exercise 4.1 solution answer key class 10

Quadratic Equations (NCERT Class 10 Chapter 4) – Exercise 4.1 Solutions

Q1. Check whether the following are quadratic equations:

  • (i) (x + 1)2 = 2(x – 3)
  • (ii) x2 – 2x = (–2)(3 – x)
  • (iii) (x – 2)(x + 1) = (x – 1)(x + 3)
  • (iv) (x – 3)(2x + 1) = x(x + 5)
  • (v) (2x – 1)(x – 3) = (x + 5)(x – 1)
  • (vi) x2 + 3x + 1 = (x – 2)2
  • (vii) (x + 2)3 = 2x(x2 – 1)
  • (viii) x3 – 4x2 – x + 1 = (x – 2)3

Solutions to Q1

  1. (x + 1)2 = 2(x – 3)
    ⇒ x2 + 2x + 1 = 2x – 6
    ⇒ x2 + 7 = 0
    Quadratic Equation
  2. x2 – 2x = (–2)(3 – x)
    ⇒ x2 – 2x = -6 + 2x
    ⇒ x2 – 4x + 6 = 0
    Quadratic Equation
  3. (x – 2)(x + 1) = (x – 1)(x + 3)
    ⇒ x2 – x – 2 = x2 + 2x – 3
    ⇒ 3x – 1 = 0
    Not Quadratic
  4. (x – 3)(2x + 1) = x(x + 5)
    ⇒ 2x2 – 5x – 3 = x2 + 5x
    ⇒ x2 – 10x – 3 = 0
    Quadratic Equation
  5. (2x – 1)(x – 3) = (x + 5)(x – 1)
    ⇒ 2x2 – 7x + 3 = x2 + 4x – 5
    ⇒ x2 – 11x + 8 = 0
    Quadratic Equation
  6. x2 + 3x + 1 = (x – 2)2
    ⇒ x2 + 3x + 1 = x2 + 4 – 4x
    ⇒ 7x – 3 = 0
    Not Quadratic
  7. (x + 2)3 = 2x(x2 – 1)
    ⇒ x3 + 8 + x2 + 12x = 2x3 – 2x
    ⇒ x3 + 14x – 6x2 – 8 = 0
    Not Quadratic
  8. x3 – 4x2 – x + 1 = (x – 2)3
    ⇒ x3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 12x
    ⇒ 2x2 – 13x + 9 = 0
    Quadratic Equation

Q2. Represent the following situations in the form of quadratic equations:

  1. The area of a rectangular plot is 528 m2. The length of the plot is one more than twice its breadth. Find the length and breadth.
  2. The product of two consecutive positive integers is 306. Find the integers.
  3. Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360. Find Rohan’s present age.
  4. A train travels a distance of 480 km at uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more to cover the same distance. Find the speed of the train.

Solutions to Q2

  1. Let breadth = x metres.
    Length = 2x + 1 m.
    Area = 528 m2:
    (2x + 1) × x = 528 ⇒ 2x2 + x – 528 = 0
  2. Let first integer = x.
    Next integer = x + 1.
    x(x + 1) = 306 ⇒ x2 + x – 306 = 0
  3. Let Rohan’s age = x years.
    Mother’s age = x + 26 years.
    Three years later: (x + 3)(x + 29) = 360 ⇒ x2 + 32x – 273 = 0
  4. Let speed = x km/h.
    Time = 480/x.
    For speed (x-8): (480/(x-8)) = (480/x) + 3
    x2 – 8x – 1280 = 0