UPSC CDS 1 2022 Math Paper

Given :
The surface area of a cube = surface area of a sphere
Length of cube =x
The radius of the sphere =y
Formula used :
1. Surface area of cube =6×(‌ side ‌)2
2. Surface area of sphere =4πr2
3. Volume of cube =side3
4. Volume of sphere =(4∕3)πr3
Formula used :
1. Surface area of cube =6×(‌ side ‌)2
2. Surface area of sphere =4πr2
3. Volume of cube =side3
4. Volume of sphere =(4∕3)πr3
Calculations :
The surface area of a cube = surface area of a sphere
Using equations (1) and (2), we get
⇒6x2=4πy2
⇒x2∕y2=4π∕6
⇒x2∕y2=(4∕6)×(22∕7)
⇒x2∕y2=44∕21
⇒‌

x

y

=‌

√‌44

√‌21

If 2x>3y then, 4x2>9y2
⇒4(√44)2>9(√‌21)2
⇒4×44>9×21
⇒176<189≠2x>3y
Statement I is incorrect
Using equations (3) and (4), we get
⇒ The volume of cube =(√‌44)3
⇒44√‌44 unit 3=88√‌11 unit ‌3
⇒ The volume of sphere =(4∕3)×(22∕7)×(√‌21)3
⇒88√21unit3
The volume of a sphere is greater than the volume of a cube
∴ Neither statement 1 nor statement 2 is correct.