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Explore the Back-StoryThis 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

If are real numbers such that

for all complex numbers z, then the value of

equals

(A) ; (B) ; (C) ; (D) ;

(*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.)

This is the generic use case of Complex Number and binomial theorem.

Note that . 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 . 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.**

Replace by . We have on the left hand side.

Now, replace by on the right hand side.

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Replacing z by on the right hand side we have

. This implies

**Think** now, what the following expression represents:

It represents the **square of the** length of point . That is simply

Hence the answer is option A.

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

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

If are real numbers such that

for all complex numbers z, then the value of

equals

(A) ; (B) ; (C) ; (D) ;

(*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.)

This is the generic use case of Complex Number and binomial theorem.

Note that . 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 . 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.**

Replace by . We have on the left hand side.

Now, replace by on the right hand side.

[/tab]

[tab]

Replacing z by on the right hand side we have

. This implies

**Think** now, what the following expression represents:

It represents the **square of the** length of point . That is simply

Hence the answer is option A.

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

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