$V_1, V_2, V_3$ and $V_4$ are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4x cm respectively. The following statements regarding these volumes are given below.
(A) $V_1 + V_2 + 2V_3 < V_4$
(B) $V_1+3V_2> V_3+ V_4$
(C) $2(V_1 +V_3) + V_2 = V_4$
(D) $V_1 + 4V_2 + V_3 < V_4$
Which of these statements is/are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (C), (D), and (A) only
Step-by-Step Verification:
(A) $V_1 + V_2 + 2V_3 < V_4$
- LHS: $1 + 8 + 2(27) = 9 + 54 = 63$
- RHS: $64$
- $63 < 64$ is Correct.
(B) $V_1 + 3V_2 > V_3 + V_4$
- LHS: $1 + 3(8) = 1 + 24 = 25$
- RHS: $27 + 64 = 91$
- $25 > 91$ is Incorrect.
(C) $2(V_1 + V_3) + V_2 = V_4$
- LHS: $2(1 + 27) + 8 = 2(28) + 8 = 56 + 8 = 64$
- RHS: $64$
- $64 = 64$ is Correct.
(D) $V_1 + 4V_2 + V_3 < V_4$
- LHS: $1 + 4(8) + 27 = 1 + 32 + 27 = 60$
- RHS: $64$
- $60 < 64$ is Correct.
Conclusion:
Statements (A), (C), and (D) are correct.
Correct Option: (C), (D), and (A) only