Which of the following relation is CORRECT?
I. `(sqrt(15) + sqrt(7)) lt (2sqrt(22))`
II. `(sqrt(17) + sqrt(5)) lt (sqrt(20) + sqrt(2))`
(a)Only I
(b)Only II
(c)Neither I nor II
(d)Both I and II

Options

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

Answer

To solve the problem, we need to evaluate the two given inequalities step by step.

### Step 1: Evaluate the first inequality
We have the first inequality:
\[ \sqrt{15} + \sqrt{7} < 2\sqrt{22} \]
To simplify this, we will square both sides of the inequality.

### Step 2: Square both sides
Squaring both sides gives us:
\[ (\sqrt{15} + \sqrt{7})^2 < (2\sqrt{22})^2 \]
Calculating both sides:
- Left side:
\[ (\sqrt{15})^2 + 2\sqrt{15}\sqrt{7} + (\sqrt{7})^2 = 15 + 2\sqrt{105} + 7 = 22 + 2\sqrt{105} \]
- Right side:
\[ (2\sqrt{22})^2 = 4 \times 22 = 88 \]
So we have:
\[ 22 + 2\sqrt{105} < 88 \]

### Step 3: Isolate the square root
Now, we isolate \(2\sqrt{105}\):
\[ 2\sqrt{105} < 88 - 22 \]
\[ 2\sqrt{105} < 66 \]
Dividing both sides by 2:
\[ \sqrt{105} < 33 \]
Squaring both sides:
\[ 105 < 1089 \]
This is true, so the first inequality is correct.

### Step 4: Evaluate the second inequality
Now we check the second inequality:
\[ \sqrt{17} + \sqrt{5} < \sqrt{20} + \sqrt{2} \]
Again, we will square both sides.

### Step 5: Square both sides
Squaring both sides gives us:
\[ (\sqrt{17} + \sqrt{5})^2 < (\sqrt{20} + \sqrt{2})^2 \]
Calculating both sides:
- Left side:
\[ (\sqrt{17})^2 + 2\sqrt{17}\sqrt{5} + (\sqrt{5})^2 = 17 + 2\sqrt{85} + 5 = 22 + 2\sqrt{85} \]
- Right side:
\[ (\sqrt{20})^2 + 2\sqrt{20}\sqrt{2} + (\sqrt{2})^2 = 20 + 2\sqrt{40} + 2 = 22 + 2\sqrt{40} \]
So we have:
\[ 22 + 2\sqrt{85} < 22 + 2\sqrt{40} \]

### Step 6: Simplify the inequality
We can cancel 22 from both sides:
\[ 2\sqrt{85} < 2\sqrt{40} \]
Dividing both sides by 2:
\[ \sqrt{85} < \sqrt{40} \]
Squaring both sides gives us:
\[ 85 < 40 \]
This is false, so the second inequality is incorrect.

### Conclusion
The first inequality is true, and the second inequality is false. Therefore, the correct answer is:
**(a) Only I**