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NCERT Class 10 Chapter 5 Quadratic Equations Official CBSE Board Sample Problems Long Answer
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.
( 9hrs36min hrs)
(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 = 5 cm; so, the length = 10 cm.
Perimeter = 30 cm.
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 90 km is hi s
Time taken to cover the total distance is 3hrs
According to the given condition,
Since speed cannot be negative, s = 50 km⟋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 ₹ 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 ₹ 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.