COMPLEX NUMBERS | CONJUGATE OF A COMPLEX NUMBER, PROPERTIES OF CONJUGATE OF A COMPLEX NUMBER, PROPERTIES OF MODULUS OF A COMPLEX NUMBER, RECIPROCAL OF A COMPLEX NUMBER, TYPE OF EXAMPLES | Conjugate of a complex no and its properties. If `z, z_1, z_2` are complex no.; then :- (i) `bar(barz)=z` (ii)`z+barz=2Re(z)`(iii)`z-barz=2i Im(z)` (iv)`z=barz hArr z` is purely real (v) `z+barz=0implies` z is purely imaginary (vi)`zbarz=[Re(z)]^2+[Im(z)]^2`, Properties of a complex no. If `z;z_1;z_2` are complex no.; then (vii)`bar(z_1+z_2)=barz_2+barz_1` (viii)`bar(z_1-z_2)=barz_1-barz_2` (ix)`bar(z_1z_2)=barz_1barz_2` (x) `(barz_1)/z_2=barz_1/barz_2` where `z_2!=0`, Modulus of a Complex Number & its properties If `z;z_1;z_2inCC` then (i)`|z|=0hArrz=0 i.e. Re(z)=Im(z)=0` (ii)`|z|=|barz|=|-z|` (iii) `-|z|leRe(z)le|z|;-|z|leIm(z)le|z|` (iv) `zbarz=|z|^2` (v)`|z_1z_2|=|z_1||z_2|` (vi)`|(z_1)/(z_2)|=|z_1|/|z_2|; z_2!=0`, If `z,z_1,z_2inCC` then (vii)`|z_1+z_2|^2=|z_1|^2+|z_2|^2+2Re(z_1barz_2)` (viii)`|z_1-z_2|^2=|z_1|^2+|z_2|^2-2Re(z_1barz_2)` (ix)`|z_1+z_2|^2+|z_1-z_2|^2=2(|z_1|^2+|z_2|^2)` (x) `|az_1-bz_2|^2+|bz_1+az_2|^2=(a^2+b^2)(|z_1|^2+|z_2|^2)` where `a,b in RR`, Find the reciprocal of a complex no., Express a complex no. in standard form `a+ib` (i) `(-5i)(1/8 i) (ii)(5i)(-3/5i) (iii) (1-i)^4`, Find the real values of x and y if (i)`(3x-7)+2iy=-5y+(5+x)i` (ii)`(x+iy)(2-3i)=(4+i)`, (i)If z is a complex no. such that `|z|=1`; prove that `(z-1)/(z+1)` is purely imaginary . what will be the conclusion if z=1 (ii) Find real `theta` such that `(3+2isintheta)/(1-2isintheta)` is purely real.