First slide

Introduction to Determinants

Question

If the system of linear equations $x + 2 a y + a z = 0$ , $x + 3 b y + b z = 0$ , $x + 4 c y + c z = 0$ ,has a non-zero solution, then $a, b, c$

Moderate

A

are in G.P
B

are in H.P.
C

satisfy $a + 2 b + 3 c = 0$
D

are in A.P.

Solution

The system will have a non-zero solution if
$Δ = |\begin{matrix} 1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c \end{matrix}| = 0$
Using $C_{2} \to C_{2} - 2 C_{3}$ , we get
$Δ = |\begin{matrix} 1 & 0 & a \\ 1 & b & b \\ 1 & 2 c & c \end{matrix}| = 0$
$\Rightarrow 1 (b c - 2 b c) + a (2 c - b) = 0 \Rightarrow 2 a c = b c + a b$
$\Rightarrow \frac{2 a c}{a + c} = b \Rightarrow a, b, c$ are in H.P.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App