Choose the incorrect relation(s) from the following :
(I ) ` sqrt(6) + sqrt(2) = sqrt(5) + sqrt(3)`
(ii) `sqrt(6) + sqrt( 2) lt sqrt(5) + sqrt (3)`
(iii) `sqrt(6) + sqrt(2) gt sqrt(5) + sqrt(3)`

Options

A. (ii) and (iii)
B. ( i)
C. (ii)
D. (i)and (iiI)

Answer

To determine the incorrect relation(s) from the given options, we will evaluate each relation step by step.

### Given Relations:
1. \( \sqrt{6} + \sqrt{2} = \sqrt{5} + \sqrt{3} \) (Relation I)
2. \( \sqrt{6} + \sqrt{2} < \sqrt{5} + \sqrt{3} \) (Relation II)
3. \( \sqrt{6} + \sqrt{2} > \sqrt{5} + \sqrt{3} \) (Relation III)

### Step 1: Calculate \( \sqrt{6} + \sqrt{2} \)
First, we will calculate the approximate values of \( \sqrt{6} \) and \( \sqrt{2} \):
- \( \sqrt{6} \approx 2.45 \)
- \( \sqrt{2} \approx 1.41 \)

Now, adding these values:
\[ \sqrt{6} + \sqrt{2} \approx 2.45 + 1.41 = 3.86 \]

### Step 2: Calculate \( \sqrt{5} + \sqrt{3} \)
Next, we will calculate the approximate values of \( \sqrt{5} \) and \( \sqrt{3} \):
- \( \sqrt{5} \approx 2.24 \)
- \( \sqrt{3} \approx 1.73 \)

Now, adding these values:
\[ \sqrt{5} + \sqrt{3} \approx 2.24 + 1.73 = 3.97 \]

### Step 3: Compare the results
Now we compare the two results:
- \( \sqrt{6} + \sqrt{2} \approx 3.86 \)
- \( \sqrt{5} + \sqrt{3} \approx 3.97 \)

From this comparison, we can see:
- \( \sqrt{6} + \sqrt{2} < \sqrt{5} + \sqrt{3} \)

### Step 4: Evaluate each relation
1. **Relation I**: \( \sqrt{6} + \sqrt{2} = \sqrt{5} + \sqrt{3} \)
- This is **incorrect** as \( 3.86 \neq 3.97 \).

2. **Relation II**: \( \sqrt{6} + \sqrt{2} < \sqrt{5} + \sqrt{3} \)
- This is **correct** as \( 3.86 < 3.97 \).

3. **Relation III**: \( \sqrt{6} + \sqrt{2} > \sqrt{5} + \sqrt{3} \)
- This is **incorrect** as \( 3.86 > 3.97 \) is false.

### Conclusion
The incorrect relations are:
- Relation I: \( \sqrt{6} + \sqrt{2} = \sqrt{5} + \sqrt{3} \)
- Relation III: \( \sqrt{6} + \sqrt{2} > \sqrt{5} + \sqrt{3} \)

Thus, the answer is that Relations I and III are incorrect.