Correct Answer - B
Let `upsilon_1 = velocity in the 1st string`
`rArr = upsilon_1 = sqrt((T)/(m))`
Because, `m_1` = mass per unit length
`=((p_1 alpha_1)/(I_1))`
` = p_1 alpha_1`
where `alpha_1` = Area of cross - section
`rArr upsilon_1 = sqrt((T)/(p_1 alpha_1)).... (1)`
Let `upsilon_2`= velocity in the second string
`rArr = upsilon_2 = sqrt((T)/(m_2))`
`rArr upsilon_2 = sqrt((T)/(p_2a_2))...(2)`
Given that,
`rArr upsilon_1 = 2upsilon_2`
`rArr = sqrt((T)/(a_1p_1)) = 2sqrt((T)/(a_2p_2))`
`rArr ((T)/(a_1p_1)) = 2((T)/(a_2p_2))`
`rArr (p_1)/(p_2) =(1)/(4)`
`rArr p_1 : p_2 = 1 :4`
`a_1 = a_2`