Question : If $a=\sqrt{6}–\sqrt{5}$, $b=\sqrt{5}–2$, and $c=2–\sqrt{3}$, then point out the correct alternative among the four alternatives given below.

Option 1: $b<a<c$

Option 2: $a<c<b$

Option 3: $b<c<a$

Option 4: $a<b<c$

Question : Let $\sqrt[3]{a}=\sqrt[3]{26}+\sqrt[3]{7}+\sqrt[3]{63}$. Then:

Option 1: $a<729$ but $a>216$

Option 2: $a<216$

Option 3: $a>729$

Option 4: $a = 729$

Question : If $x=(\sqrt{6}-1)^{\frac{1}{3}}$, then the value of $\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)$ is:

Option 1: $\frac{2 \sqrt{6}-6}{5}$

Option 2: $\frac{4 \sqrt{6}-6}{5}$

Option 3: $\frac{4 \sqrt{6}-6}{3}$

Option 4: $\frac{4 \sqrt{3}-6}{5}$

Question : If $7+3x>3+2x$; and $x+3(x-7)<5-x$; then $x$ can take which of the following values?

Option 1: –5

Option 2: 5

Option 3: 6

Option 4: –6

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: $30 \sqrt{3}$