Kaliyuga Ekalavya

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 2 (iv) Question: Represent the following situations in the form of quadratic equations :A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train. Given: Distance traveled = 480 km If speed was 8 km/h less, time taken would be 3 hours more. To Find: Quadratic equation representing the situation to find the speed of the train. Formula: WKT, Time = Distance / Speed. Solution: Let the uniform speed of the train (in km/h) = x Time taken to cover 480 km at uniform speed (t 1 in hours) = 480 x If the speed had been 8 km/h less, the new speed (in km/h) = (x - 8) Time taken to cover 480 km at reduced speed (t 2 in hours) = 480 x - 8 According to the problem, t 2 = t 1 + 3. ⇒ 480 x - 8 ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 2 (iii) Question: Represent the following situations in the form of quadratic equations :Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age. Given: Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360. To Find: Quadratic equation representing the situation to find Rohan’s present age. Formula: WKT, future age = present age + number of years. Solution: Let Rohan’s present age (in years) = x According to the given condition, Rohan’s mother’s present age = (x + 26) years. Rohan’s age 3 years from now (in years) = x + 3 Rohan’s mother’s age 3 years from now = (x + 26 + 3) = x + 29 According to the given condition, the product of their ages 3 years from now is 360. ⇒ (x + 3) × (x + 29) = 360...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 2 (ii) Question: Represent the following situations in the form of quadratic equations :The product of two consecutive positive integers is 306. We need to find the integers. Given: The product of two consecutive positive integers = 306. To Find: Quadratic equation representing the situation to find the integers. Formula: WKT, consecutive integers differ by 1. Solution: Let the first positive integer be 'x'. Since the integers are consecutive, the next positive integer will be 'x + 1'. According to the given condition, their product is 306. ⇒ x × (x + 1) = 306 ⇒ x 2 + x = 306 Rearrange the equation into the standard quadratic form ax 2 + bx + c = 0: ⇒ x 2 + x - 306 = 0 Result: The situation can be represented by the quadratic equation: x 2 + x - 306 = 0. Next Question Solution: NCERT Class X Chap...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 2 (i) Question: Represent the following situations in the form of quadratic equations :The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. Given: Area of rectangular plot = 528 m 2 Length = 1 + 2 × Breadth To Find: Quadratic equation representing the situation to find the length and breadth of the plot. Formula: WKT, Area of a rectangle = Length × Breadth. Solution: Let the breadth of the rectangular plot (in metres) = x According to the given condition, the length is one more (+) than twice (2x) its breadth. ⇒ Length (in metres) = 2x + 1 WKT, Area = Length × Breadth ⇒ 528 = (2x + 1) × x ⇒ 528 = 2x 2 + x Rearrange the equation into the standard quadratic form ax 2 + bx + c = 0: ⇒ 2x 2 + x - 528 = 0 Result:...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(viii) Question: Check whether the following are quadratic equations : x 3 – 4x 2 – x + 1 = (x – 2) 3 Given: Equation: x 3 – 4x 2 – x + 1 = (x – 2) 3 To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. WKT,   (a - b) 3 = a 3 - b 3 - 3a 2 b + 3ab 2 . Solution: The given equation: x 3 – 4x 2 – x + 1 = (x – 2) 3 Expand the right side using (a - b) 3 = a 3 - b 3 - 3a 2 b + 3ab 2 : Here, a = x and b = 2. ⇒ x 3 - 2 3 - 3(x 2 )(2) + 3(x)(2 2 ) ⇒ x 3 - 8 - 6x 2 + 12x Substitute the expanded form back into the original equation: ⇒ x 3 – 4x 2 – x + 1 = x 3 - 6x 2 + 12x - 8 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 3 - x 3 - 4x 2 + 6x 2 - x - 12x + 1 + 8 = ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1 (vii) Question: Check whether the following are quadratic equations : (x + 2) 3 = 2x (x 2 – 1) Given: Equation: (x + 2) 3 = 2x (x 2 – 1) To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. WKT, the algebraic identity (a + b) 3 = a 3 + b 3 + 3a 2 b + 3ab 2 . Solution: The given equation: (x + 2) 3 = 2x (x 2 – 1) Expand the left side using (a + b) 3 = a 3 + b 3 + 3a 2 b + 3ab 2 : ⇒ x 3 + 2 3 + 3(x 2 )(2) + 3(x)(2 2 ) ⇒ x 3 + 8 + 6x 2 + 12x Expand the right side: ⇒ 2x(x 2 ) - 2x(1) = 2x 3 - 2x Substitute the expanded forms back into the original equation: ⇒ x 3 + 6x 2 + 12x + 8 = 2x 3 - 2x Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 3 - 2x 3 + 6x 2 + 12x + 2x + 8 = 0...