Question
Quantity-I: Determine the value of (x + y -
z)3 + (x - y + z)3 - 8x3 . Quantity-II: Determine the value of 6x(z - y - x) (x - y + z) In the question, two Quantities I and II are given. You have to solve both the Quantity to establish the correct relation between Quantity-I and Quantity-II and choose the correct option.Solution
ATQ, Quantity I: Given = (x + y - z)3 + (x - y + z)3- 8x3 Or, (x + y - z)3 + (x - y + z)3 + (-2a)3 We know that, a3 + b3 + c3 = 3abc, when (a + b + c) = 0 Here, (x + y - z) = 'a' And, (x - y + z) = 'b' And, (-2x) = 'c' So, a + b + c = (x + y - z) + (x - y + z) - 2x = 0 So, required value = 3 × (x + y - z) × (x - y + z) × -2x = 6x(x + y - z) (y - x - z) So, Quantity I = 6x(x + y - z) (y - x - z) Quantity II: 6x(z - y - x) (x - y + z) Or, 6x × (-1) × (x + y - z) × (x - y + z) Or, 6x × (-1) × (-1) × (x + y - z) × (y - x - z) So, 6x(z - y - x) (x - y + z) = 6x × (x + y - z) (y - x - z) So, Quantity II = 6x × (x + y - z) (y - x - z) So, Quantity I = Quantity II
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