Assertion (A): A moving boat goes downstream at 50 km per hour and upstream at 30 km per hour. The speed of stream is 40 km per hour.

Reason (R): If the speed of boat in still water is x km per hour and speed of stream is y km per hour, then y - x = 50 and y + x = 30 ⇒ y = 40.

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are false.

Explanation

Given,

Downstream speed = 50 km/hr

Upstream speed = 30 km/hr

Speed of the stream(y) = 40 km/hr

The speed of the stream is the difference between the downstream speed and the speed of the boat in still water:

Speed of stream = $\dfrac{\text{2 Downstream speed - Upstream speed}}{2}$

= $\dfrac{50 - 30}{2}$

= $\dfrac{20}{2}$

= 10 km/hr

∴ The speed of the stream is 10 km/h, not 40 km/h.

∴ Assertion(A) is false.

Given,

Speed of the boat in still water = x km/hr

Speed of the stream = 40 km/hr

y - x = 50 and y + x = 30

Adding both equation, we get:

⇒ (y - x) + (y + x) = 50 + 30

⇒ y - x + y + x = 50 + 30

⇒ 2y = 80

⇒ y = $\dfrac{80}{2}$

⇒ y = 40

And, subtracting both equation, we get:

⇒ (y - x) - (y + x) = 50 - 30

⇒ y - x - y - x = 50 - 30

⇒ - 2x = 20

⇒ x = -$\dfrac{20}{2}$

⇒ x = - 10

The speed of the boat in still water(x) cannot be -10 km/hr, as speed cannot be negative.

∴ Reason(R) is false.

Hence, both Assertion (A) and Reason (R) are false.