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# NCERT solutions Mathematics Class 10 chapter 4 Quadratic Equations

## Chapter 4 - Quadratic Equations

#### Pages 73 - 74

Check whether the following is the quadratic equations : (x + 1)2 = 2(x - 3)

Q 1.1 | Page 73

Check whether the following is the quadratic equations: x2 - 2x = (-2)(3 - x)

Q 1.2 | Page 73

Check whether the following is the quadratic equations: (x - 2)(x + 1) = (x - 1)(x + 3)

Q 1.3 | Page 73

Check whether the following is the quadratic equations (x - 3)(2x + 1) = x(x + 5)

Q 1.4 | Page 73

Check whether the following is the quadratic equations: (2x - 1)(x - 3) = (x + 5)(x - 1)

Q 1.5 | Page 73

Check whether the following is the quadratic equations : x2 + 3x + 1 = (x - 2)2

Q 1.6 | Page 73

Check whether the following is the quadratic equations: (x + 2)3 = 2x(x2 - 1)

Q 1.7 | Page 73

Check whether the following is the quadratic equations: x3 - 4x2 - x + 1 = (x - 2)3

Q 1.8 | Page 73

Represent the following situations in the form of quadratic equations

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

Q 2.1 | Page 73

Represent the following situations in the form of quadratic equations

The product of two consecutive positive integers is 306. We need to find the integers.

Q 2.2 | Page 74

Represent the following situations in the form of quadratic equations

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.

Q 2.3 | Page 74

Represent the following situations in the form of quadratic equations.

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.

Q 2.4 | Page 74

#### Page 76

Find the roots of the following quadratic equations by factorisation x2 – 3x – 10 = 0

Q 1.1 | Page 76

Find the roots of the following quadratic equations by factorisation 2x2 + x – 6 = 0

Q 1.2 | Page 76

Find the roots of the following quadratic equations by factorisation

sqrt2 x^2 + 7x + 5sqrt2 = 0

Q 1.3 | Page 76

Find the roots of the following quadratic equations by factorisation 2x^2 – x + 1/8 = 0

Q 1.4 | Page 76

Find the roots of the following quadratic equations by factorisation: 100x2 – 20x + 1 = 0

Q 1.5 | Page 76

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

Q 3 | Page 76

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

Q 4 | Page 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.

Q 5 | Page 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.

Q 6 | Page 76

#### Pages 87 - 88

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x2 – 7x + 3  = 0

Q 1.1 | Page 87

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x2 + x – 4 =  0

Q 1.2 | Page 87

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x2 + x + 4 = 0

Q 1.3 | Page 87

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 4x^2 + 4sqrt3x + 3 = 0

Q 1.3 | Page 87

Find the roots of the quadratic equations 2x2 – 7x + 3 = 0 by applying the quadratic formula.

Q 2.1 | Page 87

Find the roots of the quadratic equations 2x2 + x – 4 = 0 by applying the quadratic formula.

Q 2.2 | Page 87

Find the roots of the quadratic equations 4x^2+4sqrt3x + 3 = 0 by applying the quadratic formula.

Q 2.3 | Page 87

Find the roots of the quadratic equations 2x2 + x + 4 = 0 by applying the quadratic formula.

Q 2.4 | Page 87

Find the roots of the following equations:

x-1/x = 3, x ≠ 0

Q 3.1 | Page 88

Find the roots of the following equations

1/(x+4) - 1/(x-7) = 11/30, x = -4, 7

Q 3.2 | Page 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.

Q 4 | Page 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

Q 5 | Page 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.

Q 6 | Page 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

Q 7 | Page 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.

Q 8 | Page 88

Two water taps together can fill a tank in 9hours. 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.

Q 9 | Page 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.

Q 10 | Page 88

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.

Q 11 | Page 88

#### Page 91

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

2x2 – 3x + 5 = 0

Q 1.1 | Page 91

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them; 3x^2 - 4sqrt3x + 4 = 0

Q 1.2 | Page 91

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

2x2 - 6x + 3 = 0

Q 1.3 | Page 91

Find the values of k for each of the following quadratic equations, so that they have two equal roots. 2x2 + kx + 3 = 0

Q 2.1 | Page 91

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

kx (x - 2) + 6 = 0

Q 2.2 | Page 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

Q 3 | Page 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.

Q 4 | Page 91

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

Q 5 | Page 91

#### Extra questions

Solve for x : 12abx2 – (9a2 – 8b2 ) x – 6ab = 0

Without solving, examine the nature of roots of the equation 2x2 + 2x + 3 = 0

Without solving, examine the nature of roots of the equation 2x2 – 7x + 3 = 0

Without solving, examine the nature of roots of the equation x2 – 5x – 2 = 0

For what value of m, are the roots of the equation (3m + 1)x2 + (11 + m) x + 9 = 0 equal?

Without solving, examine the nature of roots of the equation 4x2 – 4x + 1 = 0

Find the roots of the quadratic equation 4x2 + 4√3x + 3 = 0

Find the roots of the quadratic equation 2x2 + x + 4 = 0

Find the roots of the quadratic Equation 2x2 – 7x + 3 = 0 (if they exist) by the method of completing the square.

(i) x2  + 5x = 0         (ii) x2  = 3x          (iii) x2 = 4

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.

(i) 7x2 = 8 – 10x

(ii) 3(x2 – 4) = 5x

(iii) x(x + 1) + (x + 2) (x + 3) = 42

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

Solve  (i) x2  + 3x – 18 = 0

(ii) (x – 4) (5x + 2) = 0

(iii) 2x2 + ax – a2 = 0; where ‘a’ is a real number

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