Gujarat Board GSEB Solutions Class 10 Maths Chapter 4 Quadratic Equations Ex 4.1 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 10 Maths Chapter 4 Quadratic Equations Ex 4.1

Question 1.

Check whether the following are quadratic equations:

Solution:

âˆ´It is of the form ax^{2} + bx + c = 0, where

a = 1, b = 0, c = 7.

Hence, the given equation is a quadratic equation.

(ii) RHS = ( – 2)(3 – x) = – 6 + 2x

x^{2} – 2x = – 6 + 2x

x^{2} – 2x + 6 – 2x = 0

x^{2} – 4x + 6 = 0

It is ofthe form ax^{2} + bx + c = 0, where

a = 1, b = – 4, c = 6.

Hence, the given equation is a quadratic equation.

(iii) LHS = (x – 2)(x + 1)

– x^{2} – x – 2

RHS = (x – 1) (x + 3)

= x^{2} + 2x – 3

x^{2} – x – 2 = x^{2} + 2x – 3

– x – 2 – 2x + 3 = 0

-3x + 1 = 0

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

Hence, the given equation is not a quadratic equation

(iv) LHS = (x – 3) (2x + 1)

= 2x^{2} – 6x + x – 3

= 2x^{2} – 5x – 3

RHS = x(x + 5) = x^{2} + 5x

2x^{2} – 5x – 3 = x^{2} + 5x

= 2x^{2} – x^{2} – 5x – 5x – 3 = 0

x^{2} – 10x – 3 = 0

It is of the form ax^{2} + bx + c = 0, where

a= 1, b=10, c = -3

Hence, the given equation is a quadratic equation

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

where a = 1, b = -11, c = 8

Hence, the given equation is a quadratic equation.

(vi) RHS = (x – 2)^{2} = x^{2} – 4x + 4

x^{2} + 3x + 1 =x^{2} – 4x +4

3x + 4x + 1 – 4 = 0

7x – 3 = 0

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

Hence the given equation is not a quadratic equation.

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

Hence the given equation is not a quadratic equation.

It is not of the form ax^{2} +bx + c = 0 where a = 2, b = -13, c = 9.

Hence the given equation is not a quadratic equation.

Question 2.

Represent the following problems situations in the form of quadratic equations:

(i) The area of a rectangle plot is 528 m^{2}. The length of the plot (in metres) is one more than twice its breadth. Find the length and breadth of the plot.

(ii) The product of two consecutive positive integers is 306. Find the integers.

(iii) Rohanâ€™s mother is 26 years older than him. The product of their ages 3 years from now will be 360. We would like to find Rohanâ€™s present age.

(iv) A train travels a distance of 480 km at uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.

Solution:

(i) Let breadth of the rectangle = x m. Then,

length of the rectangle = (2x + 1) m

âˆ´ Area of the rectangle

= (2x + 1)xm^{2}

According to the given condition,

(2x + 1)x = 528

2x^{2} + x – 528 = 0

which is a quadratic equation in x.

Solving this equation by factorization

method, we get

But breadth cannot be – ve.

âˆ´ x = 16.

Hence, breadth of the rectangle = 16 m and

length of the rectangle = 2 x 16 + 1 = 33 m.

(ii) Let two consecutive positive integers are x and (x+ 1). Then

x (x + 1) = 306

x^{2} + x – 306 = 0

which is a quadratic equation in x.

Solving this equation by factorization

method, we get

But integers are +ve.

âˆ´ x = 17.

Hence, two consecutive positive integers are 17 and 18.

(iii) Let Rohanâ€™s present age = x years

Then, Rohanâ€™s mother age = (x + 26) years

Rohanâ€™s age after 3 years (x + 3) years

and Rohanâ€™s mother age after 3 years

= (x + 29) years

According to the given condition,

(x + 3) (x + 29) = 360

x^{2} + 32x + 87 = 360

x^{2} + 32x – 273 = 0

Which is a quadratic equation in r solving it by factorization method.

x^{2} + 39x – 7x – 273 = 0

x(x + 39) – 7(x + 39) = 0

(x + 39) (x – 7) = 0

= x + 39.

(or)

x = 7

But age cannot be -ve,

– x = 7

Hence, Rohanâ€™s present age a 7 years.

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

Distance = 480 km

Time taken by the tram to cover the distance of 48Okm = \(\frac { 480 }{ x } \) h.

Now, according to the given condition,

which is a quadratic equation in x.

Solving it by factorization method,

x^{2} – 40x + 32x – 1280 = 0

= x (x – 40) + 32(x – 40) = 0

(x – 40) (x + 32) = 0

x – 40 = 0

(or)

x+32 = 0

x = 40

or

x = – 32.

But speed of the train cannot be – ve.

x = 40.

Hence, the speed of the train =40 km/h.