(A)

The electric flux through left surface is,

\(\displaystyle\phi_{{1}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(-\hat{{{i}}}\right)}\right)}\)

\(\displaystyle=-{L}^{{2}}{\left({a}+{b}{x}\right)}\)

For left surface x=0.

\(\displaystyle\phi_{{1}}=-{a}{L}^{{2}}\)

Th electric flux through right surface is,

\(\displaystyle\phi_{{2}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(\hat{{{i}}}\right)}\right)}\)

\(\displaystyle={L}^{{2}}{\left({a}+{b}{x}\right)}\)

For right surface x=L

\(\displaystyle\phi_{{2}}={L}^{{2}}{\left({a}+{b}{L}\right)}\)

\(\displaystyle={a}{L}^{{2}}+{b}{L}^{{3}}\)

The electric flux through top surface is,

\(\displaystyle\phi_{{3}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(\hat{{{j}}}\right)}\right)}\)

\(\displaystyle={c}{L}^{{2}}\)

The electric flux through bottom surface is,

\(\displaystyle\phi_{{4}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(-\hat{{{j}}}\right)}\right)}\)

\(\displaystyle=-{c}{L}^{{2}}\)

Since there is no field along z-direction, the elcetric flux through front and back surface of the the cube must be zero.

The total electric flux through surface of the cube is,

\(\displaystyle\phi_{{E}}=\phi_{{1}}+\phi_{{2}}+\phi_{{3}}+\phi_{{4}}\)

\(\displaystyle=-{a}{L}^{{2}}+{a}{L}^{{2}}+{b}{L}^{{3}}+{c}{L}^{{2}}-{c}{L}^{{2}}\)

\(\displaystyle={b}{L}^{{3}}\)

Thus, the electric flux through surface of the cube is \(\displaystyle{b}{L}^{{3}}\)

(C)

According to Gauss law,

\(\displaystyle\phi_{{E}}={\frac{{{q}}}{{\epsilon_{{0}}}}}\)

\(\displaystyle{q}=\epsilon_{{0}}\phi_{{E}}\)

\(\displaystyle{q}=\epsilon_{{0}}{b}{l}^{{3}}\)

Thus, the net charge inside the cube is \(\displaystyle\epsilon_{{0}}{b}{L}^{{3}}\)

The electric flux through left surface is,

\(\displaystyle\phi_{{1}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(-\hat{{{i}}}\right)}\right)}\)

\(\displaystyle=-{L}^{{2}}{\left({a}+{b}{x}\right)}\)

For left surface x=0.

\(\displaystyle\phi_{{1}}=-{a}{L}^{{2}}\)

Th electric flux through right surface is,

\(\displaystyle\phi_{{2}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(\hat{{{i}}}\right)}\right)}\)

\(\displaystyle={L}^{{2}}{\left({a}+{b}{x}\right)}\)

For right surface x=L

\(\displaystyle\phi_{{2}}={L}^{{2}}{\left({a}+{b}{L}\right)}\)

\(\displaystyle={a}{L}^{{2}}+{b}{L}^{{3}}\)

The electric flux through top surface is,

\(\displaystyle\phi_{{3}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(\hat{{{j}}}\right)}\right)}\)

\(\displaystyle={c}{L}^{{2}}\)

The electric flux through bottom surface is,

\(\displaystyle\phi_{{4}}=\vec{{{E}}}\cdot\vec{{{A}}}\)

\(\displaystyle={\left({\left({a}+{b}{x}\right)}\hat{{{i}}}+{c}\hat{{{j}}}\right)}\cdot{\left({L}^{{2}}{\left(-\hat{{{j}}}\right)}\right)}\)

\(\displaystyle=-{c}{L}^{{2}}\)

Since there is no field along z-direction, the elcetric flux through front and back surface of the the cube must be zero.

The total electric flux through surface of the cube is,

\(\displaystyle\phi_{{E}}=\phi_{{1}}+\phi_{{2}}+\phi_{{3}}+\phi_{{4}}\)

\(\displaystyle=-{a}{L}^{{2}}+{a}{L}^{{2}}+{b}{L}^{{3}}+{c}{L}^{{2}}-{c}{L}^{{2}}\)

\(\displaystyle={b}{L}^{{3}}\)

Thus, the electric flux through surface of the cube is \(\displaystyle{b}{L}^{{3}}\)

(C)

According to Gauss law,

\(\displaystyle\phi_{{E}}={\frac{{{q}}}{{\epsilon_{{0}}}}}\)

\(\displaystyle{q}=\epsilon_{{0}}\phi_{{E}}\)

\(\displaystyle{q}=\epsilon_{{0}}{b}{l}^{{3}}\)

Thus, the net charge inside the cube is \(\displaystyle\epsilon_{{0}}{b}{L}^{{3}}\)