Q45 / 90 · KVPY 2012 (SB/SX) · Mathematics

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
  • Correct answer
    D
    Infinitely may solutions

Solution

$\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1 ; x \geq 1$ $\sqrt{(x-1)-2 \times 2 \sqrt{x-1}+4}+\sqrt{(x-1)-6 \sqrt{x-1}+9}=1$ $\begin{array}{l} 2 \sqrt{x-1}=6 \\ x=10 \\ \text { Case -II } \\ \sqrt{x-1}-2-\sqrt{x-1}+3=1 \quad 5 \leq x \leq 10 \end{array}$ Case -III $\begin{array}{l} -\sqrt{x-1}+2-\cdot \sqrt{x-1}+3=1 \\ 2 \cdot \sqrt{x-1}=4 \\ x=5 \end{array}$

Hence, the correct answer is option: 'D'.

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