117821 If α and β are different complex numbers with |β|=1, then |β−α1−α¯β| is equal to
C Givenα and β are different complex number and |β|=1 We know that- |z|2=zz¯∴|β|2=(1)2|β|2=1∴ββ¯=1 Now, |β−α1−α¯β|=|β−αββ¯−α¯β|=|β−α||β|||β−α∣=1|β| Thus, |β−α1−α¯β|=1Thus, |β−α1−α¯β|=1
117822 If z is complex number of unit modulus and argument θ,then arg(1+z1+z¯) equals
B Given, |z|=1 and arg(z)=θ(1+z)(1+z¯)=1+z1+1z=(1+z)z(1+z)=zThus the arg of the complex number (1+z1+z¯) is just the arg of z and that is θ.
117824 The modulus of [1−cosθ+isinθ]−1 is
C |11−cosθ+isinθ|=1|1−cosθ+isinθ|=1(1−cosθ)2+sin2θ=11+cos2θ−2cosθ+sin2θ=12−2cosθ=12(1−cosθ)=12⋅2sin2θ2=12|sin(θ/2)|=12|cosecθ2|.
117825 If |z1|=|z2|=…….|zn|=1, then the value of |z1+z2+…….zn|−|1z1+1z2+…….+1zn| is,
A We know that, |z|2=zz―z1z―1=z2z―2=…..=znz―n=1z―1=1z1,z―2=1z2,z―3=1z3,…..z―n=1zn∴|z1+z2+…..+zn|−|1z1+1z2+…..+1zn|=|z1+z2+….+zn|−|z―1+z―2+…..+z―n|=0
117826 The modulus of the complex number z such that |z+3−i|=1 and arg(z)=π is equal to
A Let z=x+iy∴|z+3−i|=|(x+3)+i(y−1)|=1∵argz=π⇒tan−1yx=πyx=tanπ=0⇒y=0Form equations (i) and (ii), we getx=−3,y=0,z=−3⇒|z|=|−3|=3