Number Theory T...

If $$\alpha$$ and $$\beta$$ are the roots of $${ 4x }^{ 2 }-16x+c=0,$$ c>0 such that $$1<\alpha <2<\beta <3$$, then the no.of integer values of c is 

$$\sqrt { \left( \log _ { 3 } \tan x \right) }$$  is real for:

Let P$$\left( x \right) ={ x }^{ 3 }-6{ x }^{ 2 }+Bx+C$$ has 1+5i as a zero and B,C real number, then value of (B+C) is

If $$\left| {z - 3 + 2i} \right|\, \le 4$$ then the difference between the greatest value and the least value of $$\left| z \right|$$ is :

Let  $$\left| z _ { i } \right| = i , i = 1,2,3,4$$  and  $$\left| 16 z _ { 1 } z _ { 2 } z _ { 3 } + 9 z _ { 1 } z _ { 2 } z _ { 4 } + 4 z _ { 1 } z _ { 3 } z _ { 4 } + z _ { 2 } z _ { 3 } z _ { 4 } \right| = 48 ,$$  then the value of    $$\left| \dfrac { 1 }{ \overline { z } _{ { 1 } } } +\dfrac { 4 }{ \overline { z } _{ { 2 } } } +\dfrac { 9 }{ \overline { z } _{ { 3 } } } +\dfrac { 16 }{ \overline { z } _{ { 4 } } }  \right| .$$

Find the value of $${x}^{3}+7{x}^{2}-x+16$$, when $$x=1+2i$$

Let 'z' be a complex number satisfying $$|z-2-i|\le 5,$$ Then |z-14-6i| lies in 

If  $$w = \dfrac { z } { z - \dfrac { 1 } { 3 } i }$$  and  $$| w | = 1$$  then  $$z$$  lies on

The real part of  $$\left[ 1 + \cos \left( \dfrac { \pi } { 5 } \right) + i \sin \left( \dfrac { \pi } { 5 } \right) \right] ^ { - 1 }$$  is

$$\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}=$$