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If X = (M + 1) + (M + 1)/(M + 1) - (M - 1) Then Prove that X3 - 3mx2 + 3x = M - ICSE Class 10 - Mathematics

ConceptComponendo and Dividendo Properties

Question

If x = (root (3)("m + 1") + root (3)("m - 1"))/(root (3)("m + 1") + root (3)("m - 1") then prove that x3 - 3mx2 + 3x = m

Solution

"x"/1 = (root (3)("m + 1") + root (3)("m - 1"))/(root (3)("m + 1") + root (3)("m - 1")

Applying componendo and dividendo

("x"+ 1) /("x" - 1) = (root (3)("m + 1") + root (3)("m - 1") + root (3)("m + 1") - root (3)("m - 1"))/ (root (3)("m + 1") + root (3)("m - 1") - root (3)("m + 1") + root (3)("m - 1")

=> ("x"+ 1) /("x" - 1) = (2 root (3)("m" + 1))/(2 root (3)("m" - 1))

Cubing both sides

=> ("x" + 1)^3/("x - 1")^3 = (8 ("m + 1"))/(8("m - 1"))

=> ("x"^3 + 3"x"^2 +3"x" +1)/("x"^3 - 3"x"^2 + 3"x" -1) = ("m + 1")/("m - 1")

⇒  (m - 1) (x3 + 3x2 + 3x + 1) = (m + 1 )(x3 - 3x2 + 3x - 1)

⇒ mx3 + 3mx2 + 3mx + m - x3 - 3x2 - 3x - 1 - mx3 - 3mx2 + 3mx - m + x3 - 3x2 + 3x -1

⇒ 6mx2 + 2m - 2x3 - 6x = 0

⇒ 3mx2+ m - x3 - 3x  = O

⇒  x3 - 3mx2 + 3x = m

Hence Proved.

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APPEARS IN

Frank Solution for Frank Class 10 Mathematics Part 2 (2016 to Current)
Chapter 9: Ratio and Proportion
Exercise 9.3 | Q: 13

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Solution If X = (M + 1) + (M + 1)/(M + 1) - (M - 1) Then Prove that X3 - 3mx2 + 3x = M Concept: Componendo and Dividendo Properties.
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