NCERT Solutions class 10 maths chapter 4 – Pair of Linear Equations in Two Variables

Question 1.

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

Answer

(i) Given
(x + 1)² = 2(x – 3)
(a+b)² = a²+2ab+b²
⇒ x² + 2x + 1 = 2x – 6
⇒ x² + 7 = 0
which is form of ax² + bx + c = 0. where b=0
It is a quadratic equation.


(ii) Given,
x² – 2x = (–2) (3 – x)
⇒ x² – 2x = -6 + 2x
⇒ x² – 4x + 6 = 0
which is form of ax² + bx + c = 0.
It is quadratic equation.


(iii) Given,
(x – 2)(x + 1) = (x – 1)(x + 3)
⇒ x² – x – 2 = x² + 2x – 3 Which is not in the form of ax² + bx + c = 0.
It is not a quadratic equation.


(iv) Given,
(x – 3)(2x +1) = x(x + 5)
⇒ 2x² – 5x – 3 = x² + 5x
⇒ x² – 10x – 3 = 0
which is form of ax² + bx + c = 0.
It is quadratic equation.


(v) Given,
(2x – 1)(x – 3) = (x + 5)(x – 1)
⇒ 2x² – 7x + 3 = x² + 4x – 5
⇒ x² – 11x + 8 = 0
It is form of ax²+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is quadratic equation.


(vi) Given,
x² + 3x + 1 = (x – 2)²
⇒ x² + 3x + 1 = x² + 4 – 4x
⇒ 7x – 3 = 0
It is form of ax² + bx + c = 0.
where a=0 and b,c are any real number
It is not a quadratic equation.


(vii) Given,
(x + 2)3 = 2x(x2 – 1)
⇒ x³ + 8 + x² + 12x = 2x³ – 2x
⇒ x3 + 14x – 6x2 – 8 = 0
It is not in the form of ax² + bx + c = 0.
It is not a quadratic equation.


(viii) Given, x³< – 4x²< – x + 1 = (x – 2)³< x³< – 4x²< – x + 1 = x³< – 8 – 6x²< + 12x
2x²<– 13x + 9 = 0
It is the form of ax²+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is quadratic equation.

Question 2.

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

Answer 2

(i)let the Breadth of the rectangular plot = x m
Thus, the length of the plot = (2x + 1) m
Area of rectangle = length × breadth
(2x + 1)x = 528
2x² + x =528
2x² + x – 528 = 0
which is of the form of ax²+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.


(ii) Let The first integer number = x
Thus, the next consecutive positive integer will be = x + 1
According to the question
Product of two consecutive integers = x X (x +1) = 306
x²+ x = 306
x² + x – 306 = 0
It is of the form of ax²+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.
.

(iii) Let Age of Rohan’s = x years
Rohan’s mother’s age = x + 26
After 3 years,
Age of Rohan’s = x + 3
Age of Rohan’s mother will be = x + 26 + 3 = x + 29br> According to question
(x + 3)(x + 29) = 360
x² + 29x + 3x + 87 = 360
x² + 32x + 87 – 360 = 0
x² + 32x – 273 = 0
which is of the form of ax²+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.

(iv) Let us consider, The speed of train = x km/h
Time taken to travel 480 km = 480/x km/hr
speed of train = (x – 8) km/h
Therefore, time taken to travel 480 km =

480 / (x-8)

km/h
According to the question
Speed × Time = Distance (

480 / (x-8)

) - (

480 / x

)=3
(

480x-480(x-8) / x(x-8)

)=3
480x - 480x +3840 =3x(x-8)
3840= 3x² - 24x
3x² - 24x - 3840=0
x²– 8x – 1280 = 0
x² – 8x – 1280 = 0
which is of the form of ax²+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.
.

Exercise 4.2 Exercise 4.3 Exercise 4.4