If x = b + c, y = c + a, z = a + b, then what is (x + y + z)³ - 24xyz equal to ?

Correct Answer is : 8(a3+b3+c3)8 (a³ + b³ + c³)8(a3+b3+c3)

Correct Answer is : 8(a3+b3+c3)8 (a³ + b³ + c³)8(a3+b3+c3)

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UPSC CDS 2 2022 Math Paper

Question : 35 of 100

Marks: +1, -0

x = b + c, y = c + a, z = a + b

(x + y + z)^3 - 24xyz

equal to ?

Solution:

Concept:

The problem simplifies expressions involving sums and products of variables defined in terms of

a

b

c

Explanation:

Given:

x = b + c

y = c + a

z = a + b

Compute

x + y + z = (b+c)+(c+a)+(a+b) = 2(a+b+c)

Therefore,

(x+y+z)^3 = [2(a+b+c)]^3 = 8(a+b+c)^3

Also,

xyz = (b+c)(c+a)(a+b)

Now, consider

(x+y+z)^3 - 24xyz = 8(a+b+c)^3 - 24(a+b)(b+c)(c+a)

Use the standard expansion:

(a+b+c)^3 = a^3+b^3+c^3 + 3(a+b)(b+c)(c+a)

Substitute into the expression:

8(a+b+c)^3 - 24(a+b)(b+c)(c+a) = 8[a^3+b^3+c^3 + 3(a+b)(b+c)(c+a)] - 24(a+b)(b+c)(c+a)

Expand:

8(a^3+b^3+c^3) + 24(a+b)(b+c)(c+a) - 24(a+b)(b+c)(c+a)

The terms

24(a+b)(b+c)(c+a)

cancel, leaving

8(a^3+b^3+c^3)

Answer:

8(a^3+b^3+c^3)

(Option C).

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