Class 10 Mathematics - Chapters 1-4

If $2+\sqrt{5}$ were rational, subtracting the rational number 2 would make $\sqrt{5}$ rational, which is impossible. The other choices are rational.

If the sum were rational, subtracting the original rational number would make the irrational number rational. This contradiction proves that the sum is irrational.

$\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}$. Since $\sqrt{2}$ is irrational, so is $\sqrt{18}$.

$1/\sqrt{2}=\sqrt{2}/2$. A nonzero rational multiple of $\sqrt{2}$ is irrational. The other three choices are rational.

$6^n=2^n3^n$ has no factor 5, whereas a number ending in 0 must be divisible by $10=2\times5$. In fact, every positive power of 6 ends in 6.

The greatest length that divides all three measurements is their HCF. HCF(96, 120, 168) = 24, so each piece is 24 m long.

The next common ringing time is LCM(12, 15, 20). The LCM is 60, so they next ring together after 60 minutes.

The required number is their LCM. Since $8=2^3$, $12=2^2\times3$ and $18=2\times3^2$, the LCM is $2^3\times3^2=72$.

Only 2 and 3 are common prime factors. Taking their smaller exponents gives $2^2\times3^2=36$.

Squaring gives $a^2=3b^2$, so 3 divides $a$. Write $a=3k$; then $b^2=3k^2$, so 3 also divides $b$. This contradicts the assumption that $a$ and $b$ are coprime.