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1 (vi) Question: Check whether the following are quadratic equations : x 2 + 3x + 1 = (x – 2) 2 Given: Equation: x 2 + 3x + 1 = (x – 2) 2 To Find: Whether the given equation is a quadratic equation. Formula: WKT, A quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0 WKT, (a - b) 2 = a 2 - 2ab + b 2 Solution: Start with the given equation: x 2 + 3x + 1 = (x – 2) 2 Expand the right side using (a - b) 2 = a 2 - 2ab + b 2 : ⇒ x 2 - 2(x)(2) + 2 2 = x 2 - 4x + 4 Substitute the expanded form back into the equation: ⇒ x 2 + 3x + 1 = x 2 - 4x + 4 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 2 - x 2 + 3x + 4x + 1 - 4 = 0 ⇒ 7x - 3 = 0 Compare this equation to the standard form ax 2 + bx + c = 0: Here, a = 0, b = 7, c = -3. Since the coefficient...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(v) Question: Check whether the following are quadratic equations : (2x – 1)(x – 3) = (x + 5)(x – 1) Given: Equation: (2x – 1)(x – 3) = (x + 5)(x – 1) To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. WKT, the product of two binomials (a + b)(c + d) = ac + ad + bc + bd. Solution: Start with the given equation: (2x – 1)(x – 3) = (x + 5)(x – 1) Expand the left side: ⇒ 2x(x) + 2x(-3) - 1(x) - 1(-3) = 2x 2 - 6x - x + 3 = 2x 2 - 7x + 3 Expand the right side: ⇒ x(x) + x(-1) + 5(x) + 5(-1) = x 2 - x + 5x - 5 = x 2 + 4x - 5 Substitute the expanded forms back into the equation: ⇒ 2x 2 - 7x + 3 = x 2 + 4x - 5 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ 2x 2 - x 2 - 7x - 4x + 3 +...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(iv) Question: Check whether the following are quadratic equations : (x – 3)(2x +1) = x(x + 5) Given: Equation: (x – 3)(2x +1) = x(x + 5) To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. WKT, the product of two binomials (a + b)(c + d) = ac + ad + bc + bd. Solution: Start with the given equation: (x – 3)(2x +1) = x(x + 5) Expand the left side: ⇒ x(2x) + x(1) - 3(2x) - 3(1) = 2x 2 + x - 6x - 3 = 2x 2 - 5x - 3 Expand the right side: ⇒ x(x) + x(5) = x 2 + 5x Substitute the expanded forms back into the equation: ⇒ 2x 2 - 5x - 3 = x 2 + 5x Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ 2x 2 - x 2 - 5x - 5x - 3 = 0 ⇒ x 2 - 10x - 3 = 0 Compare this equation to the standa...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(iii) Question: Check whether the following are quadratic equations : (x – 2)(x + 1) = (x – 1)(x + 3) Given: Equation: (x – 2)(x + 1) = (x – 1)(x + 3) To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. WKT, the product of two binomials (a + b)(c + d) = ac + ad + bc + bd. Solution: Start with the given equation: (x – 2)(x + 1) = (x – 1)(x + 3) Expand the left side: ⇒ x(x) + x(1) - 2(x) - 2(1) = x 2 + x - 2x - 2 = x 2 - x - 2 Expand the right side: ⇒ x(x) + x(3) - 1(x) - 1(3) = x 2 + 3x - x - 3 = x 2 + 2x - 3 Substitute the expanded forms back into the equation: ⇒ x 2 - x - 2 = x 2 + 2x - 3 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 2 - x 2 - x - 2x - 2 + 3 = 0 ⇒ -3x + 1 = 0 ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(ii) Question: Check whether the following are quadratic equations : x 2 – 2x = (–2) (3 – x) Given: Equation: x 2 – 2x = (–2) (3 – x) To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. Solution:  Start with the given equation: x 2 – 2x = (–2) (3 – x) Expand the right side: ⇒ x 2 – 2x = -6 + 2x Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 2 - 2x - 2x + 6 = 0 ⇒ x 2 - 4x + 6 = 0 Compare this equation to the standard form ax 2 + bx + c = 0: Here, a = 1, b = -4, and c = 6. Since the coefficient of x 2 (a) is 1 (which is not equal to 0), this equation is a quadratic equation. Result: The given equation x 2 – 2x = (–2) (3 – x) simplifies to x 2 - 4x + 6 = 0. Since the ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(i) Question: Check whether the following are quadratic equations : (x + 1) 2 = 2(x – 3) Given: Equation: (x + 1) 2 = 2(x – 3) To Find: Whether the given equation is a quadratic equation. Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. WKT, the algebraic identity (a + b) 2 = a 2 + 2ab + b 2 . Solution: The given equation: (x + 1) 2 = 2(x – 3) Expand the left side using (a + b) 2 = a 2 + 2ab + b 2 : ⇒ x 2 + 2(x)(1) + 1 2 = 2(x – 3) ⇒ x 2 + 2x + 1 = 2x – 6 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 2 + 2x - 2x + 1 + 6 = 0 ⇒ x 2 + 7 = 0 Compare this equation to the standard form ax 2 + bx + c = 0: Here, a = 1, b = 0, c = 7. Since the coefficient of x 2 (a) is 1 (which is not equal to 0), this equation is a quadratic equation. ...