Number Theory Test 51

Result

Number Theory Test 51

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 $${z}_{1},{z}_{2},..{z}_{n}$$ lie on the circle $$|z|=2$$ then the value of $$|{z}_{1},{z}_{2},..{z}_{n}|-4|\dfrac {1}{{z}_{1}}+\dfrac {1}{{z}_{2}}++\dfrac {1}{{z}_{n}}|=$$

A
$$0$$

B
$$n$$

C
$$-n$$

D
$$1$$

Solution
Question 2
1 / -0

What is $${ i }^{ 1000 }+{ i }^{ 1001 }+{ i }^{ 1002 }+{ i }^{ 1003 }$$ equal to (where $$i=\sqrt { -1 } $$)?

A
$$0$$

B
$$i$$

C
$$-i$$

D
$$1$$

Solution

$$i^{1000} + i^{100} + i^{1002} + i^{1003}$$
$$i^{2} = -1, i^{3} = -1, i^{4} = 1$$
$$i^{1000} = (i^{4})^{250} = 1\ i^{1002} = i^{2} - i^{1000} = -1$$
$$i^{1001} = i\cdot i^{1000} = i\ i^{1003} = i^{3}\cdot i^{1000} = -i$$
$$\therefore i^{1000} + i^{1001} + i^{1002} + i^{1003} = 1 - 1 + i - i = 0$$
Question 3
1 / -0

If $$\sqrt{3}+i(a+ib)(c+id)$$, then $$\tan^{-1}\left(\dfrac{b}{a}\right)+\tan^{-1}\left(\dfrac{d}{c}\right)$$ has the value

A
$$\dfrac{\pi}{3}+2n\pi, n \notin 1$$

B
$$n\pi+\dfrac{\pi}{6}, n \in 1$$

C
$$n\pi-\dfrac{\pi}{3}, n \in 1$$

D
$$2n\pi-\dfrac{\pi}{3}, n \in 1$$

Solution
Question 4
1 / -0

If $$Z_{1},Z_{2}$$ are two complex numbers satisfying $$|\dfrac{Z_{1}-3Z_{2}}{3-Z_{1}Z_{2}}|=1|z_{1}|\neq 3$$ then $$|z_{2}|=$$

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 5
1 / -0

A complex number z is said to be unimodular if $$|z| =1. $$. Suppose $$z_1$$ and $$z_2$$ are complex numbers such that $$\frac{z_1-2z_2}{2-z_1\overline {z}_2}$$ is unimolecolar and $$z_2$$ is not unimodular. Then the point $$z_1$$ lies on a:

A
straight line parallel to y-axis

B
circle of radius 2.

C
Circle of radius $$\sqrt2$$.

D
Staright line parallel to x-axis.

Solution
Question 6
1 / -0

For $${ { Z }_{ 1 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ 1+i\sqrt { 3 } } } };\quad { { Z }_{ 2 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ \sqrt { 3 } +i } } ;\quad { { Z }_{ 3 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ \sqrt { 3 } -i } } } }$$ which of the following holds good?

A
$$\sum { { \left| { Z }_{ 1 } \right| }^{ 2 } } =\dfrac { 3 }{ 2 } $$

B
$${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ -8 }$$

C
$$\sum { { \left| { Z }_{ 1 } \right| }^{ 3 }+ } { \left| { Z }_{ 2 } \right| }^{ 3 }={ \left| { Z }_{ 3 } \right| }^{ -6 }$$

D
$${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ 8 }$$
Question 7
1 / -0

The value of $$(z+3) (\overline{z} +3)$$ is eqquivalent to

A
$$|z +3 | ^2$$

B
$$| z- 3|$$

C
$$z^2+3$$

D
None of these

Solution

$$(2+3)(\bar{2}+3)$$
We know that $$2.\bar{2}=|2|^{2}$$
$$(2+3).(\overline{2+3})$$
$$=12+31^{2}$$
Question 8
1 / -0

The imaginary part of $$(z - 1)(\cos \, \alpha - i \, \sin \, \alpha) + (z - 1)^{-1} \times (\cos \, \alpha + i \, \sin \, \alpha ) $$ is zero, if

A
$$|z - 1 | = 2$$

B
$$\text{arg} \, (z - 1) = 2 \alpha$$

C
$$\text{arg} \, (z - 1) = \alpha$$

D
$$|z | = 1$$
Question 9
1 / -0

If $$\left| z \right| \ge 5$$ then the least value $$\left| {z + \frac{2}{z}} \right|$$ is

A
$$\frac{{23}}{5}$$

B
$$\frac{{24}}{5}$$

C
$$5$$

D
none of these

Solution
Question 10
1 / -0

For a complex number $$z$$, the minimum value of $$\left | z \right |+\left | z-\cos\alpha-i\sin\alpha \right |$$ is

A
0

B
1

C
2

D
None of these

Solution