Consider the following statements:
I. There exists no `DeltaABC` for which `SinA + SinB=SinC`
II. If the angles of a triangle are in the ratio 1:2:3, then its sides will be in the ratio `1:sqrt3:2`
Which of the above statement(s) is/are correct?

Options

A. A. Only I
B. B. Only II
C. C. Both I and II
D. D. Neither I nor II

Answer

To solve the given problem, we will analyze each statement separately and determine their validity.

### Step 1: Analyze Statement I
**Statement I:** There exists no triangle ABC for which \( \sin A + \sin B = \sin C \).

1. We know from the sine rule that in any triangle ABC, the following relationship holds:
\[ \frac{A}{\sin A} = \frac{B}{\sin B} = \frac{C}{\sin C} = 2R \]
where \( R \) is the circumradius of the triangle.

2. Rearranging the sine rule gives:
\[ \sin A = \frac{A}{R}, \quad \sin B = \frac{B}{R}, \quad \sin C = \frac{C}{R} \]

3. Substitute these into the equation \( \sin A + \sin B = \sin C \):
\[ \frac{A}{R} + \frac{B}{R} = \frac{C}{R} \]

4. Multiplying through by \( R \) (assuming \( R \neq 0 \)):
\[ A + B = C \]

5. However, in any triangle, the sum of two sides \( A + B \) must be greater than the third side \( C \) (Triangle Inequality Theorem). Therefore, \( A + B = C \) cannot hold true.

**Conclusion for Statement I:** This statement is correct; there exists no triangle ABC for which \( \sin A + \sin B = \sin C \).

### Step 2: Analyze Statement II
**Statement II:** If the angles of a triangle are in the ratio 1:2:3, then its sides will be in the ratio \( 1:\sqrt{3}:2 \).

1. Let the angles of the triangle be \( X, 2X, 3X \). According to the angle sum property of triangles:
\[ X + 2X + 3X = 180^\circ \]
This simplifies to:
\[ 6X = 180^\circ \implies X = 30^\circ \]

2. Therefore, the angles are:
- \( X = 30^\circ \)
- \( 2X = 60^\circ \)
- \( 3X = 90^\circ \)

3. Now, we can find the sides using the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
where \( A = 30^\circ, B = 60^\circ, C = 90^\circ \).

4. Calculate the sine values:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 90^\circ = 1 \)

5. The ratios of the sides will be:
\[ a : b : c = \sin 30^\circ : \sin 60^\circ : \sin 90^\circ = \frac{1}{2} : \frac{\sqrt{3}}{2} : 1 \]

6. To eliminate the fraction, multiply through by 2:
\[ 1 : \sqrt{3} : 2 \]

**Conclusion for Statement II:** This statement is also correct; if the angles of a triangle are in the ratio 1:2:3, then its sides will indeed be in the ratio \( 1:\sqrt{3}:2 \).

### Final Conclusion
Both statements I and II are correct.