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Mathematics (Maths) Class 10

Class 10 : Notes | EduRev

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Ques 1: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x2 − 3x + 5 = 0
(ii) 3x2 − 4√3 x+ 4 = 0
(iii) 2x2 − 6x + 3 = 0

Sol: (i) 2x2 - 3x + 5 = 0

Comparing the given quadratic equation with ax2 + bx + c = 0, we have:

a = 2
b = − 3
c = 5

∴ The discriminant = b2 − 4ac

= (− 3)− 4 (2) (5)
= 9 − 40
= − 31 < 0

Since b2 − 4ac is negative.

∴ The given quadratic equation has no real roots.

(ii) 3x2 - 4√3x + 4 = 0

Comparing the given quadratic equation with ax2 + bx + c = 0, we get

a = 3
b = − 4√3
c = 4
 Notes | EduRev

= (16 × 3) − 48
= 48 − 48 = 0

Thus, the given quadratic equation has two real roots which are equal. Here, the roots are:

 Notes | EduRev
 Notes | EduRev
 Notes | EduRev        Notes | EduRev
 Notes | EduRev
Thus,  Notes | EduRev

(iii) 2x2−6x + 3 = 0

Comparing it with the general quadratic equation, we have:

a = 2
b = − 6
c = 3

∴ b2 − 4ac =(− 6)2 − 4 (2) (3)
= 36 − 24
= 12 > 0

∴ The given quadratic equation has two real and distinct roots, which are given by
 Notes | EduRev
 Notes | EduRev
 Notes | EduRev
Thus, the roots are:
 Notes | EduRev

Ques 2: Find the values of k for each of the following quadratic equations, so that they have two equal roots: 
(i) 2x2 + kx + 3 = 0  
(ii) kx (x − 2) + 6 = 0
Sol: (i) 2x2 + kx + 3 = 0

Comparing the given quadratic equation with ax2 + bx + c = 0, we get

a = 2
b = k
c = 3

∴ b2 − 4ac =(− k)2 − 4 (2) (3)
= k2 − 24

∵ For a quadratic equation to have equal roots,
b− 4ac = 0

∴ k2 − 24 = 0 ⇒ k =  Notes | EduRev
 Notes | EduRev
Thus, the required values of k are

 Notes | EduRev

(ii) kx (x−2) + 6 = 0

Comparing kx (x − 2) + 6 = 0 i.e., kx2 − 2kx + 6 = 0 with ax+ bx + c = 0, we get

a = k
b = − 2k
c = 6

∴ b2 − 4ac =(− 2k)2 − 4 (k) (6)
= 4k2 − 24k

Since, the roots are real and equal,
∴ b2 − 4ac =0
⇒ 4k2 − 24k =0
⇒ 4k (k − 6) = 0
⇒ 4k = 0 or k − 6 = 0
⇒ k = 0 or k = 6

But k cannot be 0, otherwise, the given equation is no more quadratic. Thus, the required value of k = 6.

Ques 3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Sol: Let the breadth be x metres.
∴ Length = 2x metres
Now, Area = Length × Breadth
= 2x × x metre2
= 2x2 sq. metre.

According to the given condition,

2x2 = 800
 Notes | EduRev
 Notes | EduRev
Therefore, x = 20 and x = − 20

But x = − 20 is possible |(∵ breadth cannot be negative). 

∴ x = 20
⇒ 2 x = 2 × 20 = 40

Thus, length = 40 m and breadth = 20 m

Ques 4: Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Sol: Let the age of one friend = x years

∴ The age of the other friend = (20 − x) years [∵ Sum of their ages is 20 years]

Four years ago Age 

of one friend = (x − 4) years
Age of the other friend = (20 − x − 4) years
= (16 − x) years
According to the condition,
(x − 4) × (16 − x) = 48
⇒ 16x − 64 − x2 − 4x = 48
⇒− x2 − 20x − 64 − 48 = 0
⇒− x2 − 20x − 112 = 0
⇒ x2 + 20x + 112 = 0     ....(1)

Here, a = 1, b = 20 and c = 112

∴ b− 4ac = (20)2 − 4 (1) (112)
= 400 − 448
= − 48 < 0

Since b2 − 4ac is less than 0.
∴ The quadratic equation (1) has no real roots.

Thus, the given equation is not possible.

Ques 5: Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.

Sol: Let the breadth of the rectangle be x metres.

Since, the perimeter of the rectangle = 80 m.
∴ 2 [Length + Breadth] = 80
2 [Length + x] = 80
 Notes | EduRev
⇒ Length = (40 − x) metres
∴ Area of the rectangle = Length × breadth
= (40 − x) × x sq. m
= 40x − x2

Now, according to the given condition,

Area of the rectangle = 400 m2

∴ 40x − x2 = 400
⇒ − x2 + 40x − 400 = 0
⇒ x2 − 40x + 400 = 0     ...(1)

Comparing (1) with ax2 + bx + c = 0, we get

a = 1
b = − 40
c = 400

∴ b2 − 4ac =(− 40)2 − 4 (1) (400)
= 1600 − 1600 = 0

Thus, the equation (1) has two equal and real roots.
 Notes | EduRev
 Notes | EduRev
∴ Breadth, x = 20 m
∴ Length = (40 − x) = (40 − 20) m = 20 m.

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