Complex Numbers and Quadratic Equations Test 45

Result

Complex Numbers and Quadratic Equations Test 45

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Weekly Quiz Competition

Question 1
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 2
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 3
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 4
1 / -0

If $$Z=\sin \frac {6\pi}5+i(1+\cos \frac {6\pi }5)$$ then

A
$$|Z|=-2\cos \frac {3\pi}5$$

B
$$Arg(Z)=\frac {\pi}5$$

C
$$Arg(Z)=\frac {9\pi }{10}$$

D
none of these

Solution
Question 5
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 6
1 / -0

If $$z_1$$ and $$z_2$$ are two non zero complex numbers such that $$|z_1 + z_2| = |z_1| + |z_2|,$$ then arg $$z_1$$ - arg $$z_2$$ is equal

A
$$-\pi$$

B
$$\frac{-\pi}{2}$$

C
$$0$$

D
$$\frac{\pi}{2}$$

Solution
Question 7
1 / -0

If z-{1} and z-{2} are two non zero complex numbers such that $$\left| z{ }_{ 1 }+z{ }{ }_{ 2 } \right| =\left| { z }_{ 1 } \right| +\left| { z }_{ 2 } \right| $$, then arg$$z_{1}$$-arg $$z_{2}$$ is equal to:

A
$$-\pi$$

B
$$-\frac\pi{2}$$

C
$$0$$

D
$$\frac\pi{2}$$

Solution
Question 8
1 / -0

Argument and modulus of $$\left[ \frac { 1 + i } { 1 - i } \right] ^ { 2013 }$$ are respectively $$\ldots \ldots \ldots$$

A
$$\frac { - \pi } { 2 }$$ and 1

B
$$\frac { \pi } { 2 }$$ and $$\sqrt { 2 }$$

C
0 and $$\sqrt { 2 }$$

D
$$\frac { \pi } { 2 }$$ and 1

Solution
Question 9
1 / -0

If the roots of the equation $${ x }^{ 2 }-8x+({ a }^{ 2 }-6a)=0$$ are real, then

A
-2 < a < 8

B
2 < a < 8

C
$$2\le a\le 8$$

D
$$-2\le a\le 8$$

Solution
Question 10
1 / -0

Find the modules and amplitude for each of the following complex numbers

A
7-5i

B
$$\sqrt { 3 } +\sqrt { 2 } i$$

C
-8+15i

D
-3(1-i)