KVPY SA Exam 2011 Question Paper

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Question : 3 of 80

Marks: +1, -0

If

x+1/x=a,x^2+1/x^3=b

, then

x^3+1/x^2

is

$a^3+ a^2-3a-2-b$
$a^3-a^2-3a+4-b$
$a^3-a^2+3a-6-b$
$a^3+a^2+3a-16-b$

Solution:

x+1/x=a

and

x^2+1/x^3=b

(x+1/x)^2=a^2

⇒x^2+1/x^2+2=a^2

....(1)
and

(x+1/x)^3=a^3

⇒x^3+1/x^3+3x. 1/x(x+1/x)=a^3 ...(2)

adding (1) & (2)

(x^2+1/x^3)+(x^3+1/x^2)+2+3(x+1/x)=a^2+a^3

b+(x^3+1/x^2)+2+3a=a^2+a^3

∴x^3+1/x^2=a^3+a^2-3a-2-b

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