Question : What is the fourth proportional of $3 \sqrt{5}, 5 \sqrt{8}$, and $3 \sqrt{10}$?
Option 1: $10 \sqrt{5}$
Option 2: $40 \sqrt{2}$
Option 3: $30$
Option 4: $20$
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Correct Answer: $20$
Solution : We know that, fourth proportional of $a,b,$ and $c=\frac{bc}{a}$ Fourth proportional of $3 \sqrt{5}, 5 \sqrt{8}$ and $3 \sqrt{10}$ = $\frac{5 \sqrt{8} × 3 \sqrt{10}}{3 \sqrt{5}}$ = $\sqrt{5×8×10}$ = $20$ Hence, the correct answer is 20.
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Question : What is the third proportion of $2 \sqrt{3}$ and $6 \sqrt{5}$?
Option 1: $50 \sqrt{6}$
Option 2: $40 \sqrt{3}$
Option 3: $20 \sqrt{6}$
Option 4: $30 \sqrt{3}$
Question : If $x=\frac{4\sqrt{15}}{\sqrt{5}+\sqrt{3}}$, the value of $\frac{x+\sqrt{20}}{x–\sqrt{20}}+\frac{x+\sqrt{12}}{x–\sqrt{12}}$ is:
Option 1: $1$
Option 2: $2$
Option 3: $\sqrt{3}$
Option 4: $\sqrt{5}$
Question : Which of the following is true?
Option 1: $\sqrt 5 + \sqrt 3 > \sqrt 6 + \sqrt 2$
Option 2: $\sqrt 5 + \sqrt 3 < \sqrt 6 + \sqrt 2$
Option 3: $\sqrt 5 + \sqrt 3 = \sqrt 6 + \sqrt 2$
Option 4: $(\sqrt 5 + \sqrt 3 ) (\sqrt 6 + \sqrt 2 )= 1$
Question : If $x=(7+3 \sqrt{5})$, then find the value of $x^2+\frac{1}{x^2}$.
Option 1: $ \frac{580+315 \sqrt{5}}{8}$
Option 2: $\frac{799+328 \sqrt{5}}{8}$
Option 3: $\frac{799+315 \sqrt{5}}{12}$
Option 4: $\frac{799+315 \sqrt{5}}{8}$
Question : Evaluate $\sqrt{20}+\sqrt{12}+\sqrt[3]{729}-\frac{4}{\sqrt{5}-\sqrt{3}}-\sqrt{81}:$
Option 1: $\sqrt{2}$
Option 2: $\sqrt{3}$
Option 3: $0$
Option 4: $2\sqrt{2}$
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