Solve the following using identities (Part - 7)
1. 8 x^3 + 27 y^3 + 8 z^3 – 36 x y z
Sol. Using identity
x^3 + y^3 + z^3 – 3 x y z = (x + y + z) (x^2 + y^2 + z^2 – x
y – y z – z x)
8 x^3 + 27 y^3 + 8 z^3 – 36 x y z
(2 x)^3 + (3 y)^3 + (2 z)^3 – 3 * 2x * 3y * 2z
(2 x + 3 y + 2 z) {(2 x)^2 + (3 y)^2 + (2 z)^2 – 2x * 3y –
3y * 2z – 2 z * 2x}
(2 x + 3 y + 2 z) (4 x^2 + 9 y^2 + 4 z^2 – 6 x y – 6 y z – 4
z x}
2. 27 x^3 + 8 y^3 + 27 z^3 – 54 x y z
Sol. Using identity
x^3 + y^3 + z^3 – 3 x y z = (x + y + z) (x^2 + y^2 + z^2 – x
y – y z – z x)
27 x^3 + 8 y^3 + 27 z^3 – 54 x y z
(3 x)^3 + (2 y)^3 + (3 z)^3 – 3 * 3x * 2y * 3z
(3 x + 2 y + 3 z) {(3 x)^2 + (2 y)^2 + (3 z)^2 – 3 x * 2 y –
2 y * 3 z – 3 z * 3x}
(3 x + 2 y + 3 z) (9 x^2 + 4 y^2 + 9 z^2 – 6 x y – 6 y z – 9
z x}
3. 8 x^3 + 27 y^3 + 8 z^3 – 36 x y z If x + y + z = 0 so
prove that 8 x^3 + 27 y^3 + 8 z^3 = 36 x y z
Sol. Using identity
x^3 + y^3 + z^3 – 3 x y z = (x + y + z) (x^2 + y^2 + z^2 – x
y – y z – z x)
Given x + y + z = 0
x^3 + y^3 + z^3 – 3 x y z = ( 0 ) * (x^2 + y^2 + z^2 – x y –
y z – z x)
x^3 + y^3 + z^3 – 3 x y z = 0
So identity became x^3 + y^3 + z^3 = 3 x y z
8 x^3 + 27 y^3 + 8 z^3 – 36 x y z = 0
8 x^3 + 27 y^3 + 8 z^3 = 36 x y z
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| maths-ncert-solution |
4. If x + y + z = 0 so prove that x^3 + y^3 + z^3 = 3 x y z.
Sol. Using identity
x^3 + y^3 + z^3 – 3 x y z = (x + y + z) (x^2 + y^2 + z^2 – x
y – y z – z x)
Given x + y + z = 0
x^3 + y^3 + z^3 – 3 x y z = ( 0 ) * (x^2 + y^2 + z^2 – x y –
y z – z x)
x^3 + y^3 + z^3 – 3 x y z = 0
So identity became x^3 + y^3 + z^3 = 3 x y z
5. (-15)^3 + (7)^3 + (8)^3
Sol. x + y + z = 0 => -15 + 7 + 8 = 0 => 0 = 0
Using identity
x^3 + y^3 + z^3 = 3 x y z
(-15)^3 + (7)^3 + (8)^3
3 * (-15) * 7 * 8
-2520
6. (-5)^3 + (3)^3 + (2)^3
Sol. x + y + z = 0 => -5 + 3 + 2 = 0 => 0 = 0
Using identity
x^3 + y^3 + z^3 = 3 x y z
(-5)^3 + (3)^3 + (2)^3
3 * (-5) * 3 * 2
-90
7. We know that
Area = Length * Breath
a). x^2 + 6x + 8
Sol. By using identity
{(x + a) (x + b) = x^2 + (a + b)x + ab }
x^2 + 6x + 8
x^2 + (2 + 4)x + 8
(x + 2) (x + 4)
So possible length = (x + 2)
And possible breath = (x + 4)
b). x^2 + 5x + 6
Sol. By using identity
{(x + a) (x + b) = x^2 + (a + b)x + ab }
x^2 + 5x + 6
x^2 + (2 + 3)x + 6
(x + 2) (x + 3)
So possible length = (x + 2)
And possible breath = (x + 3)
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