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

Show that f(x) = |x - 5| is continuous but not differentiable at x = 5.
→
Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\text{y}=\cos^{-1}\Bigg(\frac{2\text{x}}{1+\text{x}^{2}}\Bigg), -1<\text{x}<1$
→
Suppose we have four boxes A,B,C and D containing coloured marbles as given below:
Box           Marble colour
  Red White Black
A 1 6 3
B 6 2 2
C 8 1 1
D 0 6 4
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?
→
Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=(\text{x}+1)\text{e}^{\text{x}},\text{ y}(1)=0$
→
$\int\frac{\text{x}^2+\text{x}+5}{3\text{x}+2}\text{dx}$
→
Evaluate :
$\int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x$
→
Show that the relation R defined by R = {(a, b): a - b is divisible by 3; a, b ∈ Z} is an equivalence relation.
→
Evaluate the following integrals:$\int\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)\text{dx}$
→
Find the coordinate of the point P where the line through $\text{A(3, – 4, –5) and B (2, –3, 1)}$ crosses the plane passing through three points L(2, $\text{2, 1), M(3, 0, 1) and N(4, –1, 0).}$ Also, find the ratio in which P divides the line segment AB.
 
→
Differentiate the following functions with respect to x:
$\text{e}^\text{x}\log\sin2\text{x}$
→