If `z=lamda+3+isqrt(3-lamda^2)`, for all real `lamda`, then the locus of `z` is

If z= (lamda + 3)- isqrt(5-lamda^(2)) , then the locus of z is a

If z=lambda+3+iota sqrt(5-) lamda ^(^^)2)^(@), then the locus of z is a (1) Circle (3) Parabola 85. (2) Straight line (4) None of these

If p lamda ^(4) + q lamda ^(3) + r lamda ^(2) + s lamda +t = |{:(lamda ^(2) + 3 lamda, lamda -1, lamda +3), ( lamda +1, 2 - lamda , lamda -4), ( lamda -3, lamda+4, 3lamda):}|. then the value of t is

If the following equations x+y-3z = 0, (1+lamda)x + (2+lamda) y - 8z = 0, x-(1+lamda)y + (2+lamda)z = 0 are consistent then the value of lamda is

If the following equations x+y-3z = 0, (1+lamda)x + (2+lamda) y - 8z = 0, x-(1+lamda)y + (2+lamda)z = 0 are consistent then the value of lamda is

The matrix A={:[(lamda_(1)^(2),lamda_(1)lamda_(2),lamda_(1)lamda_(3)),(lamda_(2)lamda_(1),lamda_(2)^(2),lamda_(2)lamda_(3)),(lamda_(3)lamda_(1),lamda_(3)lamda_(2),lamda_(3)^(2))]:} is idempotent if lamda_(1)^(2)+lamda_(2)^(2)+lamda_(3)^(2)=k where lamda_(1),lamda_(2),lamda_(3) are non-zero real numbers. Then the value of (10+k)^(2) is . . .

The matrix A={:[(lamda_(1)^(2),lamda_(1)lamda_(2),lamda_(1)lamda_(3)),(lamda_(2)lamda_(1),lamda_(2)^(2),lamda_(2)lamda_(3)),(lamda_(3)lamda_(1),lamda_(3)lamda_(2),lamda_(3)^(2))]:} is idempotent if lamda_(1)^(2)+lamda_(2)^(2)+lamda_(3)^(2)=k where lamda_(1),lamda_(2),lamda_(3) are non-zero real numbers. Then the value of (10+k)^(2) is . . .

A line passing through the point (1,1,1) from a tiangle of area sqrt(6) sq. unit with the lines x=y=z and x=(y)/(2)=(z)/(3) . If the point of intersection of the line with the second line be (lamda,2lamda,3lamda) , then the value of lamda is -