KVPY SA Exam 2015 Question Paper

Question : 62 of 80

Marks: +1, -0

For a real number x , let [x] denote the largest integer less than or equal to x, and let {x} = x - [x]. The number of solutions x to the equation [x]{x}=5 with 0 ≤ x ≤ 2015 is

0
3
2008
2009

Validate

Solution:

[x] . \{x\} = 5

\{x\} =5/[x]

∴

0 ≤5/[x]<1

⇒

1<5/\{x\}<∞

⇒

5 < [x] < ∞

[x] = 6, 7, 8, .....2015

∴

x = [x] + \{x\}

=\{n+5/n|n∊\{6,7,....2015\}\}

No. of values of 'x' = 2009

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