G Tewani Solutions for Chapter: Determinant, Exercise 4: DPP

The equations $(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$, $(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$ and $2x+(3\lambda +1)y +3(\lambda -1)z=0$ gives non-trivial solution for some values of $\lambda$. Then the ratio $x:y:z$ when $\lambda$ has the smallest of these values, is

The system of homogeneous equations:

$tx+\left(t+1\right)y+\left(t-1\right)z=0,$

$\left(t+1\right)x+ty+\left(t+2\right)z=0,$

$\left(t-1\right)x+\left(t+2\right)y+tz=0$ has a non-trivial solution for

If $a, b, c$ are nonzero, then the system of equations, $(\alpha +a)x+\alpha y+\alpha z=0$, $\alpha x+(\alpha +b)y+\alpha z=0$ and $\alpha x+\alpha y+(\alpha +c)z=0$ has a non-trivial solution, if

The values of $\theta$, $\lambda$ for which the following equations,

 $\sin \theta x-\cos \theta y+(\lambda +1)z=0;$

$\cos \theta x+\sin \theta y-\lambda z=0;$

$\lambda x+(\lambda +1)y+\cos \theta z=0$, have non trivial solution, is

If the system of equations, $x-2y+z=a, 2x+y-2z=b$ and $x+3y-3z=c$ have at least one solution, then

If $A,B,C$ are the angles of a triangle, the system of equations,

$\left(\sin A\right)x+y+z=\cos A,$

$x+\left(\sin B\right)y+z=\cos B,$

$x+y+\left(\sin C\right)z=1-\cos C$

has

A solution set of the equations,

  $x+2y+z=1,$

$x+3y+4z=k,$

$x+5y+10z=k^2$is

Consider the system of equations: $x\mathrm{sin\theta }-2y\mathrm{cos\theta }-az=0$, $x+2y+z=0$ and $-x+y+z=0, \theta \in R$.