`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. Some statements regarding these volumes are shown here
(1) `V_(1)+V_(2)+2V_(3)ltV_(4)`
(2) `V_(1)+4V_(2)+V_(3)ltV_(4)`
(3) ` 2(V_(1)+V_(3))+V_(2)=V_(4)`
Which of the given statements is CORRECT ?

Options

A. (1) and (2) only
B. (2) and (3) only
C. (1) and (3) only
D. (1), (2) and (3)

Answer

To solve the problem, we need to calculate the volumes of the cubes based on the given side lengths and then evaluate the three statements provided.

### Step 1: Calculate the volumes of the cubes

1. **Volume of the first cube (V1)**:
\[ V_1 = (x \, \text{cm})^3 = x^3 \, \text{cm}^3 \]

2. **Volume of the second cube (V2)**:
\[ V_2 = (2x \, \text{cm})^3 = (2^3)(x^3) = 8x^3 \, \text{cm}^3 \]

3. **Volume of the third cube (V3)**:
\[ V_3 = (3x \, \text{cm})^3 = (3^3)(x^3) = 27x^3 \, \text{cm}^3 \]

4. **Volume of the fourth cube (V4)**:
\[ V_4 = (4x \, \text{cm})^3 = (4^3)(x^3) = 64x^3 \, \text{cm}^3 \]

### Step 2: Evaluate the statements

#### Statement 1: \( V_1 + V_2 + 2V_3 < V_4 \)

- Calculate \( V_1 + V_2 + 2V_3 \):
\[ V_1 + V_2 + 2V_3 = x^3 + 8x^3 + 2(27x^3) = x^3 + 8x^3 + 54x^3 = 63x^3 \]
- Compare with \( V_4 \):
\[ 63x^3 < 64x^3 \quad \text{(True)} \]

#### Statement 2: \( V_1 + 4V_2 + V_3 < V_4 \)

- Calculate \( V_1 + 4V_2 + V_3 \):
\[ V_1 + 4V_2 + V_3 = x^3 + 4(8x^3) + 27x^3 = x^3 + 32x^3 + 27x^3 = 60x^3 \]
- Compare with \( V_4 \):
\[ 60x^3 < 64x^3 \quad \text{(True)} \]

#### Statement 3: \( 2(V_1 + V_3) + V_2 = V_4 \)

- Calculate \( 2(V_1 + V_3) + V_2 \):
\[ 2(V_1 + V_3) + V_2 = 2(x^3 + 27x^3) + 8x^3 = 2(28x^3) + 8x^3 = 56x^3 + 8x^3 = 64x^3 \]
- Compare with \( V_4 \):
\[ 64x^3 = 64x^3 \quad \text{(True)} \]

### Conclusion

All three statements are correct.

### Final Answer
All statements (1, 2, and 3) are correct.

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