Square Root of a Complex Number

Q.

Total number of complex numbers 'z', satisfying Re($z^2$) = 0, |z| = $\sqrt{3}$, is equal to :

1. 2

2. 4

3. 6

4. None

Q.

The square root of $3-4i$ is

1. $\pm (2-i)$

2. $\pm (2+i)$

3. $\pm (1-2i$

4. $\pm (1+2i)$

Q. Find the square root of $\frac{x^2}{y^2}+\frac{y^2}{x^2}-\frac{1}{i}(\frac{x}{y}-\frac{y}{x})-\frac{9}{4}$

1. $\pm [(\frac{x}{y}-\frac{y}{x})+\frac{i}{2}]$
2. $\pm [(\frac{x}{y}+\frac{y}{x})-\frac{i}{2}]$
3. $\pm [(\frac{x}{y}+\frac{y}{x})-\frac{i}{2}]$
4. $\pm [(\frac{x}{y}-\frac{y}{x})-\frac{i}{2}]$

Q.

If a=$\sqrt[2]{22i}$ then which of the following is correct?

1. a=1+i

2. $a=\sqrt{2}\times (1+i)$

3. 1

4. None of these

Q. Find the square root of $-1+2\sqrt{2}i$

1. $\pm (2+\sqrt{2}i)$
2. $\pm (3+\sqrt{2}i)$
3. $\pm (-1+\sqrt{2}i)$
4. $\pm (1+\sqrt{2}i)$

Q.

If a=$\sqrt[2]{22i}$ then which of the following is correct

1. a=1+i

2. a=1+i

3. 1

4. None of these

Q.

Check whether the following sentences are statements.

(i) Answer this question.

(ii) Everyone in this room is rich.

(iii) There is no rain without clouds.

(iv) Mathematics is fun.

(v) She is a commerce graduate.

(vi) $\sqrt{2}$ is a rational number.

(vii) Fire is always hot.

(viii) The sides of a quadrilateral have equal length.

Q. The square root of $-1+2\sqrt{2}i$ is

1. $\pm (2+\sqrt{2}i)$
2. $\pm (3+\sqrt{2}i)$
3. $\pm (-1+\sqrt{2}i)$
4. $\pm (1+\sqrt{2}i)$

Q. If $y^{1/m}=[x+\sqrt{1+x^2}]$ , then $(1+x^2)y_2+x{y}_1$ is equal to :

1. $m^2{y}^2$
2. $m^2{y}^2$
3. $m^{2}y$
4. $my$

Q.

$\frac{d}{dx}\left\{co{s}^{-}1(\frac{4x^3}{27}-x)\right\}=$

1. 

2. 

3. 

4. 

Q.

If $z=i^i$, where i = $\sqrt{-1}$, then Re(z) is

1. zero

2. None of these

3. 

4. 

Q.

$\sqrt{-8-6i}$=

1. 

2. 

3. 

4. 

Q.

Use quanlifiers to convert each of the following open sentences defined on N, into a true statement:

(i) $x+5=8$ (ii) $x^2>0$ (iii) $x+2<4$

Q. If $z$ is a complex number satisfying $z+z^{-1}=1$, then $z^n+z^{-n},n\in N$ has/have the value(s)

1. $2(-1{)}^n$ when $n$ is a multiple of $3$
2. $(-1{)}^{n+1}$ when $n$ is not a multiple of $3$
3. $(-1{)}^{n+1},$ when $n$ is a multiple of $3$
4. $0,$ when $n$ is not a multiple of $3$

Q.

The value of |$\frac{(9+i)(5+i)}{(2+3i)(4+5i)}$| is ___________

___

Q.

if a=$\sqrt[2]{22i}$ then which of the following is correct

1. a=1+i

2. a=1+i

3. 

4. None of these

Q.

$\sqrt{-8-6i}$=

1. 1$\pm$3i

2. $\pm$(1-3i)

3. $\pm$(1+3i)

4. $\pm$(3-i)

Q. $\sqrt{-8-6i}$=

1. 1$\pm$3i
2. $\pm$(1-3i)
3. $\pm$(1+3i)
4. $\pm$(3-i)

Q. If $\omega$ is a cube root of unity, and the value of
$(1-\omega )(1-{\omega }^2)(1-{\omega }^4)(1-{\omega }^8)=\alpha ,$ and
$\frac{a+b\omega +c{\omega }^2}{b+c\omega +a{\omega }^2}+\frac{a+b\omega +c{\omega }^2}{c+a\omega +b{\omega }^2}=\beta$ then

1. $\alpha +\beta =8$
2. $\alpha \beta =9$
3. $\alpha \beta =-9$
4. $\alpha +\beta =-9$

Q.

If a, b c are in HP then the expression $a(b-c)x^2$ + b(c-a)x + c(a-b)

1. None of these

2. has real and distinct factors

3. has no real factor

4. is a perfect square

Q. The value of $(a+b\omega +c{\omega }^2{)}^3+(a+b{\omega }^2+c\omega {)}^3$ is/are
[Here $\omega$ is a complex cube root of unity]

1. $(2a-b-c)(2b-a-c)(2c-a-b)$
2. $9abc,\text{if}a+b+c=0$
3. $27abc,\text{if}a+b+c=0$
4. $(2a+b+c)(2b-a-c)(2c-a-b)$