Number Theory Test 54

Result

Number Theory Test 54

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

If $$\mathrm{{z} _ { 1 }} = 10 + 6\mathrm{i} , \mathrm{{ z } _ { 2 }}= 4 + 6 \mathrm { i }$$ and $$\mathrm{ z}$$ is a complex number such that $$\operatorname { amp } \left( \dfrac { \mathrm { z } - \mathrm { z } _ { 1 } } { \mathrm { z } - \mathrm { z } _ { 2 } } \right) = \dfrac { \pi } { 4 }$$ , then the value of $$\left| \mathrm{z} - 7 - 9 \mathrm { i } \right|$$ is equal to

A
$$\sqrt { 2 }$$

B
$$2\sqrt { 2 }$$

C
$$3\sqrt { 2 }$$

D
$$2\sqrt { 3 }$$
Question 2
1 / -0

The modulus of the complex number $$z$$ such that $$\left| z + 3 - i\right | = 1$$ and $$\arg{z} = \pi$$ is equal to

A
$$1$$

B
$$2$$

C
$$4$$

D
$$3$$

Solution

$$arg(z)=\pi$$
Let $$z=x+iy$$
$$arg(x+iy)=\pi$$
$$\Rightarrow x < 0; y=0$$
$$|z+3-i|=1$$
$$|x+3-i|=1$$
$$(x+3)^2+1^2=1^2$$
$$\Rightarrow (x+3)^2=0$$
$$x=-3$$
$$|z|=|-3+0i|=3$$.
Question 3
1 / -0

The roots of the equation $$(3b+c-4a)x^2+(3c+a-4b)x+(3a+b-4c)= 0$$ are

A
Irrational

B
Rational

C
Non-real

D
Imaginary

Solution
Question 4
1 / -0

$$\frac { { z }_{ 2 }-{ 2z }_{ 2 } }{ { z }_{ 2 }-{ z }_{ 1 }{ z }_{ 2 } } $$ is unimodular then

A
$$|{ z }_{ 2 }|=2$$

B
$$|{ z }_{ 1 }|=1$$

C
Both A and B

D
None of these

Solution
Question 5
1 / -0

If z is a complex number of unit modules and argument $$\theta $$, then the real part of $$\dfrac { z(1-\bar { z } ) }{ z(1+z) } $$ is :

A
$${ -2sin }^{ 2 }\dfrac { \theta }{ 2 } $$

B
$${ 2sin }^{ 2 }\dfrac { \theta }{ 2 } $$

C
$$1+cos\dfrac { \theta }{ 2 } $$

D
$$1-cos\dfrac { \theta }{ 2 } $$
Question 6
1 / -0

This equation $$(x-5)^{11}+(x-5^{2})^{11}+....+(x-5^{11})^{11}=0$$ has

A
all the roots real

B
one real and 10 imaginary roots

C
real roots namely $$x=5,5^{2}....,5^{9},5^{10},5^{11}$$

D
none

Solution
Question 7
1 / -0

If z= $$\dfrac { 1 }{ { \left( 2+3i \right) }^{ 2 } } ,\quad then\left| z \right| $$=

A
$$\dfrac { 1 }{ 13 } $$

B
$$\dfrac { 1 }{ 5 } $$

C
$$\dfrac { 1 }{ 12 } $$

D
none of these

Solution
Question 8
1 / -0

The complex numbers $${ z }_{ 1 },{ z }_{ 2 },{ z }_{ 2 }$$ satisfying $$\dfrac { { z }_{ 1 }+{ z }_{ 3 } }{ { z }_{ 2 }-{ z }_{ 3 } } =\dfrac { 1-i\sqrt { 3 } }{ 2 } $$

A
If area zero

B
right angled isosceles

C
equilateral

D
obtuse angled
Question 9
1 / -0

If $$z$$ is a complex number such that $$| z | = 1 , z \neq 1 ,$$ then the real part of $$\frac { z - 1 } { z + 1 }$$ is

A
$$\frac { 1 } { | z + 1 | ^ { 2 } }$$

B
$$\frac {- 1 } { | z + 1 | ^ { 2 } }$$

C
$$\frac { \sqrt 2 } { | z + 1 | ^ { 2 } }$$

D
0
Question 10
1 / -0

If z be any complex number such that $$|3z-2|+|3z+2|=4$$, then locus of z is

A
An ellipse

B
A circle

C
A line-segment

D
None of these

Solution