Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Solve the following determinant equations:
$\begin{vmatrix}\text{x}+\text{a}&\text{x}&\text{x}\\\text{x}&\text{x}+\text{a}&\text{x}\\\text{x}&\text{x}&\text{x}+\text{a}\end{vmatrix}=0,\text{a}\neq0$
✓

Answer

Let $\begin{vmatrix}\text{x}+\text{a}&\text{x}&\text{x}\\\text{x}&\text{x}+\text{a}&\text{x}\\\text{x}&\text{x}&\text{x}+\text{a}\end{vmatrix}$
$=\begin{vmatrix}3\text{x}+\text{a}&\text{x}&\text{x}\\3\text{x}+\text{a}&\text{x}+\text{a}&\text{x}\\3\text{x}+\text{a}&\text{x}&\text{x}+\text{a}\end{vmatrix}$ [Applying $C_1 \rightarrow C_1 + C_2 + C_3$]
$=(3\text{x}+\text{a})\begin{vmatrix}1&\text{x}&\text{x}\\1&\text{x}+\text{a}&\text{x}\\1&\text{x}&\text{x}+\text{a}\end{vmatrix}$
$=(3\text{x}+\text{a})\begin{vmatrix}1&\text{x}&\text{x}\\0&\text{a}&0\\1&\text{x}&\text{x}+\text{a}\end{vmatrix}$ [Applying $R_2 \rightarrow R_2 - R_1$]
$=(3\text{x}+\text{a})\begin{vmatrix}1&\text{x}&\text{x}\\0&\text{a}&0\\1&0&\text{a}\end{vmatrix}$ [Applying $R_3 \rightarrow R_3 - R_1$]
$=(3\text{x}+\text{a})=(\text{a}^2-0)=0$
$\text{x}=\frac{-\text{a}}{3}$

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

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
→
A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are-as follows:
Machine
Area occupied
Labour force
Daily output ( in units)
A
$1000m^2$
12 men
60
B
$1200m^2$
8 men
40
He has maximum area of $9000 \text{m}^{2}$ available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?
→
Evaluate: $\int\frac{2}{\text{(1-x)(1+x)}^{2}}\text{dx}$
→
find the area of the region bounded by y = |x - 1| and y = 1.
→
Find the corrdinates of the points P where the line throught A(3, -4,-5) and B(2, -3, 1) crosses the plane passing throught three points L(2, 2, 1), M(3, 0, 1) and N(4, -1, 0). Also, find the ratio in which P diveides the line segment AB.
→
An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?
→
If $\text{x}=\Big(\text{t}+\frac{1}{\text{t}}\Big)^\text{a},\text{y}=\text{a}^{\text{t}+\frac{1}{\text{t}}},$ find $\frac{\text{dy}}{\text{dx}}$
→
Find the area of the region bounded by the curve $y = x - 1$ and $(y - 1)^2 = 4(x + 1).$
→
Find the maximum and the minimum values, if any, without using derivaives of the following functions:
$f(x) = 16x^2 - 16x + 28$ on R.
→
Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.
→