If abc neq 0 and the system of equations x+7a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For the system of linear equations to have a non-trivial solution,the determinant of the coefficient matrix must be zero. The system is: $x + 7ay

Vedclass · Accepted Answer

harmonic progression

Vedclass · Answer

(A) For the system of linear equations to have a non-trivial solution,the determinant of the coefficient matrix must be zero. The system is: $x + 7ay + 2az = 0$ $x + 6by + 2bz = 0$ $x + 5cy + 2cz = 0$ The determinant $D$ is given by: $D = \begin{vmatrix} 1 & 7a & 2a \ 1 & 6b & 2b \ 1 & 5c & 2c \end{vmatrix} = 0$ Expanding the determinant along the first column: $1(12bc - 10bc) - 1(14ac - 10ac) + 1(14ab - 12ab) = 0$ $2bc - 4ac + 2ab = 0$ Dividing by $2$: $bc - 2ac + ab = 0$ $2ac = ab + bc$ Dividing both sides by $abc$ (since $abc 
eq 0$): $\frac{2ac}{abc} = \frac{ab}{abc} + \frac{bc}{abc}$ $\frac{2}{b} = \frac{1}{c} + \frac{1}{a}$ This condition implies that $a, b, c$ are in Harmonic Progression. Thus,option $A$ is correct.

If $abc \neq 0$ and the system of equations $x+7ay+2az=0$,$x+6by+2bz=0$,$x+5cy+2cz=0$ has a non-trivial solution,then $a, b, c$ are in

Similar Questions

Let $A = \begin{bmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{bmatrix}$. If $|A|^2 = 25$,then $|\alpha|$ equals:

If $A$ and $B$ are square matrices of order $3$ such that $|A| = -1$ and $|B| = 3$,then find the value of $|3AB|$.

Find the equation of the line joining $A(1, 3)$ and $B(0, 0)$ using determinants and find $k$ if $D(k, 0)$ is a point such that the area of triangle $ABD$ is $3 \, \text{sq units}$.

The value of $x$ obtained from the equation $\left| \begin{array}{ccc} x + \alpha & \beta & \gamma \\ \gamma & x + \beta & \alpha \\ \alpha & \beta & x + \gamma \end{array} \right| = 0$ is:

The determinant $\left| \begin{array}{ccc} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{array} \right| = 0$,if $a, b, c$ are in

Vedclass Test Series

Exam Paper Generator

Online Exam Module