1. **Problem Statement:** (a) Raj travels 15 km at a speed of $x$ km/h. Find the time taken by Raj in minutes. (b) Rajiv travels the same distance at a speed 10 km/h slower than Raj. Find the expression for Rajiv's time in terms of $x$. (c) Rajiv takes 12 minutes longer than Raj. Form an equation in $x$ and show it simplifies to $x^2 - 10x - 750 = 0$. (d) Solve the quadratic equation and calculate Rajiv's travel time in minutes and seconds. 2. **Part (a) Time for Raj:** Time = Distance / Speed. Distance = 15 km, Speed = $x$ km/h. Time in hours = $\frac{15}{x}$. Convert hours to minutes: multiply by 60. Time in minutes = $60 \times \frac{15}{x} = \frac{900}{x}$. 3. **Part (b) Time for Rajiv:** Speed of Rajiv = $x - 10$ km/h. Time in hours = $\frac{15}{x-10}$. Time in minutes = $60 \times \frac{15}{x-10} = \frac{900}{x-10}$. 4. **Part (c) Equation from time difference:** Rajiv takes 12 minutes longer: $\frac{900}{x-10} - \frac{900}{x} = 12$. Multiply both sides by $x(x-10)$: $$900x - 900(x-10) = 12x(x-10)$$ Simplify left side: $$900x - 900x + 9000 = 12x^2 - 120x$$ So, $$9000 = 12x^2 - 120x$$ Divide all terms by 12: $$750 = x^2 - 10x$$ Rearranging: $$x^2 - 10x - 750 = 0$$ 5. **Part (d) Solve quadratic equation:** Equation: $x^2 - 10x - 750 = 0$. Using quadratic formula: $$x = \frac{10 \pm \sqrt{(-10)^2 - 4 \times 1 \times (-750)}}{2} = \frac{10 \pm \sqrt{100 + 3000}}{2} = \frac{10 \pm \sqrt{3100}}{2}$$ Calculate $\sqrt{3100} \approx 55.68$. Solutions: $$x = \frac{10 + 55.68}{2} = 32.84$$ $$x = \frac{10 - 55.68}{2} = -22.84$$ Speed cannot be negative, so $x = 32.84$ km/h (to 2 d.p). 6. **Calculate Rajiv's time:** Rajiv's speed = $x - 10 = 32.84 -10 = 22.84$ km/h. Time in minutes = $\frac{900}{22.84} \approx 39.40$ minutes. Convert 0.40 minutes to seconds: $0.40 \times 60 = 24$ seconds. **Final answers:** - $x \approx 32.84$ km/h - Rajiv's time $\approx 39$ minutes $24$ seconds