UGC-NET-Paper1

### Step 1: Define Variables
Let the speed of the man in still water be **x** km/h.
Let the speed of the river current be **y** km/h.

- **Upstream speed** = (x - y) km/h
- **Downstream speed** = (x + y) km/h

### Step 2: Form Equations
Given conditions:

1. **12 km upstream + 20 km downstream in 4 hours**
=> 12/(x - y) + 20/(x + y) = 4

2. **18 km upstream + 10 km downstream in 4 hours**
=> 18/(x - y) + 10/(x + y) = 4

### Step 3: Solve for x and y
Using elimination method:

Divide both equations by 2 to simplify:

(6/(x - y)) + (10/(x + y)) = 2
(9/(x - y)) + (5/(x + y)) = 2

Now, subtract the second equation from the first:

(6/(x - y)) - (9/(x - y)) + (10/(x + y)) - (5/(x + y)) = 0

(-3/(x - y)) + (5/(x + y)) = 0
=> 5/(x + y) = 3/(x - y)
=> 5(x - y) = 3(x + y)
=> 5x - 5y = 3x + 3y
=> 2x = 8y
=> x = 4y

### Step 4: Substitute x = 4y
Substituting in the first equation:

12/(4y - y) + 20/(4y + y) = 4
12/(3y) + 20/(5y) = 4
4/y + 4/y = 4
8/y = 4
y = 2

Now, x = 4(2) = 8

### Final Answer:
**(3) 8 km/hr**