If `(sinalpha)x^2-2x+bgeq2,` for all real values of `xlt=1a n dalpha in (0,pi/2)uu(pi//2,pi),` then possible real value of `b` is /are `a2` b. `3` c. `4` d. `5`

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To solve the inequality \((\sin \alpha)x^2 - 2x + b \geq 2\) for all real values of \(x \leq 1\) and \(\alpha \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi)\), we will follow these steps:

### Step 1: Rearranging the Inequality
We start by rearranging the given inequality:
\[ \sin \alpha \cdot x^2 - 2x + b - 2 \geq 0 \]
This can be rewritten as:
\[ \sin \alpha \cdot x^2 - 2x + (b - 2) \geq 0 \]

### Step 2: Identifying the Quadratic Form
The expression \(\sin \alpha \cdot x^2 - 2x + (b - 2)\) is a quadratic equation in the form \(ax^2 + bx + c\), where:
- \(a = \sin \alpha\)
- \(b = -2\)
- \(c = b - 2\)

### Step 3: Conditions for the Quadratic to be Non-negative
For the quadratic to be non-negative for all \(x \leq 1\), the following conditions must hold:
1. The leading coefficient \(a = \sin \alpha > 0\) (which is true since \(\alpha \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi)\)).
2. The discriminant must be less than or equal to zero:
\[ D = b^2 - 4ac \leq 0 \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-2)^2 - 4(\sin \alpha)(b - 2) \leq 0 \]
This simplifies to:
\[ 4 - 4\sin \alpha(b - 2) \leq 0 \]

### Step 4: Simplifying the Discriminant Condition
Rearranging the inequality gives:
\[ 4 \leq 4\sin \alpha(b - 2) \]
Dividing both sides by 4:
\[ 1 \leq \sin \alpha(b - 2) \]
This leads to:
\[ \frac{1}{\sin \alpha} \leq b - 2 \]
Thus:
\[ b \geq \frac{1}{\sin \alpha} + 2 \]

### Step 5: Finding the Range of \(b\)
Since \(\sin \alpha\) can take values in the interval \((0, 1)\), the minimum value of \(\frac{1}{\sin \alpha}\) approaches infinity as \(\sin \alpha\) approaches 0, and the maximum value is 1 when \(\sin \alpha = 1\). Therefore:
- As \(\sin \alpha\) approaches 1, \(b\) approaches \(3\).
- As \(\sin \alpha\) approaches 0, \(b\) approaches infinity.

Thus, the possible values for \(b\) are:
\[ b \geq 3 \]

### Step 6: Conclusion
The possible real values of \(b\) from the options given are \(3\), \(4\), and \(5\). Hence, the answer is:
\[ \text{Possible values of } b: 3, 4, 5 \]