0&2\text{x}+\lambda\\-\lambda&\text{x}+\lambda\end{vmatrix}+\text{x}\begin{vmatrix}-\lambda&0\\0&-\lambda\end{vmatrix}$
$=\lambda\big[\lambda(2\text{x}+\lambda)\big]+\text{x}\lambda^2$
$=\lambda^2(2\text{x}+\lambda+\lambda^2\text{x})$
$=3\lambda^2\text{x}+\lambda^3$
$=\lambda^2(3\text{x}+\lambda)$

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

If $\text{y}=\sqrt{\text{a}^2-\text{x}^2},$ prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=0$
→
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point $(−1, 1).$
→
If $\text{x}=\text{a}\sin\text{t}\ \text{and}\ \text{y}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2}),$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
→
Determine if f defined by:
$\text{f(x)}=\begin{cases}\text{x}^{2} \sin\frac{1}{\text{x}}, \text{if} \ \text{x}\neq0\\0, \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \text{x} = 0\end{cases}$
→
Solve the matrix equations:
$\begin{bmatrix}\text{x}&1\end{bmatrix}\begin{bmatrix}1&0\\-2&-3\end{bmatrix}\begin{bmatrix}\text{x}\\5\end{bmatrix}=0$
→
Find the area of the region bounded by the ellipse $\frac{\text{x}^2}{16}+\frac{\text{y}^2}{9}=1.$
→
If $\tan(\text{x}+\text{y})+\tan(\text{x}+\text{y})=1,$ find $\frac{\text{dy}}{\text{dx}}$
→
Let N denote the set of all natural numbers and R be the relation on$\text{N} \times \text{N}$defined by $\text{(a, b) R (c, d)}$ if ad$\text{(b + c) = bc(a + d)}$. Show that R is an equivalence relation.
→
Solve the following LPP graphically:
Maximise Z = 1000x + 600y
subject to the constraints
$\text{ }\text{x + y} \leq 200\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{x} \geq 20\\ \text{ }\text{ }\text{ }\text{ }\text{y - 4x} \geq 0\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{x, y} \geq 0. $
→
Differentiate the following functions with respect to x:
$\log(3\text{x}+2)-\text{x}^2\log(2\text{x}-1)$
→