If 8a³ + b³ + 27c³ = 18 abc, then the relation among a, b and c is:

Correct — B.

The stem is an image: if 8a³ + b³ + 27c³ = 18abc, find the relation among a, b and c.

Set x = 2a, y = b, z = 3c. Then:
x³ + y³ + z³ = 8a³ + b³ + 27c³
3xyz = 3 × 2a × b × 3c = 18abc

So the equation is exactly x³ + y³ + z³ = 3xyz, i.e. x³ + y³ + z³ − 3xyz = 0.

Factorise: (x + y + z)(x² + y² + z² − xy − yz − zx) = 0

Take the first factor, x + y + z = 0:
2a + b + 3c = 0
b + 3c = −2a → option (b).

Why the others are wrong

• (a)
  b + 3c = 2a — Option (a) drops a sign. From 2a + b + 3c = 0, moving 2a to the other side makes it −2a; b + 3c = +2a would need 4a = 0.
• (c)
  2a + 3c = b — Option (c) transposes b without changing its sign. The correct rearrangement of 2a + b + 3c = 0 is b = −2a − 3c, so 2a + 3c = −b, not +b.
• (d)
  2a + b = 3c — Option (d) makes the same slip on the 3c term. From 2a + b + 3c = 0 you get 2a + b = −3c, so the equality holds only if c = 0.

The engine here is one identity: x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx).

SSC disguises it with coefficients. 8, 1 and 27 are the cubes of 2, 1 and 3, so 8a³ + b³ + 27c³ is (2a)³ + b³ + (3c)³. The right-hand side is the tell: 3 × (2a) × b × (3c) = 18abc, which is precisely 3xyz.

With the equation reduced to x³ + y³ + z³ = 3xyz, the product of the two factors is zero, and the branch the options offer is x + y + z = 0, i.e. 2a + b + 3c = 0.

The second factor is a sum of squares in disguise — it is half of (x−y)² + (y−z)² + (z−x)² — so it vanishes only when x = y = z.

Key facts

• x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx).
• 8, 1 and 27 are the cubes of 2, 1 and 3, so 8a³ + b³ + 27c³ = (2a)³ + b³ + (3c)³.
• 3 × 2a × b × 3c = 18abc, which is the given right-hand side, so the equation reads x³ + y³ + z³ = 3xyz.
• x³ + y³ + z³ = 3xyz holds when x + y + z = 0 or when x = y = z.

Common traps

• Trying to factorise 8a³ + b³ + 27c³ directly instead of matching 18abc to 3xyz first.
• Substituting x = 2a and z = 3c but forgetting that 3xyz then carries the factor 2 × 3 = 6.
• Reaching 2a + b + 3c = 0 correctly and then transposing a term without flipping its sign.

No directly related past PYQ was found.