Statement-1: Roots of the equation (1+z)6+z6=0 are collinear. Statement-2: If z1, z2, z3 are in A.P. then points represented by z1, z2, z3 are collinear

Question

Statement-1: Roots of the equation (1+z)6+z6=0 are collinear. Statement-2: If z1, z2, z3 are in A.P. then points represented by z1, z2, z3 are collinear (A) Both the statements are true. (B) Statement-I is true, but Statement-II is false. (C) Statement-I is false, but Statement-II is true. (D) Both the statements are false.

Tardigrade Admin · Accepted Answer

Statement -1 (1+z)6=-z6 take modulus |1+z|6=|z|6 |(1+z/z)|=1 which is straight line Statement -2 z2=(z1+z3/2) ∴ z2 is mid point of line joining z1 z3. Hence z1, z2, z3 are collinear

Q. Statement-1 : Roots of the equation $(1 + z)^{6} + z^{6} = 0$ are collinear.
Statement-2 : If $z_{1}, z_{2}, z_{3}$ are in A.P. then points represented by $z_{1}, z_{2}, z_{3}$ are collinear

Both the statements are true.

Statement-I is true, but Statement-II is false.

Statement-I is false, but Statement-II is true.

Both the statements are false.

Statement -1 $(1 + z)^{6} = - z^{6}$
take modulus
$∣1 + z ∣^{6} = ∣ z ∣^{6}$
$∣ ∣ \frac{1 + z}{z} ∣ ∣ = 1$ which is straight line
Statement $- 2 z_{2} = \frac{z _{1} + z _{3}}{2}$
$∴ z_{2}$ is mid point of line joining $z_{1} & z_{3}$ .
Hence $z_{1}, z_{2}, z_{3}$ are collinear

Q. Statement-1 : Roots of the equation (1+z)6+z6=0 are collinear. Statement-2 : If z1​,z2​,z3​ are in A.P. then points represented by z1​,z2​,z3​ are collinear