# NCERT Solutions for Class 10 Maths Unit 4

### Unit 4 Quadratic Equations Exercise 4.1, 4.2, 4.3, 4.3 4.4, 4.4 Solutions

In elementary algebra, a quadratic equation is any equation having the form

a2 + bx + c = 0

where x represents an unknown, and a, b, and c represent numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.

Quadratic equations can be solved by factoring, by completing the square, by using the quadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.

### Exercise 4.1 : Solutions of Questions on Page Number : 73

Q1 :

Check whether the following are quadratic equations:

It is of the form .

Hence, the given equation is a quadratic equation.

It is of the form .

Hence, the given equation is a quadratic equation.

It is not of the form .

Hence, the given equation is not a quadratic equation.

It is of the form .

Hence, the given equation is a quadratic equation.

It is of the form .

Hence, the given equation is a quadratic equation.

It is not of the form .

Hence, the given equation is not a quadratic equation.

It is not of the form .

Hence, the given equation is not a quadratic equation.

It is of the form .

Hence, the given equation is a quadratic equation.

Q2 :

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.

(i) Let the breadth of the plot be x m.

Hence, the length of the plot is (2x + 1) m.

Area of a rectangle = Length x Breadth

∴ 528 = x (2x + 1)

(ii) Let the consecutive integers be x and x + 1.

It is given that their product is 306.

(iii) Let Rohan’s age be x.

Hence, his mother’s age = x + 26

3 years hence,

Rohan’s age = x + 3

Mother’s age = x + 26 + 3 = x + 29

It is given that the product of their ages after 3 years is 360.

(iv) Let the speed of train be x km/h.

Time taken to travel 480 km =

In second condition, let the speed of train = km/h

It is also given that the train will take 3 hours to cover the same distance.

Therefore, time taken to travel 480 km = hrs

Speed x Time = Distance

⇒480+3x-3840x-24=480
⇒3x-3840x=24
⇒3x2-24x-3840=0
⇒x2-8x-1280=0

### Exercise 4.2 : Solutions of Questions on Page Number : 76

Q1 :

Find the roots of the following quadratic equations by factorisation:

Q2 :

(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. 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. Find out the number of toys produced on that day.

Q3 :

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

Q4 :

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

Q5 :

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

Q6 :

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 : Solutions of Questions on Page Number : 87

Q1 :

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

Q2 :

Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

### Exercise 4.3 4.4 : Solutions of Questions on Page Number : 88

Q1 :

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

If the real roots exist, find them;

(I) 2x2 - 3x + 5 = 0

(II)

(III) 2x2 - 6x + 3 = 0

Q2 :

Find the roots of the following equations:

Q3 :

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

Q4 :

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.

Q5 :

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.

Q6 :

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.

Q7 :

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Q8 :

Two water taps together can fill a tank in 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.

Q9 :

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.

Q10 :

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 : Solutions of Questions on Page Number : 91

Q1 :

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

(I) 2x2 + kx + 3 = 0

(II) kx (x - 2) + 6 = 0

Q2 :

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.

Q3 :

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.

Q4 :

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