NCERT Solutions For Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3
1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x² – 3x + 5 = 0
Step 1: Calculate the discriminant (D) = b² – 4ac.
D = (-3)² – 4(2)(5)
D = 9 – 40
D = -31
Step 2: Check the nature of the roots.
Since D < 0, the roots are imaginary and not real.
(ii) 3x² – 4√3 x + 4 = 0
Step 1: Calculate the discriminant (D) = b² – 4ac.
D = (-4√3)² – 4(3)(4)
D = 48 – 48
D = 0
Step 2: Find the roots as D = 0.
x = -b/2a
x = 4√3/(2*3)
x = 2√3/3
The roots are real and equal, x = 2√3/3.
(iii) 2x² – 6x + 3 = 0
Step 1: Calculate the discriminant (D) = b² – 4ac.
D = (-6)² – 4(2)(3)
D = 36 – 24
D = 12
Step 2: Find the roots as D > 0.
x = [-b ± √D] / 2a
x = [6 ± √12] / 4
x = [6 ± 2√3] / 4
x = 3/2 ± √3/2
The roots are real and distinct, x = 3/2 ± √3/2.
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
Step 1: For equal roots, discriminant (D) = 0.
D = b² – 4ac
0 = k² – 4(2)(3)
0 = k² – 24
Step 2: Solve for k.
k² = 24
k = ±√24
k = ±2√6
(ii) kx (x – 2) + 6 = 0
Step 1: Convert to standard form ax² + bx + c = 0.
kx² – 2kx + 6 = 0
Step 2: For equal roots, discriminant (D) = 0.
D = b² – 4ac
0 = (-2k)² – 4(k)(6)
0 = 4k² – 24k
Step 3: Solve for k.
4k² – 24k = 0
k(4k – 24) = 0
Step 4: Find the values of k.
k = 0 or 4k – 24 = 0
k = 0 or k = 6
The values of k are 0 and 6.
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.
Step 1: Let the breadth of the mango grove be x meters. Therefore, the length is 2x meters.
Step 2: Use the formula for area (Area = length × breadth).
800 = 2x × x
800 = 2x²
Step 3: Formulate the quadratic equation.
2x² = 800
x² = 400
Step 4: Solve for x.
x = ±√400
x = ±20
Step 5: Find the dimensions.
Since breadth cannot be negative, x = 20 m (breadth)
Length = 2x = 40 m
The length and breadth of the mango grove are 40 m and 20 m, respectively.
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.
Step 1: Let the age of one friend be x years. Therefore, the other friend’s age is (20 – x) years.
Step 2: Use the given condition for their ages four years ago.
(x – 4)(20 – x – 4) = 48
(x – 4)(16 – x) = 48
Step 3: Expand and rearrange the equation to standard quadratic form.
x(16 – x) – 4(16 – x) = 48
16x – x² – 64 + 4x = 48
-x² + 20x – 112 = 0
Step 4: Solve for x.
x² – 20x + 112 = 0
(x – 8)(x – 14) = 0
Step 5: Find the ages.
x – 8 = 0 or x – 14 = 0
x = 8 or x = 14
The friends’ ages are either 8 and 12 years or 14 and 6 years.
5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.
Step 1: Let the length be x meters and breadth be y meters.
Step 2: Use the formula for perimeter (Perimeter = 2(length + breadth)).
80 = 2(x + y)
Step 3: Use the formula for area (Area = length × breadth).
400 = x × y
Step 4: Express y in terms of x using the perimeter equation.
80 = 2(x + y)
40 = x + y
y = 40 – x
Step 5: Substitute y in the area equation.
400 = x(40 – x)
400 = 40x – x²
Step 6: Rearrange the equation to standard quadratic form.
x² – 40x + 400 = 0
Step 7: Solve for x.
(x – 20)(x – 20) = 0
x = 20
Step 8: Find the dimensions.
Length = x = 20 m
Breadth = y = 40 – x = 20 m
The length and breadth of the park are 20 m each.