SECTION B
This section comprises 5 Very Short Answer (VSA) type questions of 2 marks each.
21. A man in a boat goes 12 km downstream and comes back to the starting point by rowing non-stop in a total time of 3 hours. If the speed of the stream is 3 km/h, find the speed with which the man can row the boat in still water.
22. (a) Solve the following differential equation:
dxdy=x+12−y
OR
(b) If
∫abx3dx=0
and
∫abx2dx=32
, then find the values of 'a' and 'b'.
(differential equation, definite integral, speed, velocity, time)
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Step-by-Step Solution
Step 1: Define variables and speeds
First, we define the variables. Let vb be the speed of the boat in still water, which is what we need to find. The speed of the stream, vs, is given as 3 km/h. When the boat travels downstream, its speed is the sum of the boat's speed and the stream's speed. When it travels upstream, its speed is the difference between the boat's speed and the stream's speed.
Step 2: Formulate time equations
The distance traveled downstream is 12 km, and the distance traveled upstream is also 12 km. Using the formula Time = Distance / Speed, we can write expressions for the time taken for the downstream journey (tdown) and the upstream journey (tup).
Step 3: Set up total time equation
The problem states that the total time for the round trip (downstream and back upstream) is 3 hours. So, we sum the individual time expressions and set them equal to 3.
Step 4: Solve the quadratic equation
To solve for vb, we first find a common denominator and simplify the equation. This leads to a quadratic equation. We then solve the quadratic equation by factoring or using the quadratic formula to find the possible values for vb.
Step 5: Select the valid speed
Since speed cannot be negative, we discard the negative solution. Therefore, the speed of the man in still water is 9 km/h.