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4-2.Quadratic Equations and Inequations
hard
In the real number system, the equation $\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$ has
A
no solution
B
exactly two distinct solutions
C
exactly four distinct solutions
D
infinitely many solutions
(KVPY-2012)
Solution
(d)
We have,
$\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$
$\Rightarrow \sqrt{(\sqrt{x-1})^2-2(2) \sqrt{x-1}+(2)^2}$
$+\sqrt{(\sqrt{x-1})^2-2 \times 3 \sqrt{x-1}+(3)^2}=1$
$\Rightarrow \quad \sqrt{(\sqrt{x-1}-2)^2}+\sqrt{(\sqrt{x-1}-3)^2}=1$
$\Rightarrow \quad|\sqrt{x-1}-2|+|\sqrt{x-1}-3|=1$
$x \in[5,10]$
$\sqrt{x-1}-2-\sqrt{x-1}+3=1$
$\therefore$
Hence, $x$ has infinite solutions in $x \in[5,10]$.
Standard 11
Mathematics
