##### NCERT-Class X-Mathematics

01-Real Numbers

02-Polynomials

03-Pair of Linear Equations in Two Variables

04-Quadratic Equations

05-Arithmetic Progression

06-Triangles

07-Coordinate Geometry

08-Introduction to Trigonometry

09-Some Applications of Trigonometry

10-Circles

11-Constructions

12-Area Related to Circles

13-Surface Areas and Volumes

14-Statistics

15-Probability

### Chapter 04-Quadratic Equations-Exercise Solutions

NCERT Book Page Number -73 Exercise 4.1

Q.1. Check whether the following are quadratic equations:

(i) $(x+1)^2$ =2(x-3)

(ii) $x^2-2x $ =(-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) $ x^{2} + 3x +1=(x-2)^{2}$

(vii) $ (x+2)^{2} =2x(x^{2}-1 )$

(viii) $ x^{3} – 4x^{2} –x +1 =(x-2)^{3}$

Answer:

(i) $(x+1)^2$ =2(x-3)

$\Rightarrow x^{2} +2x +1 = 2x-6 $

$ \Rightarrow x^{2} + 7= 0$

It is of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is a quadratic equation.

(ii) $x^2-2x $ =(-2)(3-x)

$\Rightarrow x^{2} -2x = -6+ 2x$

$\Rightarrow x^{2} -4x + 6 =0 $

It is of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is a quadratic equation.

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

$\Rightarrow x^{2} –x-2 = x^{2} +2x-3$

$\Rightarrow 3x-1 =0$

It is not of the form $ax^{2} + bx + c = 0$..

Hence, the given equation is not a quadratic equation.

(iv) (x-3)(2x+1)=x(x+5)

$\Rightarrow 2x^{2} –5x-3 = x^{2} +5x $

$\Rightarrow x^{2} –10x-3 = 0 $

It is of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is a quadratic equation.

(v) (2x-1)(x-3)=(x+5)(x-1)

$\Rightarrow 2x^{2} –7x-3 = x^{2} -11x +8 $

$\Rightarrow x^{2} –11x +8 = 0 $

It is of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is a quadratic equation.

(vi) $ x^{2} + 3x +1=(x-2)^{2}$

$\Rightarrow x^{2} +3x +1 = x^{2} +4 – 4x $

7x-3 =0

It is not of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is not a quadratic equation.

(vii) $ (x+2)^{2} =2x(x^{2}-1 )$

$\Rightarrow x^{3} +8x +6x^{2} +12x= 2x^{3}- 2x $

$\Rightarrow x^{3} -14x +6x^{2} -8 =0 $

It is not of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is not a quadratic equation.

(viii) $ x^{3} – 4x^{2} –x +1 =(x-2)^{3}$

$\Rightarrow x^{3} -4 x^{2}-x +1 = x^{3} -8 – 6x^{2} +12x $

$\Rightarrow 2x^{2} -13x + 9 =0 $

It is of the form $ax^{2} + bx + c = 0$.

Hence, the given equation is a quadratic equation.

Q.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 3 hours more to cover the same distance. We need to find the speed of the train.

Answer:

(i) Let the breadth of the plot be x m.

Hence, the length of the plot is (2x + 1) m.

Area of a rectangle = Length × Breadth

∴ 528 = x (2x + 1)

$\Rightarrow 2x^{2}+ x – 528 =0 $

(ii) Let the consecutive integers be x and x + 1.

It is given that their product is 306.

$\therefore x(x +1) = 306$

$\Rightarrow x^{2}+ x – 306 =0 $

(iii) Let Rohan’s age be x.

Hence, his mother’s age = x + 26

3 years hence,

Rohan’s age = x + 3

Mother’s age = x + 26 + 3 = x + 29

It is given that the product of their ages after 3 years is 360.

$\therefore (x+ 3)(x+29)=360$

$\Rightarrow x^{3} + 32x -273 =0 $

(iv) Let the speed of train be x km/h.

Time taken to travel 480 km = $\frac{480}{x}$ hrs

In second condition, let the speed of train = (x-8) km/h

It is also given that the train will take 3 hours to cover the same distance.

Therefore, time taken to travel 480 km = $\frac{480}{x} + 3$ hrs

Speed × Time = Distance

(x-8) $(\frac{480}{x} + 3)$ = 480

$\Rightarrow 480+3x-3840x-24=480$

$\Rightarrow 3x-3840x=24$

$\Rightarrow 3×2-24x-3840=0 $