In a 1000 m race, A beats B by 50 meters or by 6 seconds. The time taken by A to complete the race is:

114 seconds

The correct answer is Option (2) → 114 seconds

$\text{Let time taken by A} = t \text{ sec}$

$\text{A beats B by 6 sec} \Rightarrow \text{B takes } (t+6) \text{ sec}$

$\text{Speeds ratio} = \frac{1000}{t} : \frac{1000}{t+6} = \frac{t+6}{t}$

$\text{Also, A beats B by 50 m} \Rightarrow \frac{\text{speed of A}}{\text{speed of B}} = \frac{1000}{950}$

$\frac{t+6}{t} = \frac{1000}{950} = \frac{20}{19}$

$19(t+6) = 20t$

$19t + 114 = 20t$

$t = 114$

The time taken by A is $114$ seconds.