Quadratic Equations – Solved Board Questions & Speed-Time Word Problems
Solved CBSE & ICSE board examination questions on quadratic equations: discriminant conditions, upstream-downstream word problems, and train speed problems.
Quadratic Equations – Solved Board Examination Problems
A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
1. Variables & Speed Equations:
Let the speed of the stream = x km/h (where x < 18).
• Speed of boat upstream = (18 - x) km/h
• Speed of boat downstream = (18 + x) km/h [1 Mark]
2. Time Calculations:
Distance = 24 km
• Time taken upstream, t₁ = 24 / (18 - x) hours
• Time taken downstream, t₂ = 24 / (18 + x) hours
3. Setting up Quadratic Equation (t₁ - t₂ = 1 hour):
24 / (18 - x) - 24 / (18 + x) = 1
24 [ (18 + x) - (18 - x) ] / [ (18 - x)(18 + x) ] = 1
24 [ 2x ] / [ 324 - x² ] = 1
48x = 324 - x²
x² + 48x - 324 = 0 [2 Marks]
4. Solving by Factorization:
x² + 54x - 6x - 324 = 0
x(x + 54) - 6(x + 54) = 0
(x - 6)(x + 54) = 0
x = 6 or x = -54
Since speed cannot be negative, x = -54 is rejected.
Final Answer: The speed of the stream is 6 km/h. [1 Mark]
Indexed Topics & Examination Keywords
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