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UPSC CDS I 2019 Math Paper

Question Numbers: 99-100

Consider the following for the next two (02) items :

The sum of length, breadth and height of a cuboid is 22 cm and the length of its diagonal is 14 cm.

Question : 99 of 100

Marks: +1, -0

Solution:

Concept:

Use the algebraic identity

a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca)

to relate sum of cubes and volume.

Explanation:

Let the dimensions of the cuboid be

l

b

, and

h

(length, breadth, height).

Given: sum of dimensions

l + b + h = 22

cm and diagonal

\sqrt{l^2 + b^2 + h^2} = 14

cm.

Squaring the diagonal gives

l^2 + b^2 + h^2 = 196

^2

Use

(l+b+h)^2 = l^2+b^2+h^2 + 2(lb+bh+hl)

Substitute:

22^2 = 484 = 196 + 2(lb+bh+hl)

Thus

2(lb+bh+hl) = 288

, so

lb+bh+hl = 144

^2

Define

S = l^3 + b^3 + h^3

and

V = l b h

Apply the identity:

l^3+b^3+h^3 - 3lbh = (l+b+h)(l^2+b^2+h^2 - (lb+bh+hl))

Hence

S - 3V = 22 \times (196 - 144) = 22 \times 52 = 1144

^3

Answer:

1144

^3

(Option C).

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