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
![\[(1+z)^n = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n\]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-cbe1034e23bf0e9b25b083f2f1b94150_l3.png "Rendered by QuickLaTeX.com")
for all complex numbers z, then the value of
![\[(a_0 - a_2 + a_4 - a_6 + \cdots )^2 + (a_1 - a_3 + a_5 - a_7 + \cdots )^2\]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-29afd674d1897dd2cd27e36b7976dc8e_l3.png "Rendered by QuickLaTeX.com")
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
![\[(2^{n/2}, \frac{n \cdot \pi }{4} ) = a_0 + a_1 i + a_2 i^2 + a_3 I^3 \cdots + a_n I^n\]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-c7940abc80a5ceea4b7b76f1345716c8_l3.png "Rendered by QuickLaTeX.com")
. This implies
![\[ (2^{n/2}, \frac{n \cdot \pi }{4} ) = a_0 - a_2 + a_4 - \cdots + i (a_1 - a_3 + a_5 - \cdots )\]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-81ac243daa507cef9d1ab61c529cd309_l3.png "Rendered by QuickLaTeX.com")
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.
