Complex Numbers and Quadratic Equations Test -2

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Complex Numbers and Quadratic Equations Test -2

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Weekly Quiz Competition

Question 1
1 / -0

Let x, y ∈ R,then x + iy is a non real complex number if

A
x = 0

B
y = 0

C
y ≠ 0

D
x ≠ 0

Solution

If a complex number has to be a non real complex number then its imaginary part should not be zero ⇒ iy ≠ 0 ⇒ y ≠ 0
Question 2
1 / -0

Let x, y ∈ R, then x + iy is a purely imaginary number if

A
x = 0 , y = 0

B
x ≠ 0 , y = 0

C
x = 0 , y ≠ 0

D
x ≠ 0 , y ≠ 0

Solution

Purely imaginary number is a complex number which has only imaginary part ( iy)

But if y=0 the complex number iy will become 0 which is real.

Hence the condition for a number to be purely imaginary is x=0 and y ≠ 0
Question 3
1 / -0

The roots of the equation $${ \left( z+\alpha \beta \right) }^{ 3 }={ \alpha }^{ 3 }$$ represent the vertices of a triangle, one of whose sides is of length

A
$$\sqrt { 3 } \left| \alpha \beta \right|$$

B
$$\sqrt { 3 } \left| \alpha \right|$$

C
$$\sqrt { 3 } \left| \beta \right|$$

D
$$None\ of\ these$$

Solution

$$z_1=\alpha-\alpha \beta,z_2=\alpha\omega-\alpha \beta,z_3=\alpha\omega^2-\alpha \beta$$
Let vertices be denoted by $$A,B,C$$
Length of $$AB=|z_1-z_2|=|\alpha||1-\omega|=|\alpha|\left|1-\dfrac{-1-i\sqrt{3}}{2}\right|=\sqrt{3}|\alpha|$$
Length of $$BC=|z_2-z_1|=|\alpha|\left|1-\dfrac{-1+i\sqrt{3}}{2}\right|=\sqrt{3}|\alpha|$$
Length of $$BC=|z_3-z_1|=|\alpha||\omega|\left|\dfrac{-1+i\sqrt{3}}{2}\right|\left|1-\dfrac{-1+i\sqrt{3}}{2}\right|=\sqrt{3}|\alpha|$$
Hence option B is correct.
Question 4
1 / -0

For a quadratic equation if $$D < 0$$ then which of the following is true?

A
Real roots do not exist

B
Roots are real and equal

C
Roots are rational and distinct

D
Roots are real and distinct

Solution

For a quadratic equation, if $$D < 0$$ then real roots do not exist.
Question 5
1 / -0

Identify the condition for Imaginary Roots:

A
$$D = 0$$

B
$$D < 0$$

C
$$D > 0$$

D
$$D \le 0$$

Solution

For Imaginary roots, the discriminant is less than zero, that is $$D < 0$$.
Question 6
1 / -0

If the value of '$$b^2-4ac$$' is greater than zero, the quadratic equation $$ax^2+bx+c=0$$ will have

A
Two Equal Real Roots.

B
Two Distinct Real Roots.

C
No Real Roots.

D
No Roots or Solutions.

Solution

$${ b }^{ 2 }-4ac>0$$
$$\therefore \triangle >0$$
So, two equal real roots or only one real root.
Question 7
1 / -0

If $$a, b, c$$ are real and $$b^2- 4ac $$ is perfect square then the roots of the equation $$ax^2+bx+c=0$$, will be:

A
Rational & distinct

B
Real & equal

C
Irrational & distanct

D
Imaginary & distinct

Solution

Given that the discriminant $$=b^{2}-4ac $$ is a perfect square

Let $$b^{2}-4ac =k^{2}$$

$$\Rightarrow $$ Roots $$=\dfrac {-b\pm \sqrt {k^{2}}}{2a}$$

$$\Rightarrow$$ roots $$=\dfrac {-b\pm k} {2a}$$

As given $$a, b, c$$ are real
$$\Rightarrow $$ Roots are rational and distinct.
Question 8
1 / -0

If the value of '$$b^2-4ac$$' is less than zero, the quadratic equation $$ax^2+bx+c=0$$ will have

A
Two Equal Real Roots.

B
Two Distinct Real Roots.

C
No Real Roots.

D
None of the above.

Solution

$${ b }^{ 2 }-4ac<0$$
$$\therefore \triangle <0$$
So no real roots.
Question 9
1 / -0

When will the quadratic equation $$ax^2+bx+c=0$$ NOT have Real Roots?

