Mathematics

(d) $\frac{3}{5}$

Ram ate $\frac14$, so remaining is $1-\frac14=\frac34$.

Shyam ate $\frac15\times\frac34=\frac{3}{20}$.

Total eaten $=\frac14+\frac{3}{20}=\frac{5}{20}+\frac{3}{20}=\frac{8}{20}$.

Not eaten $=1-\frac{8}{20}=\frac{12}{20}=\frac35$.

(a) $AC>AB$

In a triangle, the side opposite the greater angle is greater. $AC$ is opposite $\angle B$ and $AB$ is opposite $\angle C$. Since $\angle B>\angle C$, $AC>AB$.

(d) None of them

$A=\phi$ has no elements, $B=\{0\}$ has one element $0$, and $C=\{\phi\}$ has one element $\phi$. Since $0\neq\phi$, $B\neq C$; and $A$ has no elements while $B$ and $C$ have one each, so $A\neq B$ and $A\neq C$.

(c) $\{\{1\},\{2\},\{1,2\},\phi\}$

The power set contains all subsets of $A$. For $n(A)=2$ there are $2^2=4$ subsets: $\phi$, $\{1\}$, $\{2\}$, $\{1,2\}$.