Question : Which of the following statement(s) is/are TRUE?
I. $\sqrt{11}+\sqrt7<\sqrt{10}+\sqrt8$
II. $\sqrt{17}+\sqrt{11}>\sqrt{15}+\sqrt{13}$
Option 1: Only I
Option 2: Only II
Option 3: Both I and II
Option 4: Neither I nor II
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Correct Answer: Only I
Solution :
Evaluating the given statements:
I. $\sqrt{11}+\sqrt7<\sqrt{10}+\sqrt8$
Squaring both sides we get:
⇒ $(\sqrt{11}+\sqrt7)^2<(\sqrt{10}+\sqrt8)^2$
⇒ $11+7+2\sqrt{77}<10+8+2\sqrt{80}$
⇒ $18+2\sqrt{77}<18+2\sqrt{80}$
So, statement I is true.
II. $\sqrt{17}+\sqrt{11}>\sqrt{15}+\sqrt{13}$
Squaring both sides we get:
⇒ $(\sqrt{17}+\sqrt{11})^2>(\sqrt{15}+\sqrt{13})^2$
⇒ $17+11+2\sqrt{187}>15+13+2\sqrt{195}$
⇒ $28+2\sqrt{187}>28+2\sqrt{195}$
So, statement II is wrong.
Hence, the correct answer is 'Only I'.
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