A
$$b^2 - 4ac \ge 0$$

B
$$b^2 - 4ac > 0$$

C
$$b^2 - 4ac < 0$$

D
None of these

Solution

We learnt that the quadratic equation $$ax^2+bx+c=0$$ will NOT have Real Roots when $$b^2-4ac< 0 $$. Option c is correct.
Question 10
1 / -0

If $$(1+i)^{-20}=a+ib$$, then the values of $$a$$ and $$b$$ are

A
$$a=2^{-10}, b=-2^{-10}$$

B
$$a=-2^{-10}, b=0$$

C
$$a=2^{-10}, b=0$$

D
$$none\ of\ these$$

Solution

$$(1+i)^{-20}\\((1+i)^2){-10}\\(1+i^2+2i)^{-10}\\=(2i)^{-10}\\=(\frac{2^{-10}}{i^10})\\=-2^{-10}\\\therefore \>by\>comparison\>a\>=\>-2^{-10},\>b=0$$
Question 11
1 / -0

The number of solution of $$z^2 + \bar{z} = 0$$ is

A
$$5$$

B
$$4$$

C
$$2$$

D
$$3$$

Solution

Let $$z=x+iy$$.
Now,
$$z^2+\overline{z}=0$$
or, $$x^2-y^2+2ixy+(x-iy)=0$$
or, $$(x^2-y^2+x)+i(2xy-y)=0$$
Now comparing the real and imaginary part both sides we get,
$$x^2-y^2+x=0$$.....(1) and $$2xy-y=0$$.....(2).
From (2) we get, $$x=\dfrac{1}{2}$$ or $$y=0$$.
Now $$x=\dfrac{1}{2}$$ gives from (1) we get, $$y=\pm \dfrac{\sqrt{3}}{2}$$.
And $$y=0$$ gives from (1) we get, $$x=0, 1$$.
So the solution s are $$(0,0), (1,0), \left(\dfrac{1}{2},\pm \dfrac{\sqrt{3}}{2}\right)$$.
So we have $$4$$ solutions.
Question 12
1 / -0

For any complex number $$z$$ the minimum value of $$|z|+|z-2013i|$$ is...

A
$$2010$$

B
$$2011$$

C
$$2013$$

D
$$2012$$

Solution

Q:- minimum value of $$\left | z \right |+\left | z-2013i \right |$$
Solution For ZEG
$$ \left | 2013i \right | = \left | z+(2013i-z) \right | \leqslant \left | z \right |+\left | 2013i-z \right |$$
$$ \Rightarrow 2013 \leqslant \left | z \right |+\left | z-2013i \right |$$
$$ \Rightarrow $$ minimum value of $$ \left | z \right |+\left | z-2013i \right | $$ is
2013
which is attained when $$ z = i$$
so, c is correct option.
$$ x^{2}+y^{2} \leqslant 1 $$ & $$ y^2 \leqslant 1-x . $$

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Complex Numbers and Quadratic Equations Test -2

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Complex Numbers and Quadratic Equations Test -2

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Question 1

x = 0

y = 0

y ≠ 0

x ≠ 0

Solution

Question 2

x = 0 , y = 0

x ≠ 0 , y = 0

x = 0 , y ≠ 0

x ≠ 0 , y ≠ 0

Solution

Question 3

$$\sqrt { 3 } \left| \alpha \beta \right|$$

$$\sqrt { 3 } \left| \alpha \right|$$

$$\sqrt { 3 } \left| \beta \right|$$

$$None\ of\ these$$

Solution

Question 4

Real roots do not exist

Roots are real and equal

Roots are rational and distinct

Roots are real and distinct

Solution

Question 5

$$D = 0$$

$$D < 0$$

$$D > 0$$

$$D \le 0$$

Solution

Question 6

Two Equal Real Roots.

Two Distinct Real Roots.

No Real Roots.

No Roots or Solutions.

Solution

Question 7

Rational & distinct

Real & equal

Irrational & distanct

Imaginary & distinct

Solution

Question 8

Two Equal Real Roots.

Two Distinct Real Roots.

No Real Roots.

None of the above.

Solution

Question 9

$$b^2 - 4ac \ge 0$$

$$b^2 - 4ac > 0$$

$$b^2 - 4ac < 0$$

None of these

Solution

Question 10

$$a=2^{-10}, b=-2^{-10}$$

$$a=-2^{-10}, b=0$$

$$a=2^{-10}, b=0$$

$$none\ of\ these$$

Solution

Question 11

$$5$$

$$4$$

$$2$$

$$3$$

Solution

Question 12

$$2010$$

$$2011$$

$$2013$$

$$2012$$

Solution