Assertion (A): If 2x - 3y = 10 and xy = 16, then value of - 27y^3 + 8x^3 = 3880. Reason (R): x^3 - y^3 = (x - y) (x^2 + xy + y^2). 1. A is true, R is false. 2. A is false, R is true. 3. Both A and R are true. 4. Both A and R are false.

Both A and R are true. Explanation Given, 2x - 3y = 10 xy = 16 Cubing both sides of equation, ⇒ 2x - 3y = 10 ⇒ (2x - 3y)3 = 103 ⇒ (2x)3 - (3y)3 - 3 2x 3y(2x - 3y) = 103 ⇒ 8x3 - 27y3 - 18xy(2x - 3y) = 1000 ⇒ 8x3 - 27y3 - 18 x 16 x 10 = 1000 ⇒ 8x3 - 27y3 - 2880 = 1000 ⇒ 8x3 - 27y3 = 1000 + 2880 ⇒ 8x3 - 27y3 = 3880 ∴ Assertion (A) is true. Given, R.H.S. = (x - y) (x2 + xy + y2) = x(x2 + xy + y2) - y(x2 + xy + y2) = x3 + x2y + xy2 - x2y - xy2 + y3 = x3 + y3 ∴ L.H.S. = R.H.S. ∴ Reason (R) is true. Hence, both Assertion (A) and Reason (R) are true.

Mathematics

Assertion (A): If 2x - 3y = 10 and xy = 16, then value of - 27y³ + 8x³ = 3880.
Reason (R): x³ - y³ = (x - y) (x² + xy + y²).
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Expansions

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Answer

Both A and R are true.

Explanation

Given,

2x - 3y = 10

xy = 16

Cubing both sides of equation,

⇒ 2x - 3y = 10

⇒ (2x - 3y)³ = 10³

⇒ (2x)³ - (3y)³ - 3 $\times$ 2x $\times$ 3y(2x - 3y) = 10³

⇒ 8x³ - 27y³ - 18xy(2x - 3y) = 1000

⇒ 8x³ - 27y³ - 18 x 16 x 10 = 1000

⇒ 8x³ - 27y³ - 2880 = 1000

⇒ 8x³ - 27y³ = 1000 + 2880

⇒ 8x³ - 27y³ = 3880

∴ Assertion (A) is true.

Given,

R.H.S. = (x - y) (x² + xy + y²)

= x(x² + xy + y²) - y(x² + xy + y²)

= x³ + x²y + xy² - x²y - xy² + y³

= x³ + y³

∴ L.H.S. = R.H.S.

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.

Answered By

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Mathematics

Assertion (A): If 2x - 3y = 10 and xy = 16, then value of - 27y³ + 8x³ = 3880.
Reason (R): x³ - y³ = (x - y) (x² + xy + y²).
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Expansions

Answer

Related Questions

Assertion (A): If a = 2, b = -6 and c = 4, then $\dfrac{3abc - a^3 - b^3 - c^3}{ab + bc + ca - a^2 - b^2 - c^2}= 0$
Reason (R): If a + b + c = 0 then a³ + b³ + c³ + 3abc = 0
A is true, R is false.
A is false, R is true.
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Assertion (A): 5x⁵ - 20x³ in the form of factors is 5x³(x² - 4).
Reason (R):
5x⁵ - 20x³
= 5x³(x² - 4)
= 5x³(x + 2)(x - 2).
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Mathematics

Assertion (A): If 2x - 3y = 10 and xy = 16, then value of - 27y3 + 8x3 = 3880.Reason (R): x3 - y3 = (x - y) (x2 + xy + y2).A is true, R is false.A is false, R is true.Both A and R are true.Both A and R are false.

Expansions

Answer

Related Questions

Assertion (A): 5x5 - 20x3 in the form of factors is 5x3(x2 - 4).Reason (R):5x5 - 20x3= 5x3(x2 - 4)= 5x3(x + 2)(x - 2).A is true, R is false.A is false, R is true.Both A and R are true.Both A and R are false.

Assertion (A): If 2x - 3y = 10 and xy = 16, then value of - 27y³ + 8x³ = 3880.
Reason (R): x³ - y³ = (x - y) (x² + xy + y²).
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Assertion (A): 5x⁵ - 20x³ in the form of factors is 5x³(x² - 4).
Reason (R):
5x⁵ - 20x³
= 5x³(x² - 4)
= 5x³(x + 2)(x - 2).
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.