Complex Numbers and Quadratic Equations Test 42

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Complex Numbers and Quadratic Equations Test 42

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Question 1
1 / -0

If $$z_1+ z_2 + z_3 = 0$$ and $$|z_1|=|z_2|=|z_3|= 1$$, then area of triangle whose vertices are $$z_1, z_2$$ and $$z_3$$ is:

A
$$\dfrac{3 \sqrt{3}}{4}$$

B
$$\dfrac{\sqrt{3}}{4}$$

C
$$1$$

D
$$2$$

Solution

$${|z^{}_{1}+z^{}_{2}|}^2+{|z^{}_{1}-z^{}_{2}|}^2=2({|z^{}_{1}|}^2+{|z^{}_{2}|}^2)$$ ....... $$(1)$$
$$z^{}_{1}+z^{}_{2}+z^{}_{3}=0$$
$$z^{}_{3}=-z^{}_{2}-z^{}_{1}$$
By taking modulus both sides
$$|-z^{}_{3}|=|z^{}_{2}+z^{}_{1}|$$ ......... $$(2)$$
Using $$(2)$$ in $$(1),$$ we have
$${|-z_{3}|}^2+{|z^{}_{1}-z^{}_{2}|^2}=2{(1+1)}$$
$$1+{|z^{}_{1}-z^{}_{2}|^2}=2{(1+1)}$$
$${|z^{}_{1}-z^{}_{2}|^2}=3$$
$$|z^{}_{1}-z^{}_{2}|=\sqrt{3}$$
Similarily $$|z^{}_{2}-z^{}_{3}|$$=$$|z^{}_{3}-z^{}_{1}|=\sqrt{3}$$
Hence sides of triangle formed are $$\sqrt{3},$$ $$\sqrt{3},$$ $$\sqrt{3}$$ and are equal.
So triangle formed is an equilateral triangle
Area of an equilateral triangle $$=\dfrac{\sqrt{3}}{4}s^2$$
where $$s$$ is side of equilateral triangle
So area of triangle formed by $$z^{}_{1} ,z^{}_{2},z^{}_{3}=3\dfrac{\sqrt{3}}{4}$$
Question 2
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 3
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 4
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
Question 5
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 6
1 / -0

$$|z-4| < |z-2|$$ represents the region given by?

A
$$Re(z) > 3$$

B
$$Re(z) < 0$$

C
$$Re(z) > 2$$

D
None of these

Solution
Question 7
1 / -0

If $$P(x)=a{x}^{2}+bx+c$$ and $$Q(x)=-a{x}^{2}+dx+c$$ where $$ac\ne 0$$, then $$P(x).Q(x)=0$$ has

A
exactly one real root

B
atleast two real roots

C
exactly three real roots

D
all four are real roots

Solution
Question 8
1 / -0

The modulus of $$\overline { 6+{ i }^{ 3 } } +\overline { 6+{ i } }+\overline { 6+{ i }^{ 2 } } $$ is

A
$$17$$

B
$$\sqrt{533}$$

C
$$\sqrt{456}$$

D
$$49$$

Solution
Question 9
1 / -0

If $$\left| z \right| =1$$ and $$\left| \omega -1 \right| =1$$ where $$z,\omega \in C$$ then the largest set of values of $${ \left| 2z-1 \right| }^{ 2 }+{ \left| 2\omega -1 \right| }^{ 2 }$$ equals

A
$$\left[1,9\right]$$

B
$$\left[2,6\right]$$

C
$$\left[2,12\right]$$

D
$$\left[2,18\right]$$

Solution
Question 10
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 $$2.\bar{2}=|2|^{2}$$
$$(2+3).(\overline{2+3})$$
$$=12+31^{2}$$