Complex Plane

Q.

12.11.10.9.8.7 can be written in factorial notation as :

1. 6!

2. 

3. 2!

4. 12!

Q. 138.If an=n(n!), then sigma(r=1 to 100 ar)

Q.

Express the following in standard form: (8 – 4i)-(-2 – 3i)+(-10 + 3i)

1. 2+2i

2. 3+3i

3. 0+ 2i

4. 1+ 2i

Q. ntFind the value of determinant 1 3 9 27n nt 3 9 27 1n nt 9 27 1 3n nt 27 1 3 9n

Q. For any complex number Z, the min value of |Z|+|Z+3i| is A)2 B)4 C)9 D)3^(1/2)

Q. Prove that: $\frac{(2n+1)!}{n!}$ = 2ⁿ [1 · 3 · 5 ... (2n − 1) (2n + 1)]

Q. 21. Let M be the maximum value of (6x−3y−8z), subject to 2x²+3y²+4z²= 1.Find [M].

Q. $Z_1=3+4i$ and $|Z_2|=1$. Then, the maximum value of $|Z_1+Z_2|$ is and the minimum value of $|Z_1+Z_2|$ is .

1. 7
2. 6
3. 4
4. 3

Q. For any complex number , the minimum value of |z| +|z-1| is

1. 1
2. 0
3. 
4. 

Q. If x ≠ 1 and $f\left(x\right)=\frac{x+1}{x-1}$is a real function, then f(f(f(2))) is
(a) 1
(b) 2
(c) 3
(d) 4

Q. 1.2 + 2.2² + 3.2³ + ... + n.2ⁿ = (n − 1) 2ⁿ⁺¹+2

Q. If z is any complex number satisfying $|z-3-2i|\le 2,$ then minimum value of $|2z-6+5i|$ is___

1. 4.5
2. 5
3. 2.5
4. 9

Q. If $z_1$ and $z_2$ be two complex number, then Re $\left(z_1{z}_2\right)$ =

1. Re $\left(z_1\right).Re\left(z_2\right)$
2. Re $\left(z_1\right).Im\left(z_2\right)$
3. Im $\left(z_1\right).Re\left(z_2\right)$
4. None of these

Q.

If i$z^3$ + $z^2$ - z + i = 0, where i = $\sqrt{-1}$ then |z| is equal to :

1. 1

2. 1/2

3. 1/4

4. None

Q.

Express the following in standard form: i²⁰ + (1-2i)³

1. -1+2i

2. -10+4i

3. -10+2i

4. -10+i

Q. If z and z' are complex numbers such that z.z' = z, then z' =

1. 0+i0
2. 1+0i
3. 0+i
4. 1+i

Q.

For any two complex numbers $z_1$ and $z_2$ prove that

$Re\left(z_1{z}_2\right)=Re\left(z_1\right)Re\left(z_2\right)-Im\left(z_1\right)Im\left(z_2\right)$

Q. 1 + 3 + 5 + ... + (2n − 1) = n² i.e., the sum of first n odd natural numbers is n².

Q. If $z=\left(\frac{1+i}{1-i}\right)$, then z⁴ equals
(a) 1
(b) −1
(c) 0
(d) none of these

Q. Re $\frac{(1+i{)}^2}{3-i}=$

1. 1/5

2. -1/5

3. 1/10
4. –1/10

Q.

The number of terms in the expansion (1+2x+x²)²⁰ are.

1. 21

2. 20

3. 41

4. 40

Q.

In the expansion of (a+b)ⁿ, n Є N the middle term is given by

1. 

2. 

3. 

4. 

Q. Express the given complex number in the form $a+ib:$$[(\frac{1}{3}+i\frac{7}{3})+(4+i\frac{1}{3})]-(-\frac{4}{3}+i)$

Q.

Express the following in standard form: (2 + 3i)(2 - 3i)(1 - i)²

1. 5 + 12i

2. 4 – 9 i

3. - 26 i

4. 6 + 12i

Q.

If ${}^{48}{C}_{3r-2}={}^{48}{C}_{2}r$ Find the value of r

1. 20

2. 2

3. 10

4. 1

Q.

Express the following in standard form : (2 - 3i)²

1. 8 + 2i

2. - 5 + 12i

3. 8 + 12i

4. 5 + 2i

Q.

Express the following in standard form: (2+3i)(2-3i)(1+i)²

1. 0 + 26 i

2. 6 + 12i

3. 5 + 12i

4. 4 + 9 i

Q.

Express the following in standard form: (2 – 3i)²

1. 8 + 2i

2. -5 - 12i

3. 5 + 2i

4. 8 + 12i

Q.

Two complex numbers are given as,

$Z_1$ = $a_1$ + i$b_1$

$Z_2$ = $a_2$ + i$b_2$

For the complex numbers $Z_1$ and $Z_2$ to be equal, the necessary conditions are

1. Re ( Z₁ ) = Re( Z₂)

2. Im( Z₁ ) = Im (Z₂)

3. Both a & b

4. All a, b & c

Q. if log 4 to the base 5 is equal to α and log 18 to the base 12 is equal to β then what is the value of log 24 to the base 5