Complex Numbers

Q.

$\underset{x\to \infty }{lim}\frac{(2+x{)}^{40}(4+x{)}^5}{(2-x{)}^{45}}$

1. $1$
2. $16$
3. $-1$
4. $32$

Q. The sum ${\sum }_{r=1}^{n}r.{}^{2n}{C}_r$ is equal to

1. 
2. 
3. 
4. 

Q. 47. Sum and product of two complex number are real if and only if they are conjugate of each other.

Q. 25. Least value |z|+|z-1|+|z-i|+|z-3-4i| Where z is a complex number

Q.

Express the complex number $\frac{1}{(1+i)}$ in a + ib form where a and b are real. Find the value of $a^2+b^2$.

__

Q. Non-zero complex number z is such that Z+ 1/Z is Z purely real then sum of possible values of arg(z) is k, then k equals

Q.

If ${}^n{P}_r=5040({}^{n-1}{C}_{r-1}+{}^{n-1}{C}_r)$, then r =

1. None

2. 6

3. 7

4. 5

Q.

If ${}^8{C}_r$ = ${}^8{C}_{r+2}$, then the value of ${}^r{C}_2$ is

1. 8

2. 3

3. 5

4. 2

Q.

Which of the following is not a real valued function?

1. f :N → N

2. f :R → R

3. f :{1+i, i, , 2i} → C where i is √-1 and C is the set of complex numbers

4. f :Q → Q

Q.

if $i^2=-1$, then the value of $\sum _{n=1}^{200}{i}^n$ is

1. 50

2. -50

3. 0

4. 100

Q.

The value of ${}^n{C}_0$ - ${}^n{C}_2$ + ${}^n{C}_4$ - ${}^n{C}_6$ . . . . .

1. 

2. 

3. 

4. 

Q.

If n is an odd natural number , then $\sum _{r=0}^n\frac{(-1{)}^r}{{}^r{C}_r}$ equals :

1. 1/n

2. n/2ⁿ

3. 0

4. None

Q.

The principal argument of the complex number 1

1. 0

2. 

3. 1

4. 

Q. Write the component statements of the statement. Also, check whether the statement is correct or not.
'If x and y are integers, then xy is a rational number.

Q. The value of $(1+i{)}^8+(1-i{)}^8$ is
[RPET 2001; KCET 2001]

1. 16

2. -16
3. 32
4. -32

Q. Express the following expression in the form of a + ib .

Q.

If A = ф, n(B) = 4 then n(A x B) is

1. 4

2. 1

3. 0

4. 5

Q.

Express the following in the form a + ib where a, b Є R.

1. 

2. 

3. 

4. 

Q.

$\text{If}y=2^{2^x},then\frac{dy}{dx}=$

1. 

2. 

3. 

4. 

Q.

The argument of the complex number -1-i√3

1. 

2. 

3. 

4. 

Q. If ( x + iy ) 3 = u + iv , then show that .

Q.

In how many different ways the letters of the word HEXAGON can be permuted?

1. 4050

2. 3240

3. 2040

4. 5040

Q. 5.(1-i)-(-1 + i6)

Q.

Let $\alpha ,\beta$ be real and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line Re(z) = 1, then it is necessary that

1. $\beta ϵ(1,\infty )$

2. $\beta ϵ(0,1)$

3. $\beta ϵ(-1,0)$

4. $|\beta |=1$

Q. 4. 3(7 i7) (7+ i7)

Q.

a + ib > c + id can be explained only when

1. a = 0, d = 0

2. b = 0, c = 0

3. a = 0, c = 0

4. b = 0, d = 0

Q.

${}^{47}{C}_4+\sum _{r=1}^5{}^{52-r}{C}_3$ =

1. 

2. 

3. 

4. 

Q. If z1 and z2 are complex numbers such that |(z1-3z2) / (3-z1ˉz2)| = 1 and z2 not equal to 1 then find |z1|

Q.

The component statements 'p' and 'q' of the given compound statement
'All integers are either even or odd.; are

1. P : All integers are even, q: All integers are not odd

2. P : All integers are even, q: All integers are odd

3. P : All non integers are even, q: All non integers are odd

4. P : All integers are not even, q: All integers are notodd

Q. Find the value of ${}^{47}{C}_4+\sum _{r=1}^5$ ${}^{52-r}{C}_3=?$

1. 
2. 
3. 
4.