NIMCET 2023 Question Paper with solutions

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Question : 46 of 120

Marks: +1, -0

If

{|x-6|}={|x^2-4 x |}-{|x^2-5 x+6 |}

, where

x

is a real variable, then choose the correct option:

$x=(2,5)$
$x=[2,3] ∪[6, ∞)$
$x= {\R}-[2,6]$
None of these

Solution:

|x-6|={| x^2-4 x |}+|(x-3)(x-2)|

Its one solution is

\;x(x-4) ≥ 0

and

x^2-5 x+6 ≥ 0

and

x ≥ 6

\;x ≤ 0

or

x ≥ 4

\;x ≤ 2

and

\;x ≥ 3

Its intersection

x ≥ 6

⇒ x ∈[6, ∞)

Its second solution

x(x-4) ≤ 0

and

x^2-5 x+6 ≤ 0

0 ≤ x ≤ 4

and

2 ≤ x ≤ 3

and

x< 6

and

x ≤ 6

Its intersection is

2 ≤ x ≤ 3

⇒ x ∈[2,3]

Union of both solutions is

x ∈[2,3] ∪[6, ∞)

.

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