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Algebra

Blazing goIITian

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6 Jun 2009 10:11:58 IST

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576

functions all over the place

None

Q1. If f(x) is a polynomial satisfying the condition f(x).f(1/x) = f(x) + f(1/x) and f(2) = 9 then

(a) 2f(4) = 3f(6) ..... (b) 14f(1) = f(3) ....... (c) 9f(3) = 2f(5) ........ (d) f(10) = f(11)

Q2. Let g(x) be a polynomial satisfying g(x).g(y) = g(x)+g(y)+g(xy)-2 for all x,y belong to R and g(1) is no tequal to 1. If g(3) = 10 then find the value of g(5)

Q3. Let f be a real valued function. If for a given positive constant p,

f(x+p) = 7 + [2007 - 147f(x) + 21(f(x))² - (f(x))³]. Prove that f(x) is periodic and find its period

Q4. The function f(x) satisfies f(x+p) + f(x-p) = 3^1/2f(x), for all x belonging to R, p>0 . prove that f(x) is periodic and find its period

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Comments (13)

akash93

Cool goIITian

Joined: 21 Nov 2008

Posts: 53

6 Jun 2009 12:42:07 IST

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replace x by x+p n then by x-p

add (1) n (2)

.....(3)

replace x by x+2p

add (3) n (4)

replace x by x+2p

..... (5)

replace x by x+12p

from 5

hence the period of f(x) is 12p

PLz rate

Swapnika.... finally into a branded college....NIT m cuming..!!!

Blazing goIITian

Joined: 7 Aug 2008

Posts: 1047

6 Jun 2009 12:47:24 IST

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for first two questions use RHD method n substitute the condition given in question to solve it...

gaiger

New kid on the Block

Joined: 6 Jun 2009

Posts: 3

6 Jun 2009 13:03:19 IST

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1) f(x)= a_nxⁿ + a_n-1x^n-1 +....a₀

we have, (a_nxⁿ + a_n-1x^n-1 +....a₀)*(a_n(1/x)ⁿ + a_n-1(1/x)^n-1 +....a₀)

= (a_nxⁿ + a_n-1x^n-1 +....a₀) + (a_n(1/x)ⁿ + a_n-1(1/x)^n-1 +....a₀)

comparing the coeffs of x^n and x^0

we have a_n=+ or -1

a₀=1

and for coefficient of x^n-1

we have a_n*a₁+a_n-1*a₁=a_n-1

or a_n-1=0

similarly,

a_i=0 where i={n-1,n-2,n-3....,1}

or f(x)= 1+or - xⁿ

or b)

gaiger

New kid on the Block

Joined: 6 Jun 2009

Posts: 3

6 Jun 2009 13:03:56 IST

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1) f(x)= a_nxⁿ + a_n-1x^n-1 +....a₀

we have, (a_nxⁿ + a_n-1x^n-1 +....a₀)*(a_n(1/x)ⁿ + a_n-1(1/x)^n-1 +....a₀)

= (a_nxⁿ + a_n-1x^n-1 +....a₀) + (a_n(1/x)ⁿ + a_n-1(1/x)^n-1 +....a₀)

comparing the coeffs of x^n and x^0

we have a_n=+ or -1

a₀=1

and for coefficient of x^n-1

we have a_n*a₁+a_n-1*a₁=a_n-1

or a_n-1=0

similarly,

a_i=0 where i={n-1,n-2,n-3....,1}

or f(x)= 1+or - xⁿ

or b)

gaiger

New kid on the Block

Joined: 6 Jun 2009

Posts: 3

6 Jun 2009 13:05:02 IST

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we have a_n=+ or -1

a₀=1

and for coefficient of x^n-1

we have a_n*a₁+a_n-1*a₁=a_n-1

or a_n-1=0

similarly,

a_i=0 where i={n-1,n-2,n-3....,1}

or f(x)= 1+or - xⁿ

or b)

AIR 538 Asish

Blazing goIITian

Joined: 29 Nov 2008

Posts: 1137

6 Jun 2009 17:09:48 IST

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what is the RHD method? new to this chapter

Mirka

Blazing goIITian

Joined: 13 Aug 2008

Posts: 1313

7 Jun 2009 15:32:08 IST

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For the first one, remember the following results :

Given that f(2) = 9

This means that the sign between 1 and xⁿ is +ve and n = 3

So, f(x) = 1 + x³

Answer will be Option B .... 14 f(1) = f(3)

Mirka

Blazing goIITian

Joined: 13 Aug 2008

Posts: 1313

7 Jun 2009 15:46:37 IST

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Second Qsn

x = y = 1 :

let y = 1/x

sign must be +ve and n = 2

Thus, g(x) = 1 + x²

g(5) = 1 + 25 = 26

AIR 538 Asish

Blazing goIITian

Joined: 29 Nov 2008

Posts: 1137

7 Jun 2009 15:47:43 IST

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thx mirka

Mirka

Blazing goIITian

Joined: 13 Aug 2008

Posts: 1313

7 Jun 2009 15:49:22 IST

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ur welcome ....

4^th one

Similar to this Qsn (with 1 in place of p) but method d same –

http://www.goiit.com/posts/list/trignometry-period-913117.htm

See the trigo method by ankit there....

Mirka

Blazing goIITian

Joined: 13 Aug 2008

Posts: 1313

7 Jun 2009 17:19:06 IST

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The third one, i simplified upto a point ....

hmm... now, how do we proceed ??

Soumik

Blazing goIITian

Joined: 31 Jul 2008

Posts: 1266

7 Jun 2009 23:05:19 IST

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Is the 3rd qsn's period 2p?

AIR 538 Asish

Blazing goIITian

Joined: 29 Nov 2008

Posts: 1137

8 Jun 2009 09:04:04 IST

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yes soumik

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