Given that `=(10)/(sin x+ sqrt3 cos x)` the minimum positive value of y is :-

Options

A. 1
B. 2
C. 5
D. `(10)/(sqrt3)`

Explanation

To find the minimum positive value of \( y = \frac{10}{\sin x + \sqrt{3} \cos x} \), we can follow these steps:

### Step 1: Identify the Denominator
The denominator of \( y \) is \( d = \sin x + \sqrt{3} \cos x \). To minimize \( y \), we need to maximize \( d \).

### Step 2: Differentiate the Denominator
To find the maximum value of \( d \), we differentiate it with respect to \( x \):
\[ \frac{dd}{dx} = \cos x - \sqrt{3} \sin x \]
Setting the derivative equal to zero for maximization:
\[ \cos x - \sqrt{3} \sin x = 0 \]

### Step 3: Solve for \( x \)
Rearranging the equation gives:
\[ \cos x = \sqrt{3} \sin x \]
Dividing both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)):
\[ 1 = \sqrt{3} \tan x \]
Thus,
\[ \tan x = \frac{1}{\sqrt{3}} \]
This implies:
\[ x = 30^\circ \quad \text{or} \quad x = \frac{\pi}{6} \text{ radians} \]

### Step 4: Substitute \( x \) Back into the Denominator
Now we substitute \( x = 30^\circ \) back into the denominator:
\[ d = \sin(30^\circ) + \sqrt{3} \cos(30^\circ) \]
Calculating the trigonometric values:
\[ \sin(30^\circ) = \frac{1}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
So,
\[ d = \frac{1}{2} + \sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{1}{2} + \frac{3}{2} = 2 \]

### Step 5: Calculate the Minimum Value of \( y \)
Now substituting \( d \) back into the equation for \( y \):
\[ y = \frac{10}{d} = \frac{10}{2} = 5 \]

### Final Answer
Thus, the minimum positive value of \( y \) is:
\[ \boxed{5} \]