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![[Post New]](/templates/default/images/icon_minipost_new.gif) 25 Jun 2008 22:35:46 IST
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1.Find all the pairs (a,b) of real numbers such that whenever  is a root of  is also a root of the equation.
2.How many 6-digit numbers are there such that:
a)the digits of each number are all from the set {1,2,3,4,5};
b)Any digit that appears in the number appears at least twice?
(Example: 225252 is an admissible number, while 222133 is not.)
Please don't ask for answers as I don't have it.
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(\alpha-2)=0\\\alpha=-1,\alpha=2\\Observe\;a=-2\alpha,b=\alpha^2\\Thus,(a,b)=(2,1)or(-4,4))
\alpha=\alpha^2-2\;and\;\beta=\beta^2-2\\(II)\alpha=\beta^2-2\;and\;\beta=\alpha^2-2\\(III)\alpha=\alpha^2-2=\beta^2-2\;and\;\alpha\ne\beta\\(IV)\beta=\alpha^2-2=\beta^2-2\;and\;\alpha\ne\beta)
Solving\;get\;(\alpha,\beta)=(2,-1)or(-1,2),hence\;(a,b)=(-(\alpha%2B\beta),\alpha\beta)=(-1,-2)\\(II)\alpha-\beta=-(\alpha^2-\beta^2)\Rightarrow \alpha%2B\beta=-1\\\alpha%2B\beta=\alpha^2%2B\beta^2-4\\\alpha%2B\beta=(\alpha%2B\beta)^2-2\alpha\beta-4\\\Rightarrow \alpha\beta=-1\Rightarrow(a,b)=(-(\alpha%2B\beta),\alpha\beta)=(1,-1)\\(III)\alpha=-\beta\;will\;give\;soln\Rightarrow\alpha=2,\beta=-2\;or\alpha=-1,\beta=1\\\Rightarrow (a,b)=(0,-4)\;or\;(0,-1)\\(IV)Same\;as\;(III)\;so\;we\;get\;same\;pairs\\hence\;soln\;is\;\\(a,b)=(-4,4),(2,1),(-1,-2),(1,-1),(0,-4),(0,-1))
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25 Jun 2008 22:42:29 IST
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OK. Think it is correct.
Try the second one ...
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25 Jun 2008 22:49:02 IST
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hey rudra see this,it will answer all your queries
http://www.isid.ac.in/~rbb/crmosol_07.pdf
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27 Jun 2008 08:08:33 IST
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edited
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B.Tech CSE, ISMU |
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