Prove that the complex numbers z1, z2 origin form an equilateral triangle only if
My Self Assessment
Fill in the blank
If the product of the roots of the equation is 7 Then the roots are real for ………….
The solution of the equation is …………..
If the quadratic equation and have a common root then the numerical value of a + b is …………
There are exactly two distinct linear functions ………. and ………. which map {−1, 1} onto {0, 2}.
If α, β, γ are the cube roots of P, P < 0, then for any x, y, z, ………..
If x < 0, y < 0, and then x ……….. and y ………..
The sum of the real roots of the equation is ………..
The least value of the expression for x > 1 is
a) 10
b) 2
c) −0.01
d) none of these
If then x lies in the interval
a) (2, ∞)
b) (1, 2)
c) (−2, −1)
d) None of these
The number is
a) an integer
b) a rational number
c) an irrational number
d) a prime number
(Subjective problem)
Solve where a > 0, b = a2x.
Find |z| and Arg (z) when
Show that equation of the form where a and r are real constants and represents a straight line if a = 0 and circle if a ≠ 0.
Find |z| and Arg (z) when and – π < θ < 0
Find |z| and Arg (z) when and 0 < θ < π
Prove that Re (iz) = −Im(z) and Im (iz) = Re (z)
If |z1| = |z2| = . . . = |zn| = 1 then show that
Let and are two non-zero complex numbers. If both + and are real then either and are real or
Let z1 and z2 are two non-zero complex numbers. If |z1 + z2| = |z1 – z2| then show that i.e. the ratio z1/z2 is purely imaginary.
Find complex numbers z which simultaneously satisfy the equations and
Given that ≠ 0 show that if and only if arg () – arg () = 2nπ for any integer n.
If α + iβ is a root of the polynomial Pn (z) = zn + zn – 1 + . . . + z + = 0
Then show that α – iβ is also a root of Pn(z) = 0 where the coefficients , , , . . ., are real.
