If a system of three linear equations `x+4a y+a=0,x+3b y+b=0, and x+2c y+c=0` is consistent, then prove that `a ,b , c` are in H.P.

If the equation 2x+3y+1=0,3x+y-2=0, and a x+2y-b=0 are consistent, then prove that a-b=2.

If the equation 2x+3y+1=0,3x+y-2=0,and ax+2y-b=0 are consistent, then prove that a-b=2.

If c<1 and the system of equations x+y-1=0,2x-y-c=0,and -bx+3b y-c=0 is consistent, then the possible real values of b are

If c<1 and the system of equations x+y-1=0,2x-y-c=0,a n d -b x+3b y-c=0 is consistent, then the possible real values of b are b ∈(-3 3/4) b. b ∈(-3/2,4) c. b∈ (-3/4,3) d. none of these

If the lines x+a y+a=0,\ b x+y+b=0\ a n d\ c x+c y+1=0 are concurrent, then write the value of 2a b c-a b-b c-c adot

If the system of linear equations x+a y+a z=0,\ x+b y+b z=0,\ x+c y+c z=0 has a non zero solution then (a) System is always non trivial solutions (b) System is consistent only when a=b=c (c) If a!=b!=c then x=0,\ y=t ,\ z=-t\ AAt in R (d) If a=b=c\ t h e n\ y=t_1, z=t_2,\ x=-a(t_1+t_2)AAt_1, t_2 in R

If x =a, y=b, z=c is a solution of the system of linear equations x + 8y + 7z =0, 9 x + 2y + 3z =0, x + y+z=0 such that point (a,b,c ) lies on the plane x + 2y + z=6, then 2a+ b+c equals

If the lines a x+12 y+1=0,\ b x+13 y+1=0 and c x+14 y+1=0 are concurrent, then a , b , c are in a. H.P. b. G.P. c. A.P. d. none of these

If the system of equations ax + by + c = 0,bx + cy +a = 0, cx + ay + b = 0 has infinitely many solutions then the system of equations (b + c) x +(c + a)y + (a + b) z = 0 (c + a) x + (a+b) y + (b + c) z = 0 (a + b) x + (b + c) y +(c + a) z = 0 has

x=2, y=−1 is a solution of the linear equation (a) x+2y=0 (b) x+2y=4 (c) 2x+y=0 (d) 2x+y=5