NCERT Solutions is the best guide for the students. NCERT Solutions Class 10 Maths Chapter 4- Quadratic Equations Exercise 4.1 has all the solutions to the questions provided in the exercise on page no. 73. The subject experts have provided accurate stepwise solutions to the questions.
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Exercise 4.2 Solutionsâ€“ 6 Questions
Exercise 4.3 Solutionsâ€“ 11 Questions
Exercise 4.4 Solutionsâ€“ 5 Questions
NCERT Solutions for Class 10 Maths Chapter 4- Quadratic Equations Exercise 4.1
Exercise 4.1 Page: 73
1. Check whether the following are quadratic equations:
(i) (x + 1)^{2} = 2(x â€“ 3)
(ii) x^{2} â€“ 2x = (â€“2) (3 â€“ x)
(iii) (x â€“ 2)(x + 1) = (x â€“ 1)(x + 3)
(iv) (x â€“ 3)(2x +1) = x(x + 5)
(v) (2x â€“ 1)(x â€“ 3) = (x + 5)(x â€“ 1)
(vi) x^{2} + 3x + 1 = (x â€“ 2)^{2}
(vii) (x + 2)^{3} = 2x (x^{2} â€“ 1)
(viii) x^{3} â€“ 4x^{2} â€“ x + 1 = (x â€“ 2)^{3 }
Solutions:
(i) Given,
(x + 1)^{2} = 2(x â€“ 3)
By using the formula for (a+b)^{2 }= a^{2}+2ab+b^{2}
â‡’Â x^{2}Â + 2xÂ + 1 = 2xÂ â€“ 6
â‡’Â x^{2}Â + 7 = 0
Since the above equation is in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ quadratic equation.
(ii)Given, x^{2} â€“ 2x = (â€“2) (3 â€“ x)
By using the formula for (a+b)^{2 }= a^{2}+2ab+b^{2}
â‡’Â x^{2Â }â€“^{Â }2x = -6Â + 2xÂ
â‡’Â x^{2Â }â€“ 4xÂ + 6 = 0
Since the above equation is in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ quadratic equation.
(iii)Given, (x â€“ 2)(x + 1) = (x â€“ 1)(x + 3)
By using the formula for (a+b)^{2 }= a^{2}+2ab+b^{2}
â‡’Â x^{2Â }â€“Â xÂ â€“ 2 =Â x^{2Â }+ 2xÂ â€“ 3
â‡’ 3xÂ â€“ 1 = 0
Since the above equation is not in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ not a quadratic equation.
(iv)Given, (x â€“ 3)(2x +1) = x(x + 5)
By using the formula for (a+b)^{2}=a^{2}+2ab+b^{2}
â‡’ 2x^{2Â }â€“ 5xÂ â€“ 3 =Â x^{2Â }+ 5x
â‡’ Â x^{2Â }â€“ 10xÂ â€“ 3 = 0
Since the above equation is in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ quadratic equation.
(v)Given, (2xÂ â€“ 1)(xÂ â€“ 3) = (xÂ + 5)(xÂ â€“ 1)
By using the formula for (a+b)^{2}=a^{2}+2ab+b^{2}
â‡’ 2x^{2Â }â€“ 7xÂ +Â 3 =Â x^{2Â }+ 4xÂ â€“ 5
â‡’Â x^{2Â }â€“ 11xÂ +Â 8 = 0
Since the above equation is in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ quadratic equation.
(vi)Given, x^{2}Â + 3xÂ + 1 = (xÂ â€“ 2)^{2}
By using the formula for (a+b)^{2}=a^{2}+2ab+b^{2}
â‡’Â x^{2}Â + 3xÂ + 1Â =Â x^{2}Â + 4Â â€“ 4x
â‡’ 7xÂ â€“ 3 = 0
Since the above equation is not in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ not a quadratic equation.
(vii)Given, (xÂ + 2)^{3}Â = 2x(x^{2}Â â€“ 1)
By using the formula for (a+b)^{2 }= a^{2}+2ab+b^{2}
â‡’Â x^{3}Â + 8Â +Â x^{2}Â + 12xÂ = 2x^{3}Â â€“ 2x
â‡’Â x^{3}Â + 14xÂ â€“ 6x^{2}Â â€“ 8 = 0
Since the above equation is not in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ not a quadratic equation.
