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RRB Technician Grade 1 Model Paper 5

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Question : 62 of 100

Marks: +1, -0

If

a + 2b + 3c = 0

, then what is the value of

(a^{3} + 8b^{3} + 27c^{3} )

?

18 abc
12 abc
6 abc
0

Solution:

Concept:

Use the algebraic identity for the sum of three cubes:

x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx)

Explanation:

Let

x = a

,

y = 2b

, and

z = 3c

.

Then

x^3 = a^3

,

y^3 = 8b^3

,

z^3 = 27c^3

.

Also,

xyz = a(2b)(3c) = 6abc

, so

3xyz = 18abc

.

Substitute these into the identity:

a^3 + 8b^3 + 27c^3 - 18abc = (a+2b+3c)(a^2 + 4b^2 + 9c^2 - 2ab - 6bc - 3ca)

Given

a + 2b + 3c = 0

, the right-hand side becomes

0

:

a^3 + 8b^3 + 27c^3 - 18abc = 0

Therefore,

a^3 + 8b^3 + 27c^3 = 18abc

Answer:

The value of

a^3 + 8b^3 + 27c^3

is

18abc

.

Correct option: A.

18abc

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