This is a Test of Mathematics Solution (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also see: Cheenta I.S.I. & C.M.I. Entrance Course

## Problem

If \(a_0, a_1, \cdots, a_n \) are real numbers such that $$ (1+z)^n = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n $$ for all complex numbers z, then the value of $$ (a_0 – a_2 + a_4 – a_6 + \cdots )^2 + (a_1 – a_3 + a_5 – a_7 + \cdots )^2 $$ equals

(A) \( 2^n \) ; (B) \( a_0^2 + a_1^2 + \cdots + a_n^2 \) ; (C) \( 2^{n^2} \) ; (D) \( 2n^2 \) ;

## Sequential Hints

(*How to use this discussion:* **Do not read the entire solution at one go. First, **read more on the **Key Idea, **then give the problem a try. **Next, **look into **Step 1 **and give it another try and so on.)

### Key Idea

This is the generic use case of Complex Number \( \iota =\sqrt {-1} \) and binomial theorem.

### Step 1

Note that \( i^2 = -1 \). Also, geometrically speaking, i = (0,1). Hence adding (1,0) to i (=(0,1)) gives us the point (1, 1). Polar coordinate of this point is \( ( \sqrt 2, \frac{pi}{4} ) \). Here is a picture:

Try the problem with this hint before looking into step 2. Remember, no one learnt mathematics by looking at solutions.

**At Cheenta we are busy with Complex Number and Geometry module. Additionally I.S.I. Entrance Mock Test 1 is also active now.**

## More Resources

Look into the following notes on Complex Number and Geometry module.