In the real number system, the equation {x+3-4{x-1}}+{x+8-6{x-1}}=1 has

Correct Answer is : Infinitely many solution

Correct Answer is : Infinitely many solution

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BITSAT 2019 Slot 2 Solved Paper

Section: Mathematics

Question : 111 of 150

Marks: +1, -0

\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1

has

Solution:

We have,

\sqrt{x+3-4\sqrt{x-1}}

+\sqrt{x+8-6\sqrt{x-1}}=1

\sqrt{(\sqrt{x-1})^{2}-2\times2\sqrt{x-1}+4}

+\sqrt{(\sqrt{x-1})^{2}-2\times3\sqrt{x-1}+9}=1

\Rightarrow\sqrt{(\sqrt{x-1}-2)^{2}}+\sqrt{(\sqrt{x-1}-3)^{2}}=1

\Rightarrow|\sqrt{x-1}-2|+|\sqrt{x-1}-3|=1

\Rightarrow|\sqrt{x-1}-2|+|\sqrt{x-1}-3|

=(\sqrt{x-1}-2)-(\sqrt{x-1}-3)

We know that,

|x-a|+|x-b|=(x-a)-(x-b)

then,

(x-a)(x-b)<0

∴

(\sqrt{x-1}-2)(\sqrt{x-1}-3)<0

\Rightarrow 2 < \sqrt{x-1}<3\Rightarrow5 < x<10

\therefore

Equation have infinite many solutions.

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