**Example 1.**

If is the imaginary cube root of unity, then value of the expression

is

(a)

(b)

(c)

(d)

**Answer. (a).**

**Solution:**

rth term of the given expression is

because .

Thus, the value of the expression is given by

More later,

Nalin Pithwa

Mathematics demystified

August 28, 2015 – 11:06 pm

**Example 1.**

If is the imaginary cube root of unity, then value of the expression

is

(a)

(b)

(c)

(d)

**Answer. (a).**

**Solution:**

rth term of the given expression is

because .

Thus, the value of the expression is given by

More later,

Nalin Pithwa

August 26, 2015 – 2:23 pm

**Example.**

If are complex numbers such that

, then is

(a) equal to 1

(b) less than 1

(c) greater than 3

(d) equal to 3

**Solution:**

Since , we get

Now,

August 11, 2015 – 9:38 am

**Question:**

If z lies on the circle , then equals

(a) 0

(b) 2

(c) -1

(d) none of these.

**Solution:**

Note that represents a circle with the segment joining and as a diameter. (*draw this circle for yourself!)*

If z lies on this circle, then is purely imaginary.

Ans. (d).

**Question:**

If and (where z \neq -1), then equals

(a) 0

(b)

(c)

(d)

**Solution:**

This shows that w is equidistant from -1 and 1. Hence, w lies on the perpendicular bisector of the segment joining -1 and 1, that is, w lies on the imaginary axis.

Hence, .

More later,

Nalin Pithwa

August 2, 2015 – 8:13 am

**Example. **

For any complex number z, the minimum value of is:

a) 0

b) 1

c) 2

d) none of these.

**Solution.**

We have for ,

Thus, the required minimum value is 2 and it is attained for any z lying on the segment joining and .

**Answer. Option C.**

More later,

Nalin Pithwa

August 1, 2015 – 6:23 am

**Question:**

If , then find the value of .

**Solution:**

We have .

But,

Thus,

but, which equals

that is, .

Thus, .

Hope you are finding it useful,

More later,

Nalin Pithwa

July 31, 2015 – 1:59 pm

If , then equals

(a) 4

(b) 3

(c) 2

(d) 1.

**Solution.**

We can write the given equation as

, or

and

and

**Answer. Option d.**

More later,

Nalin Pithwa

July 29, 2015 – 1:18 pm

**Problem.**

If lie on the unit circle , then value of

is

(a) 0

(b) n

(c) -n

(d) none of these.

**Solution.**

As lie on the circle , for

Thus, for

Hence, , which in turn equals

, that is,

.

(since ).

**Answer. Option a.**

More later,

Nalin Pithwa

July 28, 2015 – 11:23 pm

**Question:**

If , then

equals

(a) -1

(b) 1

(c) -i

(d) i

**Solution.**

Using De Moivre’s theorem,

which in turn equals

Hence, .

More complex stuff to be continued in next blog (pun intended) 🙂

Nalin Pithwa

July 28, 2015 – 2:10 am

**Find the number of solutions of the equation** .

**Solution.**

Given that . Hence, .

.Hence, we get

(since )

If , we get .

Thus,

Thus, , that is,

for . Therefore, the given equation has five solutions.