Ans. (i)  

Comparing this equation with general equation ,

We get a = 2, b = −3 and c = 5

Discriminant =  (2) (5)

= 9 – 40 = −31

Discriminant is less than 0 which means equation has no real roots.

(ii) 

Comparing this equation with general equation ,

We get a = 3, b = and c = 4

Discriminant =  = − 4 (3) (4)

= 48 – 48 = 0

Discriminant is equal to zero which means equations has equal real roots.

Applying quadratic formula   to find roots,

Because, equation has two equal roots, it means 

(iii) 

Comparing equation with general equation ,

We get a = 2, b = −6, and c = 3

Discriminant =  (2) (3)

= 36 – 24 = 12

Value of discriminant is greater than zero.

Therefore, equation has distinct and real roots.

Applying quadratic formula  to find roots,

⇒ 

⇒ 

Ans. (i)  

We know that quadratic equation has two equal roots only when the value of discriminant is equal to zero.

Comparing equation with general quadratic equation , we get a = 2, b = k and c = 3

Discriminant =  (2) (3) = 

Putting discriminant equal to zero

⇒ 

⇒ 

(ii) kx (x − 2) + 6 = 0

⇒ 

Comparing quadratic equation  with general form , we get a = k, b = −2k and c = 6

Discriminant =  (k) (6)

= 

We know that two roots of quadratic equation are equal only if discriminant is equal to zero.

Putting discriminant equal to zero

⇒ 4k (k − 6) = 0⇒ k = 0, 6

The basic definition of quadratic equation says that quadratic equation is the equation of the form , where a ≠ 0.

Therefore, in equation , we cannot have k = 0.

Therefore, we discard k = 0.

Hence the answer is k = 6.

Ans. Let breadth of rectangular mango grove = x metres 

Let length of rectangular mango grove = 2x metres

Area of rectangle = = 

According to given condition:

=> 

⇒ = 0

= 0

Comparing equation with general form of quadratic equation , we get a = 1, b = 0 and c = −400

Discriminant  (1) (−400) = 1600

Discriminant is greater than 0 means that equation has two disctinct real roots.

Therefore, it is possible to design a rectangular grove.

Applying quadratic formula,  to solve equation,

⇒ x = 20, −20

We discard negative value of x because breadth of rectangle cannot be in negative.

Therefore, x = breadth of rectangle = 20 metres

Length of rectangle = 2x = = 40 metres

Ans. Let age of first friend = x years 

then age of second friend = (20 − x) years

Four years ago, age of first friend = (x − 4) years

Four years ago, age of second friend = (20 − x) − 4 = (16 − x) years

According to given condition,

=> (x − 4) (16 − x) = 48

⇒ = 48

⇒ 

⇒ 

Comparing equation,  with general quadratic equation , we get a = 1, b = −20 and c = 112

Discriminant = (1) (112) = 400 – 448 = −48 < 0

Discriminant is less than zero which means we have no real roots for this equation.

Therefore, the give situation is not possible.

Ans. Let length of park = x metres 

We are given area of rectangular park = 

Therefore, breadth of park =  metres{Area of rectangle = length × breadth}

Perimeter of rectangular park = 2 (length + breath) = metres

We are given perimeter of rectangle = 80 metres

According to condition:

=> 

⇒ 

⇒ 

⇒ 

⇒ 

Comparing equation,  with general quadratic equation , we get a = 1, b = −40 and c = 400

Discriminant =  (1) (400) = 1600 – 1600 = 0

Discriminant is equal to 0.

Therefore, two roots of equation are real and equal which means that it is possible to design a rectangular park of perimeter 80 metres and area .

Using quadratic formula  to solve equation,

Here, both the roots are equal to 20.

Therefore, length of rectangular park = 20 metres

Breadth of rectangular park =