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Question
Choose the correct alternative.
If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?
Options
8
4
5
20
Solution
8
Explanation:
x4. y5 = (x + y)m + 1 ...(i)
∴ `"d"/"dx" ("x"^4. "y"^5) = "d"/"dx" ("x" + "y")^("m" + 1)`
∴ `"x"^4 "d"/"dx" "y"^5 + "y"^5 "d"/"dx" "x"^4 = ("m" + 1)("x" + "y")^("m" + 1 − 1) . "d"/"dx" ("x" + "y")`
∴ `"x"^4 . 5"y"^4 "d"/"dx" "y"+ "y"^5 4"x"^3 "d"/"dx" "x" = ("m" + 1)("x" + "y")^"m" ["d"/"dx" "x" + "d"/"dx" "y"]`
∴ `5"x"^4"y"^4 "dy"/"dx" + 4"x"^3 "y"^5 . 1 = ("m" + 1)("x" + "y")^"m" [1 + "dy"/"dx"]`
∴ `5"x"^4"y"^4 "dy"/"dx" + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [1 + "dy"/"dx"]`
Put `"dy"/"dx" = "y"/"x"`
∴ `5"x"^((cancel4)3)"y"^4 . "y"/cancel"x" + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [1 + "y"/"x"]`
∴ `5"x"^3"y"^4 . "y" + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [("x" + "y")/"x"]`
∴ `5"x"^3"y"^5 + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [("x" + "y")/"x"]`
∴ `9"x"^3"y"^5 = ("m" + 1)/"x" [("x" + "y")^("m" + 1)]`
∴ `9"x"^3"y"^5 = ("m" + 1)/cancel"x" "x"^((cancel4)3)."y"^5`
∴ `9cancel("x"^3"y"^5) = ("m" + 1) cancel("x"^3"y"^5)`
∴ 9 = m + 1
∴ m = 9 - 1
∴ m = 8
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