Identities for irrational numbers

Q. Find the sum of the squares of the following: $\frac{\sqrt{3}}{\sqrt{2}+1},\frac{\sqrt{3}}{\sqrt{2}-1},\frac{\sqrt{2}}{\sqrt{3}}$

1. $\frac{56}{4}$
2. $18\frac{2}{3}$
3. $2\frac{18}{3}$
4. $3\frac{18}{2}$

Q. If $x=\sqrt{\frac{5+2\sqrt{6}}{5-2\sqrt{6}}},\text{then}{x}^2(x-10{)}^2=$

1. 5
2. $5^2$
3. 1
4. $2^2-1$

Q. If $2^a=3^b=6^c$, then $c=$ .

1. $\frac{ab}{a+b}$
2. $\frac{a}{a+b}$
3. $\frac{a}{1+b}$
4. $\frac{abc}{a+b}$

Q. $(2+\sqrt{3}{)}^2+(2-\sqrt{3}{)}^2=$

1. 7
2. 5
3. 14
4. 12

Q. Match the following:

1. 1
2. $2-\sqrt{3}$
3. $3+2\sqrt{2}$
4. $\frac{3-\sqrt{5}}{2}$

Q. If $y=\sqrt{\sqrt{7}+7+\sqrt{8+2\sqrt{7}}}-\sqrt{7}$, then find the value of $y$.

1. 1
2. $\sqrt{7}$
3. $8-2\sqrt{7}$
4. $\sqrt{7}-1$

Q. ${64}^{-\frac{1}{3}}[{64}^{\frac{1}{3}}-{64}^{\frac{2}{3}}]=$ .

1. $\frac{1}{3}$
2. $-3$
3. $\frac{2}{3}$

Q. Sum of two irrational numbers is always an irrational number.

1. True
2. False

Q. $(\sqrt{4}+\sqrt{3})\times (\sqrt{4}-\sqrt{3})=$

1. 1
2. 2
3. 7
4. 3

Q. The product of two irrational numbers is always an irrational number.

1. True
2. False