V₁, V₂, V₃ and V₄ are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4 cm respectively. Some statements regarding these volumes are given below :
(i) V₁ + V₂ + 2V₃ < V₄
(ii) V₁ + 4V₂ + V₃ < V₄
(iii) 2(V₁ + V₃) + V₂ = V₄
Which of these statements area correct ?

A. 1 and 2

B. 2 and 3

C. 1 and 3

D. 1, 2 and 3

Answer: Option D

Solution(By Examveda Team)

Clearly, we have :
V₁ = x³,
V₂ = (2x)³ = 8x³
V₃ = (3x)³ = 27x³
V₄ = (4x)³ = 64x³
(i) V₁ + V₂ + 2V₃
= x³ + 8x³ + 2 × 27x³
= 63x³ < V₄

(ii) V₁ + 4V₂ + V₃
= x³ + 4 × 8x³ + 27 x³
= 60x³ < V₄

(iii) 2(V₁ + V₃) + V₂
= 2 (x³ + 27x³) + 8x³
= 64x³ = V₄