(viii)Given,Â x^{3}Â â€“ 4x^{2}Â â€“Â xÂ + 1 = (xÂ â€“ 2)^{3}
By using the formula for (a+b)^{2 }= a^{2}+2ab+b^{2}
â‡’ Â x^{3}Â â€“ 4x^{2}Â â€“Â xÂ + 1Â =Â x^{3}Â â€“ 8 â€“ 6x^{2Â }Â + 12x
â‡’ 2x^{2}Â â€“ 13xÂ + 9 = 0
Since the above equation is in the form of ax^{2}Â +Â bxÂ +Â cÂ = 0.
Therefore, the given equation isÂ quadratic equation.
2. Represent the following situations in the form of quadratic equations:
- The area of a rectangular plot is 528 m^{2}. 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.
- The product of two consecutive positive integers is 306. We need to find the integers.
- Rohanâ€™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 Rohanâ€™s present age.
- 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
Solutions:
(1)Let us consider,
Breadth of the rectangular plot =Â xÂ m
Thus, the length of the plot = (2xÂ + 1) m.
As we know,
Area of rectangle = lengthÂ Ã—Â breadth = 528 m^{2}
Putting the value of length and breadth of the plot in the formula, we get,
(2xÂ + 1) Ã—Â xÂ = 528
â‡’ 2x^{2}Â +Â xÂ =528
â‡’ 2x^{2}Â +Â xÂ â€“ 528 = 0
Therefore, the length and breadth of plot, satisfies the quadratic equation, 2x^{2}Â +Â xÂ â€“ 528 = 0, which is the required representation of the problem mathematically.
(2)Let us consider,
The first integer number =Â x
Thus, the next consecutive positive integer will be =Â xÂ + 1
Product of two consecutive integers =Â xÂ Ã—Â (xÂ +1) = 306 â‡’Â x^{2Â }+Â xÂ = 306
â‡’Â x^{2Â }+Â xÂ â€“ 306 = 0
Therefore, the two integers x and x+1, satisfies the quadratic equation, x^{2Â }+Â xÂ â€“ 306 = 0, which is the required representation of the problem mathematically.
3) Let us consider,
Age of Rohanâ€™s =Â xÂ years
Therefore, as per the given question,
Rohanâ€™s motherâ€™s age =Â xÂ + 26
After 3 years,
Age of Rohanâ€™s =Â xÂ + 3
Age of Rohanâ€™s mother will be =Â xÂ + 26 + 3 =Â xÂ + 29
The product of their ages after 3 years will be equal to 360, such that
(xÂ + 3)(xÂ + 29) = 360
â‡’Â x^{2}Â + 29xÂ + 3xÂ + 87 = 360
â‡’Â x^{2}Â + 32xÂ + 87 â€“ 360 = 0
â‡’Â x^{2}Â + 32xÂ â€“ 273 = 0
Therefore, the age of Rohan and his mother, satisfies the quadratic equation, x^{2}Â + 32xÂ â€“ 273 = 0, which is the required representation of the problem mathematically.
4) Let us consider,
The speed of train = xÂ km/h
And
Time taken to travel 480 km = 480/x km/hr
As per second condition, the speed of train = (xÂ â€“ 8) km/h
Also given, the train will take 3 hours to cover the same distance.
Therefore, time taken to travel 480 km = 480/(x+3) km/h
As we know,
Speed Ã— Time = Distance
Therefore,
(xÂ â€“ 8)(480/(xÂ + 3) = 480
â‡’ 480Â + 3xÂ â€“ 3840/xÂ â€“ 24 = 480
â‡’ 3xÂ â€“ 3840/xÂ = 24
â‡’ 3x^{2Â }â€“ 8xÂ â€“ 1280 = 0
Therefore, the speed of the train, satisfies the quadratic equation, 3x^{2Â }â€“ 8xÂ â€“ 1280 = 0, which is the required representation of the problem mathematically.
A quadratic equation is an equation in the form:
ax^{2 }+ bx+ c = 0
Where x is an unknown value and a, b, c are real numbers.
The NCERT Solutions Class 10 Maths Chapter 4- Quadratic Equations Exercise 4.1 contains solutions to a total of 2 questions. The first question is provided with 8 options. The students are asked to find out whether the equations are quadratic or not. The second question is divided into 4 parts. Each part is in the form of a situation. The students are asked to represent the situations in the form of quadratic equations.
Key Features of NCERT Solutions Class 10 Maths Chapter 4- Quadratic Equations Exercise 4.1
- The solutions are provided by subject experts.
- The answers are accurate.
- Each question in NCERT Solutions Class 10 Maths Chapter 4- Quadratic Equations Exercise 4.1 is explained properly stepwise.
- The questions are prepared from the examination perspective.