B

[latex]\frac {3}{4}[/latex] × (x − 2) x = (x + 7)(x − 10)
⇒[latex]{x}^{2}[/latex] − 6x − 280 = 0
⇒ x = 20 and x =-14
so, the acceptable values is x = 20
Therefore, Total work = (x - 2)x = 18 x 20 = 360 unit
Now 360 = 30 x k
⇒ k = 12 days

A

Ratio of times taken by A and B = 1 : 2.
The time difference is (2 - 1) 1 day while B take 2 days and A takes 1 day.
If difference of time is 1 day, B takes 2 days.
If difference of time is 30 days, B takes 2 x 30 = 60 days.
So, A takes 30 days to do the work.
A's 1 day's work = [latex]\frac {1}{30}[/latex]
B's 1 day's work = [latex]\frac {1}{60}[/latex]
(A + B)'s 1 day's work = [latex]\frac {1}{30} + \frac {1}{60} = \frac {1}{120}[/latex]
A and B together can do the work in 20 days.

B

Ratio of rates of working of A and B = 2 : 1. So, ratio of times taken = 1:2
Therefore, A's 1 day's work = [latex]\frac {1}{9}[/latex]
B's 1 day's work = [latex]\frac {1}{18}[/latex]
(A+B)'s 1 day's work= [latex]\frac {1}{9} + \frac {1}{18} = \frac {1}{6}[/latex]
so, A and B together can finish the work in 6 days

D

(A + B)'s 1 day's work = [latex]\frac {1}{30}[/latex]
C's 1 day work = [latex]\frac {1}{60}[/latex]
(A + B + C)'s 1 day's work = [latex]\frac {1}{20} + \frac {1}{60} = \frac {1}{15}[/latex]
Also A's 1 day's work = (B + C)'s 1 day's work
Therefore, we get: 2 x (A's 1 day 's work) = [latex] \frac {1}{15}[/latex]
[latex] \Rightarrow[/latex] A's 1 day's work = [latex] \frac {1}{30}[/latex]
Therefore, B's 1 day's work = [latex]\frac {1}{20} - \frac {1}{30} = \frac {1}{60}[/latex]
So, B alone could do the work in 60 days.

A

Ratio of efficiency A : B : C
Ratio of efficiency 3 : 1 : 2
Ratio of No.of days [latex]\frac {1}{3}[/latex] : [latex]\frac {1}{1}[/latex] : [latex]\frac {1}{2}[/latex]
or 2 : 6 :3
Hence A is correct.