Hi 9th mate,
Pls answer 15,16,17,18
Answers
Let us take x² = a and y² = b and z² = c
So (x² - y²)³ + (y² - z²)³ + (z² - x²)³ =
(a - b)³ + (b - c)³ + ( c - a)³
= a³ - b³ - 3ab(a-b) + b³ - c³ - 3bc(b-c) + c³ - a³ - 3ca(c-a)
= -3ab(a-b) - 3bc(b-c) - 3ca(c-a)
= -3x²y²(x²- y²) - 3y²z²(y²-z²) - 3z²x²(z² - x²)
= -3x⁴y² + 3x²y⁴ - 3y⁴z² + 3y²z⁴ - 3z⁴x² + 3z²x⁴ ...............(1)
(x-y)³ + (y - z)³ + (z - x)³
= x³ - y³ - 3x²y + 3xy² + y³ - z³ - 3y²z + 3yz² + z³ - x³ - 3z²x + 3zx²
= - 3x²y + 3xy² - 3y²z + 3yz² - 3z²x + 3zx² ..........(2)
(x² - y²)³ + (y² - z²)³ + (z² - x²)³ =
(x-y)³ + (y - z)³ + (z - x)³
= -3x⁴y² + 3x²y⁴ - 3y⁴z² + 3y²z⁴ - 3z⁴x² + 3z²x⁴
- 3x²y + 3xy² - 3y²z + 3yz² - 3z²x + 3zx²
= - x⁴y² + x²y⁴ - y⁴z² + y²z⁴ - z⁴x² + z²x⁴
- x²y + xy² - y²z + yz² - z²x + zx²
or
= x⁴y² - x²y⁴ + y⁴z² - y²z⁴ + z⁴x² - z²x⁴
x²y - xy² + y²z - yz²+ z²x - zx²
16
(ab + bc +ca - a² - b² - c²)
= 4*5 + 5*6 + 6*4 - 4² - 5² - 6²
= 20 + 30 + 24 - 16 - 25 - 36
= - 3 ............(1)
3abc - a³ - b³ - c³
= 3*4*5*6 - 4³ - 5³ - 6³
= 360 - 64 - 125 - 216
= - 45 ................(2)
(ab + bc +ca - a² - b² - c²) = - 3 = 1/15
( 3abc - a² - b² - c²) -45
17.
RHS
(x-y)² + (y - z)² + (z - x)²
= x² + y² - 2xy + y² + z³ - 2yz + z² + x² - 2zx
= 2x² + 2y² + 2z²- 2xy - 2yz - 2zx
= 2(x² + y² + z²- xy - yz - zx) ..........(1)
1/2 (x+ y +z) [(x-y)² + (y - z)² + (z - x)²]
= 1/2 (x+y+z)* 2(x² + y² + z²- xy - yz - zx)
= (x+y+z) (x² + y² + z²- xy - yz - zx)
= x³ + y³ + z³ - 3xyz = LHS
18.
8x³ + 27y³ + 64z³ - 3*2x*3y*4z = (2x+3y+4z) (4x² + 9y² + 16z² - 6xy - 12yz - 8zx)
8x³ + 27y³ + 64z³ - 72xyz = 0
8x³ + 27y³ + 64z³ = 72xyz
The question you gave wrong
Answer:
Him 9th mate,
playing answer 15,16,17,18