$${E_1}(r) = {1 \over {4\pi {\varepsilon _0}}}{Q \over {{r^2}}}$$

$${E_2}(r) = {\lambda \over {2\pi {\varepsilon _0}r}}$$

$${E_3}(r) = {\sigma \over {2{\varepsilon _0}}}$$

At $$r = {r_0}$$,

$${E_1}({r_0}) = {1 \over {4\pi {\varepsilon _0}}}{Q \over {r_0^2}}$$

$${E_2}({r_0}) = {\lambda \over {2\pi {\varepsilon _0}{r_0}}}$$

$${E_3}({r_0}) = {\sigma \over {2{\varepsilon _0}}}$$

As $${E_1}({r_0}) = {E_2}({r_0}) = {E_3}({r_0})$$ (Given)

$$\therefore$$ $${1 \over {4\pi {\varepsilon _0}}}{Q \over {r_0^2}} = {\lambda \over {2\pi {\varepsilon _0}{r_0}}} = {\sigma \over {2{\varepsilon _0}}}$$

Then

$${1 \over {4\pi {\varepsilon _0}}}{Q \over {r_0^2}} = {\sigma \over {2{\varepsilon _0}}}$$

or $$Q = \sigma \pi r_0^2$$

Hence, option (a) is incorrect.

Now, $${\lambda \over {2\pi {\varepsilon _0}{r_0}}} = {\sigma \over {2{\varepsilon _0}}}$$ or $${r_0} = {\lambda \over {\pi \sigma }}$$

Hence, option (b) is incorrect.

At $$r = {r_0}/2$$,

$${E_1}({r_0}/2) = {1 \over {4\pi {\varepsilon _0}}}{Q \over {{{({r_0}/2)}^2}}} = {4 \over {4\pi {\varepsilon _0}}}{Q \over {r_0^2}} = 4{E_1}({r_0})$$

or, $${E_1}({r_0}) = {1 \over 4}{E_1}({r_0}/2)$$

$${E_2}({r_0}/2) = {\lambda \over {2\pi {\varepsilon _0}({r_0}/2)}} = {{2\lambda } \over {2\pi {\varepsilon _0}{r_0}}} = 2{E_2}({r_0})$$

or, $${E_2}({r_0}) = {1 \over 2}{E_2}({r_0}/2)$$

$$\because $$$${E_1}({r_0}) = {E_2}({r_0})$$

$$\therefore$$ $${1 \over 4}{E_1}({r_0}/2) = {1 \over 2}{E_2}({r_0}/2)$$

or, $${E_1}({r_0}/2) = 2{E_2}({r_0}/2)$$

Hence, option (c) is correct.

$${E_3}({r_0}/2) = {\sigma \over {2{\varepsilon _0}}} = {E_3}({r_0})$$

$$\because$$ $${E_2}({r_0}) = {E_3}({r_0})$$

$$\therefore$$ $${1 \over 2}{E_2}({{{r_0}} \over 2}) = {E_3}({{{r_0}} \over 2})$$

or, $${E_2}({r_0}/2) = 2{E_3}({r_0}/2)$$

Hence, option (d) is incorrect.