Kaliyuga Ekalavya

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 5 Question: Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2? If so, find its length and breadth. Given: A rectangular park with Perimeter (P) = 80 m Area (A) = 400 m 2 To Find: 1. Is it possible to design such a rectangular park? 2. If yes, find its length and breadth. Formula: Perimeter of a rectangle is given by: P = 2(Length + Breadth) Area of a rectangle is given by: A = Length × Breadth For a quadratic equation of the form ax 2 + bx + c = 0, real solutions exist if and only if its discriminant (Δ) is greater than or equal to zero. Δ = b 2 - 4ac ≥ 0 Solution: Let the length of the rectangular park be 'l' meters Let the breadth of the rectangular park be 'b' meters. Given Perimeter (P) = 80 m WKT, P = 2(l + b) ⇒  2(l + b) = 80 ⇒  l + b = 40 (E...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 4 Question: 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. Given: The sum of the present ages of two friends is 20 years. Four years ago, the product of their ages was 48. To Find: 1. Is the given situation possible? 2. If yes, determine their present ages. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, real solutions exist if and only if its discriminant (Δ) is greater than or equal to zero. Δ = b 2 - 4ac ≥ 0 Solution: Let, The present age of the first friend (in years) = x Since the sum of their ages is 20 years, The present age of the second friend(in years) = 20 - x Four years ago: Age of the first friend (in years) = x - 4 Age of the second friend (in years...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 3 Question: Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m^2? If so, find its length and breadth. Given: A rectangular mango grove of Length (L) is twice its breadth (B) ⇒ L = 2B Area (A) = 800 m 2 To Find: 1. Is it possible to design such a rectangular mango grove? 2. If yes, find its length and breadth. Formula: Area of a rectangle is given by: Area = Length × Breadth (A = L × B) A quadratic equation of the form ax 2 + bx + c = 0, for real solutions to exist, the discriminant (Δ) must be greater than or equal to zero. Δ = b 2 - 4ac ≥ 0 Solution: Let the breadth of the rectangular mango grove be 'x' meters. According to the problem, the length is twice its breadth: Length = 2x meters. The area of the rectangular mango grove is given as 800 m 2 . WKT,...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 2 (ii) Question: Find the values of k for each of the following quadratic equations, so that they have two equal roots : kx (x – 2) + 6 = 0. Given: The equation: kx (x – 2) + 6 = 0 To Find: The value(s) of 'k' such that the equation has two equal roots. Formula: A quadratic equation of the form ax 2 + bx + c = 0 to have two equal roots, its discriminant (Δ) must be equal to zero. Δ = b 2 - 4ac = 0 Solution: The given equation is kx (x – 2) + 6 = 0. First, rewrite the equation in the standard quadratic form ax 2 + bx + c = 0: ⇒ kx 2 - 2kx + 6 = 0 Comparing it with the standard form ax 2 + bx + c = 0, we have: a = k b = -2k c = 6 For the equation to be a quadratic equation, the coefficient 'a' cannot be zero, i.e., k ≠ 0. (Condition 1) For two equal roots, the discriminant must be zero (Δ = 0): ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 2 (i) Question: Find the values of k for each of the following quadratic equations, so that they have two equal roots: 2x 2 + kx + 3 = 0. Given: The quadratic equation: 2x 2 + kx + 3 = 0 To Find: The value(s) of 'k' such that the equation has two equal roots. Formula: For quadratic equation of the form ax 2 + bx + c = 0 to have two equal roots, its discriminant (Δ) must be equal to zero. Δ = b 2 - 4ac = 0 Solution: The given quadratic equation is 2x 2 + kx + 3 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2 b = k c = 3 For two equal roots, the discriminant must be zero (Δ = 0): ⇒ b 2 - 4ac = 0 ⇒ (k) 2 - 4(2)(3) = 0 ⇒ k 2 - 24 = 0 ⇒ k 2 = 24 ⇒ k = ±√24 ⇒ k = ±√(4 × 6) ⇒ k = ±2√6 Result: The values of k for which the quadratic equation 2x 2 + kx + ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 1 (iii) Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : 2x 2 – 6x + 3 = 0 Given: The quadratic equation: 2x 2 – 6x + 3 = 0 To Find: The nature of the roots. If real roots exist, find them. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots. WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 2x 2 – 6x + 3 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2,...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 1 (ii) Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : 3x 2 – 4√3x + 4 = 0. Given: The quadratic equation: 3x 2 – 4√3x + 4 = 0 To Find: The nature of the roots. If real roots exist, find them. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots. WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 3x 2 – 4√3x + 4 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 1 (i) Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : 2x 2 – 3x + 5 = 0. Given: The quadratic equation: 2x 2 – 3x + 5 = 0 To Find: The nature of the roots. If real roots exist, find them. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots. WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 2x 2 – 3x + 5 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2 ...