Modulus of Complex Number MCQ Quiz in தமிழ் - Objective Question with Answer for Modulus of Complex Number - இலவச PDF ஐப் பதிவிறக்கவும்

CONCEPT:

• If z = x + iy then $|z|=\sqrt{x^2+y^2}$ 

CALCULATION:

Let z = 3 + i

As we know that, if z = x + iy then $|z|=\sqrt{x^2+y^2}$ 

Here, x = 3 and y = 1

⇒ $|z|=\sqrt{3^2+1^2}=\sqrt{10}$ 

Hence, correct option is 1.

CONCEPT:

Let z = a + i b be a complex number. Then, the modulus of z, denoted by |z|, is defined to be the non-negative real number $\sqrt{a^2+b^2}.$

CALCULATION:

Given expression is z = $\sqrt{6}+i\sqrt{3}$

Concept:

The modulus of a complex number is the distance of the complex number from the origin in the argand plane.

If z = x + iy is a complex number where x and y are real and i = √-1,

Then the non-negative value $\sqrt{x^2+y^2}$ is called the modulus of complex number z (or x + iy).

The modulus of a complex number is also called the absolute value of the complex number.

Calculation:

We have,

z = (5 + √-11)(1 + √-5) = (5 + √11i)(1 + √5i)

Now, let v = (5 + √11i) and w = (1 + √5i)

$\Rightarrow \left|v\right|=\sqrt{5^2+(\surd 11{)}^2}=\sqrt{25+11}=6$

$\Rightarrow \left|w\right|=\sqrt{1^2+(\surd 5{)}^2}=\sqrt{1+5}=\surd 6$

Modulus of the complex number z is given by

$\left|z\right|=\left|vw\right|=\left|v\right|\left|w\right|$

$\therefore \left|z\right|=6\sqrt{6}$

Hence, the modulus is 6√6.

Concept: 

Modulus of complex no. z =  a + ib is given by |z| = $\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$ . 

Property of complex number:

$\left|\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right|=\frac{|{\mathrm{z}}_1|}{|{\mathrm{z}}_2|}$

Calculation:

Let z = $\frac{\cos 2\theta -\mathrm{i}\sin 2\theta }{\cos 2\theta +\mathrm{i}\sin 2\theta }$

Taking modulus on both sides, we get

⇒ |z| = $\left|\frac{\cos 2\theta -\mathrm{i}\sin 2\theta }{\cos 2\theta +\mathrm{i}\sin 2\theta }\right|$

= $\frac{|\cos 2\theta -\mathrm{i}\sin 2\theta |}{|\cos 2\theta +\mathrm{i}\sin 2\theta |}$

= $\frac{\sqrt{{\cos }^{2}2\theta +{\sin }^{2}2\theta }}{\sqrt{{\cos }^{2}2\theta +{\sin }^{2}2\theta }}$

= 1

Concept;

Modulus of a complex number z = x + iy

|z| = $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$

Calculation:

Given z = 5√2 + i 5√2 

|z| = $\sqrt{(5\sqrt{2}{)}^2+(5\sqrt{2}{)}^2}$

|z| = $\sqrt{50+50}$ 

|z| = $\sqrt{100}$ = 10

Concept;

Modulus of a complex number z = x + iy

|z| = $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$

Calculation:

Let z = $\frac{1+\mathrm{i}}{2+\mathrm{i}}$

Multiplying by the conjugate of the denominator

z = $\frac{1+\mathrm{i}}{2+\mathrm{i}}\times \frac{2-\mathrm{i}}{2-\mathrm{i}}$

z = $\frac{2+2\mathrm{i}-\mathrm{i}-{\mathrm{i}}^2}{4-(-1)}$

z = $\frac{2+\mathrm{i}+1}{4-(-1)}$

z = $\frac{3+\mathrm{i}}{5}$

Now magnitude of z

|z| = $\sqrt{{\left(\frac{3}{5}\right)}^2+{\left(\frac{1}{5}\right)}^2}$

|z| = $\sqrt{\left(\frac{9+1}{25}\right)}=\sqrt{\frac{10}{25}}$

|z| = $\frac{\sqrt{10}}{5}$

Concept;

