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**Q.1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:****(i) 2 x^{2} – 7x +3 = 0**

⇒ 2

Dividing by 2 on both sides, we get

⇒ x

⇒ x

On adding (7/4)

⇒ (x)

⇒ (x-7/4)

⇒(x-7/4)

⇒(x-7/4)

⇒

⇒

⇒ x = 12/4 or x = 2/4

⇒ x = 3 or x = 1/2

**(ii) 2x ^{2} + x – 4 = 0**

⇒ 2x

Dividing both sides of the equation by 2, we get

⇒ x

Now on adding (1/4)

⇒ (x)

⇒ (x + 1/4)

⇒ x + 1/4 = ± √33/4

⇒ x = ± √33/4 – 1/4

⇒ x = ± √33-1/4

Therefore, either x = √33-1/4 or x = -√33-1/4

**(iii) 4x ^{2} + 4√3x + 3 = 0**

Converting the equation into a

⇒ (2x)

⇒ (2x + √3)

⇒ (2x + √3) = 0 and (2x + √3) = 0

Therefore, either x = -√3/2 or x = -√3/2.

**(iv) 2x ^{2} + x + 4 = 0**

⇒ 2x

Dividing both sides of the equation by 2, we get

⇒ x

⇒ x

By adding (1/4)

⇒ (x)

⇒ (x + 1/4)

⇒ (x + 1/4)

As we know, the square of numbers cannot be negative.

Therefore, there is no real root for the given equation, 2x

⇒ x = (7±√(49 – 24))/4

⇒ x = (7±√25)/4

⇒ x = (7±5)/4

⇒ x = (7+5)/4 or x = (7-5)/4

⇒ x = 12/4 or 2/4

∴ x = 3 or 1/2

a = 2, b = 1 and c = -4

By using quadratic formula, we get,

⇒x = -1±√1+32/4

⇒x = -1±√33/4

∴ x = -1+√33/4 or x = -1-√33/4

**(iii) 4x ^{2} + 4√3x + 3 = 0**

On comparing the given equation with ax

a = 4, b = 4√3 and c = 3

By using quadratic formula, we get,

⇒ x = -4√3±√48-48/8

⇒ x = -4√3±0/8

∴ x = -√3/2 or x = -√3/2

On comparing the given equation with ax

a = 2, b = 1 and c = 4

By using quadratic formula, we get

⇒ x = -1±√1-32/4

⇒ x = -1±√-31/4

As we know, the square of a number can never be negative. Therefore, there is no real solution for the given equation.

⇒ x-7-x-4/(x+4)(x-7) = 11/30

⇒ -11/(x+4)(x-7) = 11/30

⇒ (x+4)(x-7) = -30

⇒ x

⇒ x

We can solve this equation by factorization method now,

⇒ x

⇒ x(x – 2) – 1(x – 2) = 0

⇒ x = 1 or 2

Let us say, present age of Rahman is

Three years ago, Rehman’s age was (

Five years after, his age will be (

Given, the sum of the reciprocals of Rehman’s ages 3 years ago and after 5 years is equal to 1/3.

∴ 1/

(2

⇒ 3(2

⇒ 6

⇒

⇒

⇒

⇒ (

⇒

As we know, age cannot be negative.

Therefore, Rahman’s present age is 7 years.

Let us say, the marks of Shefali in Maths be x.

Then, the marks in English will be 30 – x.

As per the given question,

(x + 2)(30 – x – 3) = 210

(x + 2)(27 – x) = 210

⇒ -x

⇒ x

⇒ x

⇒ x(x – 12) -13(x – 12) = 0

⇒ (x – 12)(x – 13) = 0

⇒ x = 12, 13

Therefore, if the marks in Maths are 12, then marks in English will be 30 – 12 = 18 and the marks in Maths are 13, then marks in English will be 30 – 13 = 17

Let us say, the shorter side of the rectangle be x m.

Then, larger side of the rectangle = (x + 30) m

As given, the length of the diagonal is = x + 30 m

Therefore,

⇒

⇒

⇒

⇒

⇒

⇒ (

⇒

However, side of the field cannot be negative. Therefore, the length of the shorter side will be 90 m.

and the length of the larger side will be (90 + 30) m = 120 m.

Let us say, the larger and smaller number be

As per the question given,

⇒

⇒

⇒

⇒

⇒ (

⇒

However, the larger number cannot considered as negative number, as 8 times of the larger number will be negative and hence, the square of the smaller number will be negative which is not possible.

Therefore, the larger number will be 18 only.

∴

⇒

∴ Smaller number = ±12

Therefore, the numbers are 18 and 12 or 18 and -12.

Let us say, the speed of the train be

Time taken to cover 360 km = 360/

As per the question given,

⇒ (

⇒ 360 –

⇒

⇒

⇒ (

⇒

As we know, the value of speed cannot be negative.

Therefore, the speed of train is 40 km/h.

Let the time taken by the smaller pipe to fill the tank = x hr.

Time taken by the larger pipe = (

Part of tank filled by smaller pipe in 1 hour = 1/

Part of tank filled by larger pipe in 1 hour = 1/(

As given, the tank can be filled in = 75/8 hours by both the pipes together.

Therefore,

1/

⇒ 2

⇒ 75(2

⇒ 150

⇒ 8

⇒ 8

⇒ 8

⇒ (

⇒

Time taken by the smaller pipe cannot be 30/8 = 3.75 hours, as the time taken by the larger pipe will become negative, which is logically not possible.

Therefore, time taken individually by the smaller pipe and the larger pipe will be 25 and 25 – 10 =15 hours respectively.

Let us say, the average speed of passenger train =

Average speed of express train = (

Given, time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance. Therefore,

(132/x) – (132/(x+11)) = 1

132(x+11-x)/(x(x+11)) = 1

132 × 11 /(x(x+11)) = 1

⇒ 132 × 11 =

⇒

⇒

⇒

⇒ (

⇒

As we know, Speed cannot be negative.

Therefore, the speed of the passenger train will be 33 km/h and thus, the speed of the express train will be 33 + 11 = 44 km/h.

Let the sides of the two squares be

Therefore, their perimeter will be 4

And area of the squares will be

Given,

4

Also,

⇒ (6 +

⇒ 36 +

⇒ 2

⇒

⇒

⇒

⇒ (

⇒

As we know, the side of a square cannot be negative.

Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m.

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