Algebra
- If a = 2, b = –3 then the value of 27 a3 – 54 a2b + 36 ab2 – 8b3 is
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27a3 – 54a2b + 36ab2 – 8b3
= (3a)3 – 3(3a)2 (2b) + 3 × 3a × (2b)2 – (2b)3
= (3a – 2b)3
= (3 × 2 –2 (–3))3 = (6 + 6)3
= (12)3 = 1728Correct Option: D
27a3 – 54a2b + 36ab2 – 8b3
= (3a)3 – 3(3a)2 (2b) + 3 × 3a × (2b)2 – (2b)3
= (3a – 2b)3
= (3 × 2 –2 (–3))3 = (6 + 6)3
= (12)3 = 1728
- If x + y + z = 9 then the value of (x – 4)3 + (y – 2)3 + (z – 3)3 – 3 (x – 4) (y – 2) (z – 3) is
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If a + b + c = 0, then a3 + b3 + c3 – 3abc = 0
Here, x – 4 + y – 2 + z – 3
= x + y + z – 9 = 9 – 9 = 0
∴ (x – 4)3 + (y – 2)3 + (z – 3)3 – 3
(x – 4) (y – 2) (z – 3) = 0Correct Option: C
If a + b + c = 0, then a3 + b3 + c3 – 3abc = 0
Here, x – 4 + y – 2 + z – 3
= x + y + z – 9 = 9 – 9 = 0
∴ (x – 4)3 + (y – 2)3 + (z – 3)3 – 3
(x – 4) (y – 2) (z – 3) = 0
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If x3 + 3y2x = 35 , what is x = y3 + 3x2y 19 y
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x3 + 3y2x = 35 y3 + 3x2y 19
By componendo and dividendo,x3 + 3y2x + y3 + 3x2y = 35 + 19 x3 + 3y2x − y3 − 3x2y 35 − 19 = 54 16 ⇒ (x + y)3 = 27 = 
3 
3 (x − y)3 8 2 ⇒ x + y = 3 x − y 2
By componendo and dividendo again⇒ x + y + x − y = 3 + 2 ⇒ x = 5 x + y − x + y 3 − 2 y Correct Option: C
x3 + 3y2x = 35 y3 + 3x2y 19
By componendo and dividendo,x3 + 3y2x + y3 + 3x2y = 35 + 19 x3 + 3y2x − y3 − 3x2y 35 − 19 = 54 16 ⇒ (x + y)3 = 27 = 3 3 (x − y)3 8 2 ⇒ x + y = 3 x − y 2
By componendo and dividendo again⇒ x + y + x − y = 3 + 2 ⇒ x = 5 x + y − x + y 3 − 2 y
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What is (x² − y²)3 + (y² − z²)3 + (z² − x²)3 (x − y)3 + (y − z)3 +(z − x)3
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x2 – y2 + y2 – z2 + z2 – x2 = 0
∴ (x² − y²)3 + (y² − z²)3 + (z² − x²)3
= 3(x² − y²)(y² − z²)(z² − x²)
[If a + b + c = 0, a3 + b3 + c3 = 3abc]
Similarly,
x – y + y – z + z – x = 0
∴ (x – y)3 + (y – z)3 + (z – x)3
= 3 (x – y) (y – z) (z – x)∴ (x² − y²)3 + (y² − z²)3 + (z² − x²)3 (x − y)3 + (y − z)3 +(z − x)3 = 3(x² − y²)(y² − z²)(z² − x²) 3 (x – y) (y – z) (z – x)
= (x + y) (y + z) (z + x)Correct Option: C
x2 – y2 + y2 – z2 + z2 – x2 = 0
∴ (x² − y²)3 + (y² − z²)3 + (z² − x²)3
= 3(x² − y²)(y² − z²)(z² − x²)
[If a + b + c = 0, a3 + b3 + c3 = 3abc]
Similarly,
x – y + y – z + z – x = 0
∴ (x – y)3 + (y – z)3 + (z – x)3
= 3 (x – y) (y – z) (z – x)∴ (x² − y²)3 + (y² − z²)3 + (z² − x²)3 (x − y)3 + (y − z)3 +(z − x)3 = 3(x² − y²)(y² − z²)(z² − x²) 3 (x – y) (y – z) (z – x)
= (x + y) (y + z) (z + x)
- What will be the value of x3 + y3 + z3 – 3xyz when x + y + z = 9 and x2 + y2 + z2 = 31?
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x + y + z = 9
x2 + y2 + z2 = 31
(x + y + z)2 = x2 + y2 + z2 + 2 (xy + yz + zx)
⇒ 81 = 31 + 2 (xy + yz + zx)
⇒ 2 (xy + yz + zx)
= 81 – 31
= 50
⇒ xy + yz + zx = 25
∴ x3 + y3 + z3 – 3xyz
= (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
= 9 (31 – 25)
= 9 × 6 = 54Correct Option: C
x + y + z = 9
x2 + y2 + z2 = 31
(x + y + z)2 = x2 + y2 + z2 + 2 (xy + yz + zx)
⇒ 81 = 31 + 2 (xy + yz + zx)
⇒ 2 (xy + yz + zx)
= 81 – 31
= 50
⇒ xy + yz + zx = 25
∴ x3 + y3 + z3 – 3xyz
= (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
= 9 (31 – 25)
= 9 × 6 = 54
