# NCERT Class 10 Chapter 5 Quadratic Equations Official CBSE Board Sample Problems Long Answer

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## Question

**Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60. Find the numbers.**

### Solution

Let three consecutive natural numbers are ,x and .

According to question,

(, rejected)

Hence, the numbers arc 9. 10 and 11.

## Question

**Two water taps together can fill a tank in 9 hours 36 minutes. The tap of larger diameter takes 8 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.**

### Solution

Let x be the time taken by larger diameter tap.

be the time taken by smaller diameter tap.

According to question.

(9hrs36minhrs)

(Rejected)

Hence, time taken by Larger and smaller ta arc 16 hrs and 24 hrs respectively.

## Question

**Solve the given quadratic equation for** **.**

### Solution

Roots of above quadratic equation are given by quadratic formula,

## Question

**The length of a rectangle is 5 cm more than its width and the area is** **. Find the length, width and the perimeter.**

### Solution

Let the width be x. Then the length.

is impossible to be a width

Width = 5cm; so, the length = 10cm.

Perimeter = 30cm.

## Question

**Two pipes running together can fill a cistern in** *minutes.* If one pipe takes 3 minutes more than the other to fill the cistern, find the time in which each pipe will fill the cistern.

### Solution

Suppose the time taken by the taps are x minutes and minutes

(Since x cannot be negative)

Times taken by the pipes are 5 and 8 minutes respectively

## Question

**A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away on time, it had to increase its speed by 250 km per hour from its normal speed. Find the normal speed.**

### Solution

Let normal speed of the plane be x km/hr

Time take hrs

Increased speed hrs

Normal speed is 750 km/hr

## Question

**A motor boat whose speed is 20 km/h in still water takes 1 hour more to go 48 km upstream than to return downstream to the same spot. Find the speed of the stream.**

### Solution

Let the speed of the stream be x km/hr

Speed upstream km/hr and speed downstream km/hr

Speed of the stream is 4 km/hr

## Question

**A bus travels at a certain average speed for a distance of 75 km and then travels a distance of 90 km at an average speed of 10 km/h more than the initial speed. It takes 3 hours to complete the total journey. Find the initial speed.**

### Solution

Let the speed of the bus be s km/hr

Time taken by the bus to cover a distance of 75 km is *hrs*

Time taken by the bus to cover a distance of 90km is *hi’s*

Time taken to cover the total distance is 3hrs

According to the given condition,

*Since speed cannot be negative, s* = *50km/hr*

## Question

**Some students planned a picnic. The total budget for food was Rs 2000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by Rs.20. How many students attended the picnic and how much did each student pay for the food?**

### Solution

Suppose x students were to attend

25 students attended the picnic

Each student paid Rs. 80 for the picnic.

## Question

### Solution

=

## Question

**An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speeds of the express train is 11 km/h more than that of the passenger train find the average speed of the two trains.**

### Solution

Let the speed of passenger train be x km/hour. Then speed of express train will be km/hour

Time taken by the passenger train hours

Time taken by the express train = hours

According to question

Thus speed of passenger train is 33 km/hour and that of express train = 44 km/hour.