CAT Exam Model Paper 3 with solutions for free online practice

Question : 96 of 100

Marks: +1, -0

In the adjoining figure, ABC is an equilateral triangle inscribing a square of maximum possible area. Again in the square there is an equilateral triangle whose side is same as that of the square. Further the smaller equilateral triangle inscribes a square of maximum possible area. What is the area of the innermost square if the each side of the outermost triangle be 0.01 m?

$( 873 - 504 √ { 3 } ) { \cm } ^ { 2 }$
$( 738 - 504 √ { 3 } ) { \cm } ^ { 2 }$
$( 873 - 405√ { 2 } ) { \cm } ^ { 2 }$
None of these

Validate

Solution:

(a)

PCQ is also an equilateral triangle
∴ PC = PQ = PM = a
∴

a/{\PA}=\sin60°=√3/2

∴PA =

{2a}/√3

∴ AC = AP + PC =

{2a}/√3+a=1\cm

⇒ a =

√3/{2+√3}=√3(2-√3)

Now, in figure (iii)
PM = MT = a
Let the each side of square RSYX be K, then RT = K also (since RTS is an equilateral triangle)

∴

{ K } /{ R M } = {√{ 3 } } /{ 2 }

∴

R M = { 2 K }/ { √ { 3 } }

∴

M T = R T + R M = K + { 2 K }/ {√ { 3 } }

M T = { ( √{ 3 } + 2 ) } /{ √ { 3 } } K

But MT = a
∴

a = ( { √ { 3 } + 2 }/ {√ { 3 } } )

∴ = { √ { 3 } a }/ { (√ { 3 } + 2 ) }

But

a = √ { 3 } ( 2 - √ { 3 } )

∴ K = { √ { 3 } }/ { ( √ { 3 } + 2 ) } [ √ { 3 } ( 2 - \√ { 3 } ) ]

K = { 3 ( 2 - √ { 3 } ) }/ { ( 2 + √ { 3 } ) } ×{ ( 2 - √ { 3 } ) }/ { ( 2 - √{ 3 } ) }

⇒K = { 3 ( 2 - √{ 3 } ) ^ { 2 } } /{ 1 } = 3 ( 7 - 4 √ { 3 } )

∴ Area of square RSYX =

K ^ { 2 } = [ 3 ( 7 - 4 √ { 3 } ) ] ^ { 2 }

K ^ { 2 } = 9 ( 49 + 48 - 56 √ { 3 } )

K ^ { 2 } = ( 873 - 504 √ { 3 } ) { \cm } ^ { 2 }

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