Consider the function
`f(x)={{:((alphacosx)/(pi-2x),If,xne(pi)/(2)),(3,If,x=(pi)/(2)):}`
Which is continuous at `x=(pi)/(2)` where `alpha` is a constant.
What is the value of `alpha`?

Options

A. 6
B. 3
C. 2
D. 1

Explanation

To determine the value of \( \alpha \) for the function

\[ f(x) = \begin{cases} \frac{\alpha \cos x}{\pi - 2x} & \text{if } x \neq \frac{\pi}{2} \\ 3 & \text{if } x = \frac{\pi}{2} \end{cases} \]

to be continuous at \( x = \frac{\pi}{2} \), we need to ensure that the left-hand limit and right-hand limit at \( x = \frac{\pi}{2} \) are equal to \( f\left(\frac{\pi}{2}\right) \).

### Step 1: Calculate the Left-Hand Limit

The left-hand limit as \( x \) approaches \( \frac{\pi}{2} \) from the left is given by:

\[ \lim_{x \to \frac{\pi}{2}^-} f(x) = \lim_{x \to \frac{\pi}{2}^-} \frac{\alpha \cos x}{\pi - 2x} \]

Substituting \( x = \frac{\pi}{2} - h \) where \( h \to 0^+ \):

\[ \lim_{h \to 0} \frac{\alpha \cos\left(\frac{\pi}{2} - h\right)}{\pi - 2\left(\frac{\pi}{2} - h\right)} = \lim_{h \to 0} \frac{\alpha \sin h}{2h} \]

Using the limit \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \):

\[ = \lim_{h \to 0} \frac{\alpha \sin h}{2h} = \frac{\alpha}{2} \]

### Step 2: Calculate the Right-Hand Limit

The right-hand limit as \( x \) approaches \( \frac{\pi}{2} \) from the right is given by:

\[ \lim_{x \to \frac{\pi}{2}^+} f(x) = \lim_{x \to \frac{\pi}{2}^+} \frac{\alpha \cos x}{\pi - 2x} \]

Substituting \( x = \frac{\pi}{2} + h \) where \( h \to 0^+ \):

\[ \lim_{h \to 0} \frac{\alpha \cos\left(\frac{\pi}{2} + h\right)}{\pi - 2\left(\frac{\pi}{2} + h\right)} = \lim_{h \to 0} \frac{\alpha (-\sin h)}{-2h} = \lim_{h \to 0} \frac{\alpha \sin h}{2h} \]

Again using the limit \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \):

\[ = \frac{\alpha}{2} \]

### Step 3: Set Left-Hand Limit Equal to Right-Hand Limit

For the function to be continuous at \( x = \frac{\pi}{2} \):

\[ \lim_{x \to \frac{\pi}{2}^-} f(x) = \lim_{x \to \frac{\pi}{2}^+} f(x) = f\left(\frac{\pi}{2}\right) \]

Thus, we have:

\[ \frac{\alpha}{2} = 3 \]

### Step 4: Solve for \( \alpha \)

Multiplying both sides by 2:

\[ \alpha = 6 \]

### Conclusion

The value of \( \alpha \) for which the function is continuous at \( x = \frac{\pi}{2} \) is:

\[ \boxed{6} \]