Class 10 Mathematics Chapter 4: Quadratic Equations - NCERT Solutions

Welcome to the complete NCERT solutions for Class 10 Mathematics Chapter 4: Quadratic Equations. In this section, we provide detailed, easy-to-understand solutions for all the questions from this chapter. Whether you're preparing for exams or seeking a deeper understanding of the subject, these Quadratic Equations question answers will offer you valuable insights and explanations. Each solution is crafted to ensure conceptual clarity and step-by-step problem-solving methods, enabling students to grasp the core themes and excel in their academics.

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Exercise 1 ( Page No. : 73 )

Q1	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
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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.
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Exercise 2 ( Page No. : 76 )

Q1	Find the roots of the following quadratic equations by factorisation:
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Q2	Solve the problems given in Example 1.
Ans:	(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

Q3	Find two numbers whose sum is 27 and product is 182.
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Q4	Find two consecutive positive integers, sum of whose squares is 365.
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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.
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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.
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Exercise 3 ( Page No. : 88 )

Q1	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
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Q2	Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
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Q3	Find the roots of the following equations:
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Q4	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.
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Q5	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.
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Q6	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.
Ans:	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

Q7	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.
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Q8	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.
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Q9	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.
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Q10	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.
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Q11	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.
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Exercise 4 ( Page No. : 91 )

Q1	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
Ans:	(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.

Q2	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
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Q3	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.
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Q4	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.
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Q5	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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