Find the values x if :
`|[x-3,0,0],[0,x-4,sqrt3],[0,sqrt3,x-6]|=0`

3x^2-4sqrt3x+4=0

Find the general value of x tan 2x - sqrt3 = 0

Find the angle between the lines y-sqrt(3)x-5=0 and sqrt(3)y-x+6=0 .

The locus of the point of intersection of the lines sqrt3 x- y-4sqrt3 t= 0 & sqrt3 tx +ty-4 sqrt3=0 (where t is a parameter) is a hyperbola whose eccentricity is:

Find the roots of the equations: sqrt(3)x^(2)-sqrt(2)x+3sqrt(3)=0 .

Solve x^(2)-(sqrt3+1)x+sqrt3=0 .

If |[x,-6,1],[2x,-3,3],[0,1,2]| =0 find the value of x

Prove that the locus of the point of intersection of the lines sqrt3 x -y -4 sqrt3 k =0 and sqrt3 k x+ ky - 4 sqrt3 =0, for different values of k, is a hyperbola whose eccentricity is 2.

The equations of lines passing through the point (1,0) and at distance of (sqrt(3))/(2) units from the origin are (i) sqrt(3)x+y-sqrt(3)=0,sqrt(3)x-ysqrt(3)=0 (ii) sqrt(3)x+y+sqrt(3)=0,sqrt(3)x-y+sqrt(3)=0 (iii) x+sqrt(3)y-sqrt(3)=0,x-sqrt(3)y-sqrt(3)=0 (iv) none of these

Find the distance between the parallel lines x+4sqrt(3)y+10=0 and x+4sqrt(3)y-18=0 .