Practice Exercise 4.3

Nature of Roots & Discriminant Analysis

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Q1(i) Q1(ii) Q1(iii) Q2(i) Value of k Q2(ii) Value of k Q3 Rect. Design Q4 Ages Q5 Park

Q1. Find Nature of Roots

Practice Question 1(i)

Find nature of roots: 3x² - 5x + 4 = 0

View Step-by-Step Solution

Step 1: Identify Coefficients

a = 3, b = -5, c = 4

Step 2: Find Discriminant (D)

D = b² - 4ac
D = (-5)² - 4(3)(4) = 25 - 48 = -23

Step 3: Conclusion

Since D < 0, roots are not real.

No Real Roots Exist

Practice Question 1(ii)

Find nature of roots: 4x² - 12x + 9 = 0

View Step-by-Step Solution

Step 1: Identify Coefficients

a = 4, b = -12, c = 9

Step 2: Find Discriminant (D)

D = (-12)² - 4(4)(9) = 144 - 144 = 0
D = 0, so two equal real roots exist.

Step 3: Find Roots

x = -b / 2a = 12 / 8 = 3/2

Roots: 3/2, 3/2 (Real and Equal)

Practice Question 1(iii)

Find nature of roots: 2x² - 4x + 1 = 0

View Step-by-Step Solution

Step 1: Identify Coefficients

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

Step 2: Find Discriminant (D)

D = (-4)² - 4(2)(1) = 16 - 8 = 8
D > 0, so two distinct real roots exist.

Step 3: Find Roots

x = (-b ± √D) / 2a = (4 ± √8) / 4
x = (4 ± 2√2) / 4 = (2 ± √2) / 2

Roots: (2+√2)/2 and (2-√2)/2

Q2. Find Value of k

Practice Question 2(i)

Find k for equal roots: kx² - 12x + 9 = 0

View Step-by-Step Solution

Step 1: Condition for Equal Roots

D = b² - 4ac = 0

Step 2: Substitute Values

(-12)² - 4(k)(9) = 0
144 - 36k = 0
36k = 144

Step 3: Solve for k

k = 144 / 36 = 4

k = 4

Practice Question 2(ii)

Find k for equal roots: x(x - k) + 4 = 0

View Step-by-Step Solution

Step 1: Standard Form

x² - kx + 4 = 0
a = 1, b = -k, c = 4

Step 2: Condition D = 0

(-k)² - 4(1)(4) = 0
k² - 16 = 0

Step 3: Solve for k

k² = 16
k = ±4

k = 4 or k = -4

Word Problems (Check Possibility)

Practice Question 3

Is it possible to design a rectangular garden whose length is three times its breadth, and the area is 300 m²? If so, find its dimensions.

View Step-by-Step Solution

Step 1: Form Equation

Let Breadth = x. Length = 3x.
Area = x(3x) = 300
3x² = 300 → x² = 100

Step 2: Check Possibility

x² - 100 = 0. Discriminant D = 0² - 4(1)(-100) = 400 > 0.
Yes, possible.

Step 3: Solve

x = ±10. Since breadth > 0, x = 10.

Breadth = 10m, Length = 30m

Practice Question 4

Is this possible? The sum of ages of two friends is 25 years. Five years ago, the product of their ages was 50.

View Step-by-Step Solution

Step 1: Form Equation

Friend 1 = x. Friend 2 = 25 - x.
5 years ago: (x - 5) and (20 - x).
(x - 5)(20 - x) = 50
20x - x² - 100 + 5x = 50
-x² + 25x - 150 = 0 → x² - 25x + 150 = 0

Step 2: Check Discriminant

D = (-25)² - 4(1)(150) = 625 - 600 = 25
D > 0, so real roots exist.

Step 3: Solve

x = (25 ± 5)/2. x = 15 or x = 10.

Yes, possible. Ages are 15 years and 10 years.

Practice Question 5

Is it possible to design a rectangular park of perimeter 60 m and area 250 m²? If so, find its length and breadth.

View Step-by-Step Solution

Step 1: Form Equation

Perimeter = 2(L + B) = 60 → L + B = 30 → B = 30 - L.
Area = L × B = 250
L(30 - L) = 250
30L - L² = 250 → L² - 30L + 250 = 0

Step 2: Check Discriminant

D = (-30)² - 4(1)(250) = 900 - 1000 = -100.
Since D < 0, no real roots exist.

No, this situation is not possible.

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