Directions: Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question while the data in statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. A boat takes a total time of three hours to travel downstream from P to Q and upstream back from Q to P. What is the speed of the boat in still water? I. The speed of the river current is 1 km per hour. II. The distance between P and Q is 4 km.

### Concept & Total Time Equation When given the total time for a round trip, we use the sum of the time taken for the downstream and upstream journeys. $$Total\ Time = \frac{Distance}{Speed_{downstream}} + \frac{Distance}{Speed_{upstream}}$$ ### Step-by-Step Solution * Given in the main question: Total time = $3$ hours. Let boat speed in still water be $u$. * From Statement I: Speed of current $v = 1$ km/hr. Equation becomes $\frac{Distance}{u+1} + \frac{Distance}{u-1} = 3$. This has two variables, so it is insufficient. * From Statement II: Distance $= 4$ km. Equation becomes $\frac{4}{u+v} + \frac{4}{u-v} = 3$. This has two variables, so it is insufficient. * Combining both statements: We can plug both knowns into our main equation. * $\frac{4}{u+1} + \frac{4}{u-1} = 3$ * $4(u-1) + 4(u+1) = 3(u^2 - 1) \Rightarrow 8u = 3u^2 - 3 \Rightarrow 3u^2 - 8u - 3 = 0$. * This quadratic equation will yield one positive root for $u$, thus giving us a unique speed for the boat. ### Exam Strategy & Shortcut For Data Sufficiency, you do not need to solve the final quadratic equation. Once you have a single equation with one variable (where the physics of the problem dictates a single positive root), you can safely declare the data sufficient. ### Common Pitfall A common pitfall is forgetting that a quadratic equation might yield two positive roots. However, in standard distance-speed-time problems of this nature, you will typically get one positive and one negative root, making the positive root the unique valid answer. ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question.**.