# NCERT Solutions for class 10th Maths Chapter 4 Quadratic Equations

#### Exercise 4.1

Question 1. Check whether the following are quardratic equation:

Note: The Degree of a quadratic polynomials is 2.

Question 2. Represent the following situations in the form of quadratic equations:

• (i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
• (ii) The product of two consecutive positive integers is 306.We need to find the integers.
• (iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
• (iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Note: (i) Consecutive even numbers would be in the form 2x,2x+2,2x+4,and so on.
(ii) Consecutive odd numbers would be in the form 2x-1,2x+1,2x+3,and so on

Note: Since distance is constant (same), there is an inverse relationship between speed and time. Speed increases as time decreases and speed decreases as time increases

#### Exercise 4.2

Question 1. Find the roots of the following quardratic eqations by factorization:

Question 2. Represent the following situations mathematically and solve the equations.

(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was rs.750. We would like to find out the number of toys produced on that day.

Sol. (i) Let the number of marbles John had be x.
Then, the number of marbles Jivanti had =(45 -x).
Then, the number of marbles left with John, when he lost 5 marbles = x-5 Number of marbles left with Jivanti, when she lost 5 marbles = 45-x-5=(40-x)
Therefore, either John had 9 marbles and Jivanti had 36 marbles or vice-versa.

(ii) Let the number of toys produced on that day be x.
Cost of production of each toy that day = (55 -x).
So, total cost of production that day = x (55 -x)
i.e. 750=x(55-x) =>750=55x-x2 => x2-55x+750=0
x2-30x-25x+750 = 0 (Splitting the middle term)
x(x-30)-25(x-30)=0 => (x-25)(x-30)=0 => x=25 or 30
Hence, the cost of production for 25 toys is same as in the case when 30 toys are made.

Question 3. Find two numbers whose sum is 27 and product is 182.

Sol. Let one number be x
Sum of two numbers = 27 (Given)
then,other number = (27-x)
Now,product of two numbers = 182 (Given)
or xx(27-x)=l82 => x2-27x+l82=0
x2-l4x-l3x + 182 = 0 (Splitting the middle term)
(x-13)(x-14)=0
(x-13)=0 x=l3 and (x-14)=0 x=l4
Hence, the two numbers are 13 and 14.

Question 4. Find two consecutive positive integers, sum of whose squares is 365.

Sol. Let the consecutive positive integers be x and x + 1.
According to the question,
Sum of squares of two consecutive positive integers= 365 (Given)

Questions 5. The altitude of a right triangle is 7 cm less than its base.If the hypotenuse is 13 cm, find the other two sides.

Question 6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day.If the total cost of production on that day was rs. 90, find the number of articles produced and the cost of each article.

#### Exercise 4.3

Note: As per the latest syllabus provided by the CBSE for 2019-2020,topic-“completing the square” of chapter.”Quadratic Equations” has been removed.

Question 1. Find the roots of the following quadratic equations,if they exist,by the method of completing the square:

Question 2. Find the roots of the quadratic equation given in Q.1 above by applying the quadratic formula.

Question 3. Find the roots of the following equations:

Question 4. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1/3.Find his present age.

Question 5. In a class test,the sum of Shefali’s marks in Mathematics and English, is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210.Find her marks in the two subjects.

Question 6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Question 7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number.Find the two numbers.

Question 9. Two water taps together can fill a tank in 9 3/8 hours.The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately.Find the time in which each tap can separately fill the tank.

Question 10. 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 speed of the express train is ll km/h more than that of the passenger train. Find the average speed of the two trains.

Question 11. Sum of the areas of two squares is 468 m2.If the difference of their perimeters is 24 m,find the sides of the two squares.

#### Exercise 4.4

Question 1. Find the nature of the roots of the following quadratic equations.

Question 2. Find the values of k for each of the following quadratic equations, so that they have two equal roots

Question 3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2 ? If so, find its length and breadth

Question 4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Sol. Let the age of one friend = x years
Sum of the ages of two friends = 20 years
:. Age of other friend = (20-x) years Four years ago,
Age of one friend =(x-4) years Age of other friend =(16 – x) years According to the question,
Product of their ages, 4 years ago= 48 years
=> (x-4)x(l6-x)=48 =>16x-x2-64+4x=48 => -x2+20x-64-48=0
=> -x2+20x-112=0 =>x2-20x+ll2=0
On comparing with ax2 +bx+ c =0, we get a= I,b =-20,c = 112 Discriminant,D = b2-4ac=400-448 = -48 => D<0
Thus, no real roots are possible.

Question 5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so,find its length and breadth.

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