SSC GD 2024 PYQ — Question 58

The correct answer is option 4 i.e. 6 km/hr.

the original speed of the boat be ( x ) km/h.

Time = Distance / Speed

Time₁ = 54 / x

the speed is increased by 3 km/h, the new speed is $x + 3$ km/h.

Time₂ = 54 / x + 3

the boat takes 3 hours less when the speed is increased by 3 km/h. 

Time₁ - Time₂ = 3

⇒ 54 / x - 54 / x + 3 = 3

⇒ $\frac{54(x + 3) - 54x}{x(x + 3)} = 3$

⇒$\frac{54x + 162 - 54x}{x(x + 3)} = 3$

⇒ $\frac{162}{x(x + 3)} = 3$

⇒ 162 = 3x(x + 3)

⇒162 = 3x²+ 9x

⇒ 54 = x² + 3x

⇒ ​​​​​x² + 3x - 54 = 0

⇒x = $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

( x² + 3x - 54 = 0 ), ( a = 1 ), ( b = 3 ), and ( c = -54 ). 

⇒x = $\frac{-3 \pm \sqrt{3^2 - 4(1)(-54)}}{2(1)}$

⇒ x = $\frac{-3 \pm \sqrt{9 + 216}}{2}$

⇒ x = $\frac{-3 \pm \sqrt{225}}{2}$

⇒ x = $\frac{-3 \pm 15}{2}$

The two possible solutions are:

⇒ x = $\frac{-3 + 15}{2} = \frac{12}{2} = 6$

⇒ x = $\frac{-3 - 15}{2} = \frac{-18}{2} = -9$

speed cannot be negative, the original speed of the boat is ( x = 6 ) km/h.