**Q) **Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took 4 hours more than Ajay to complete the journey of 400 km.

1. What will be the distance covered by Ajay’s car in two hours?

a) 2(x +5) km b) (x – 5) km c) 2( x + 10) km d) (2x + 5) km

2. Which of the following quadratic equation describe the speed of Raj’s car?

a) x2 – 5 x – 500 = 0 b) x2 + 4x – 400 = 0 c) x2 + 5x – 500 = 0 d) x2 – 4x + 400 = 0

3. What is the speed of Raj’s car?

a) 20 km/hour b) 15 km/hour c) 25 km/hour d) 10 km/hour

4. How much time took Ajay to travel 400 km?

a) 20 hour b) 40 hour c) 25 hour d) 16 hour

**Ans: 1. Distance covered by Ajay’s car in two hours:**

**∵ **It is given that Speed of Raj’s car = X km/ h

and also given that Ajays car is 5 km/h faster

∴ Speed of Ajay’s car = x + 5 km/h

We know that the the distance D in km is given by:

D = S x T

where, S = Speed in km/h

and T = time taken in hr

Hence, the distance travelled by Ajay’s care in 2 hrs:

= 2 (X + 5) km/h

**Therefore option (a) is correct.**

**2. Quadratic equation for the speed of Raj’s car:**

It is given that the Raj took 4 hours more than Ajay to complete the journey of 400 km.

We know that the the time T in hrs is given by:

T =

where, D = distance travelled in km

and S = Speed in km/h

∵ Speed of Raj’s Car = X km\h

∴ Time taken to travel 400 km by Raj’s car =

∵ Speed of Raj’s Car = X + 5 km\h

∴ Time taken to travel 400 km by Ajay’s car =

By given condition, Time taken by Raj – Time taken by Ajay = 4 hrs

∴ = 4

∴ 400 = 4

∴ 100 = 1

∴ 100 (5) = X (X +5)

∴ X^{2 }+ 5 X – 500 = 0

**Therefore option (c) is correct.**

**3. Speed of Raj’s car:**

To get value of X, we need to solve the quadratic equation:

X^{2 }+ 5 X – 500 = 0

X + 25 X – 20 X – 500 = 0

(X + 25) (X – 20) = 0

∴ X = – 25 and X = 20

Since value of speed can not be negative X – 25; Therefore, X = 20 km/h

**Therefore option (a) is correct.**

**4. Time taken by Ajay to travel 400 km:**

Given that the speed of Ajay’s car = X + 5

∵ The speed of Raj’s car, X = 20 km/h

∴ The speed of Ajay’s car = X + 5 = 20 + 5 = 25 km/h

∴ Time taken to travel 400 km by Ajay’s car = = 16 hrs

**Therefore option (d) is correct.**