Chapter 4 : Quadratic Equations

At Saralstudy, we are providing you with the solution of Class 10 Mathematics Quadratic Equations according to the latest NCERT (CBSE) Book guidelines prepared by expert teachers. Here we are trying to give you a detailed answer to the questions of the entire topic of this chapter so that you can get more marks in your examinations by preparing the answers based on this lesson. We are trying our best to give you detailed answers to all the questions of all the topics of Class 10 Mathematics Quadratic Equations so that you can prepare for the exam according to your own pace and your speed.

Exercise 1 ( Page No. : 73 )

Check whether the following are quadratic equations :
(i) (x + 1)² = 2(x – 3) (ii) x² – 2x = (–2) (3 – x) (iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)

(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x²+ 3x + 1 = (x – 2)² (vii) (x + 2)³ = 2x (x2 – 1) (viii) x³ – 4x² – x + 1 = (x – 2)³

Exercise 1 ( Page No. : 73 )

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.

Exercise 2 ( Page No. : 76 )

Find the roots of the following quadratic equations by factorisation:

Exercise 2 ( Page No. : 76 )

Solve the problems given in Example 1.

(i) let x and y are the no of marbles of Romil and Somi respectively.

Acc. to ques,

X + y = 45................... (1)

When they both lost their 5 marbles

(x – 5) (y – 5) = 124.................... (2)

From equation (1), we have

y = 45 – x

Put this value of x in equation (2), we get

(x – 5)(45 – x – 5) = 124

(x – 5)(40 – x) = 124

40x – x² – 200 + 5x = 128

x² - 45x + 324 = 0

x² - 36x - 9x + 324 = 0

x (x – 36) – 9 (x – 36) = 0

(x – 9) (x – 36) = 0

x = 9 or x = 36

Exercise 2 ( Page No. : 76 )

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

Exercise 2 ( Page No. : 76 )

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

Exercise 2 ( Page No. : 76 )

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

Exercise 2 ( Page No. : 76 )

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 3 ( Page No. : 88 )

Find the roots of the following quadratic equations, if they exist, by the method of
completing the square:
(i) 2x²– 7x + 3 = 0 (ii) 2x²+ x – 4 = 0 (iv) 2x² + x + 4 = 0

Exercise 3 ( Page No. : 88 )

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

Exercise 3 ( Page No. : 88 )

Find the roots of the following equations:

Exercise 3 ( Page No. : 88 )

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.

Exercise 3 ( Page No. : 88 )

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.

Exercise 3 ( Page No. : 88 )

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.

Let the shorter side of rectangular field be = x meter

The diagonal of the field = (x + 60) m

And, longer side is 30m more than shorter side = (x + 30) m

According to question;

Using Pythagoras theorem;

(x + 60)² = x² + (x + 30)²

x (x - 90) + 30 (x - 90) = 0

(x + 30) (x - 90) = 0

(x + 30) = 0 or (x - 90) = 0

Either, x = -30 or x = 90

But x ≠ -30 because length can never be negative.

Hence, shorter side of field is 90 m and longer side is (x + 30) = 120 m

Exercise 3 ( Page No. : 88 )

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.

Exercise 3 ( Page No. : 88 )

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.

Exercise 3 ( Page No. : 88 )

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.

Exercise 3 ( Page No. : 88 )

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 11km/h more than that of the passenger train, find the average speed of the two trains.

Exercise 3 ( Page No. : 88 )

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

Exercise 4 ( Page No. : 91 )

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x²– 3x + 5 = 0 (iii) 2x²– 6x + 3 = 0

(i) 2x² – 3x + 5 = 0

On comparing given equation ax² + bx + c = 0

We get,

a = 2, b = -3 and c = 5

We know discriminant (D) = b² – 4ac

D = (-3)² – 4 2 5

D = 9 – 40

D = -31

As, b² – 4ac < 0

Therefore, no real roots exist for the given equation.

Exercise 4 ( Page No. : 91 )

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

Exercise 4 ( Page No. : 91 )

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.

Exercise 4 ( Page No. : 91 )

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.

Exercise 4 ( Page No. : 91 )

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

Chapter 4 : Quadratic Equations

Exercise 1 ( Page No. : 73 )

Exercise 2 ( Page No. : 76 )

Exercise 2 ( Page No. : 76 )

Exercise 2 ( Page No. : 76 )

Exercise 2 ( Page No. : 76 )

Exercise 2 ( Page No. : 76 )

Exercise 2 ( Page No. : 76 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 3 ( Page No. : 88 )

Exercise 4 ( Page No. : 91 )

Exercise 4 ( Page No. : 91 )

Exercise 4 ( Page No. : 91 )

Exercise 4 ( Page No. : 91 )

Exercise 4 ( Page No. : 91 )

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