Modulus of a complex number z = x + iy

|z| = $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$

Calculation:

Given z = 12 - 5i

|z| = $\sqrt{(12{)}^2+(-5{)}^2}$

|z| = $\sqrt{144+25}$ 

|z| = $\sqrt{169}$ = 13

Concept:

If z = x + iy 

⇒ Modulus of z = $|z|=\sqrt{x^2+y^2}$

Calculation:

Given, 

z = $\frac{1+\text{i}\sin \theta }{1-\text{i}\sin \theta }$ where i = $\sqrt{-1}$

Multiply numerator and denominator by (1 + i sin θ),

$\Rightarrow z=\frac{1+\text{i}\sin \theta }{1-\text{i}\sin \theta }\times \frac{1+\text{i}\sin \theta }{1+\text{i}\sin \theta }$,

$\Rightarrow z=\frac{(1+\text{i}\sin \theta {)}^2}{1^2-(\text{i}\sin \theta {)}^2}$

$\Rightarrow z=\frac{(1+2i\sin \theta -{\sin }^2\theta )}{1+{\sin }^2\theta }$

$\Rightarrow z=\frac{(1-{\sin }^2\theta )}{1+{\sin }^2\theta }+i\frac{2\sin \theta }{1+{\sin }^2\theta }$

$\Rightarrow z=\frac{{\cos }^2\theta }{1+{\sin }^2\theta }+i\frac{2\sin \theta }{1+{\sin }^2\theta }$

$\Rightarrow |z|=\sqrt{(\frac{{\cos }^2\theta }{1+{\sin }^2\theta }{)}^2+(\frac{2\sin \theta }{1+{\sin }^2\theta }{)}^2}$

$\Rightarrow |z|=\frac{1}{1+{\sin }^2\theta }\sqrt{{\cos }^4\theta +4{\sin }^2\theta }$

$\Rightarrow |z|=\frac{1}{1+{\sin }^2\theta }\sqrt{(1-{\sin }^2\theta {)}^2+4{\sin }^2\theta }$

$\Rightarrow |z|=\frac{1}{1+{\sin }^2\theta }\sqrt{1-2{\sin }^2\theta +{\sin }^4\theta +4{\sin }^2\theta }$

$\Rightarrow |z|=\frac{1}{1+{\sin }^2\theta }\sqrt{1+2{\sin }^2\theta +{\sin }^4\theta }$

$\Rightarrow |z|=\frac{1+{\sin }^2\theta }{1+{\sin }^2\theta }=1$

∴ The correct answer is option (1).

Concept;

Modulus of a complex number z = x + iy

|z| = $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$

Calculation:

Given: z = (3 + 4i)(4 - 5i)

z = 12 - 15i + 16i - 20i²

z = 12 + 20 + i = 32 + i                                           (∵  i2 = -1)

|z| = $\sqrt{(32{)}^2+(1{)}^2}=\sqrt{1024+1}=\sqrt{1025}$

|z| = 5√41 

If z = z1.z2 so, modulus of |z| = |z1|.|z2|

z = (3 + 4i)(4 - 5i)

|z| = $(\sqrt{(3{)}^2+(4{)}^2}).(\sqrt{(4{)}^2+(5{)}^2})$

|z| = 5 × √41  = 5√41   

CONCEPT:

Let z = a + ib be a complex number. Then, the modulus of z, denoted by |z|, is defined to be the non-negative real number $\sqrt{a^2+b^2}.$

CALCULATION:

Given expression is $\frac{4+\sqrt{2}i}{3-\sqrt{2}i}$

$\Rightarrow \frac{4+\sqrt{2}i}{3-\sqrt{2}i}=\frac{4+\sqrt{2}i}{3-\sqrt{2}i}\times \frac{3+\sqrt{2}i}{3+\sqrt{2}i}=\frac{12+4\sqrt{2}i+3\sqrt{2}i-2}{3^2-{\left(\sqrt{2}i\right)}^2}$

$\Rightarrow \frac{10+7\sqrt{2}i}{11}=\frac{10}{11}+\frac{7\sqrt{2}}{11}i$