The complete set of values of a so that equation `sin^4 x+ a sin^2 x+ 4=0` has at least one real root is (A) `(- oo, -5]` (B) `(- oo , 4] uu [ 4, oo)` (C) `(-oo, -4]` (D) `[4, oo)`

Options

A. `(oo, -5]`
B. `(-oo, 4] uu [4, oo)`
C. `(-oo. -4]`
D. `[4,oo)`

Answer

To solve the equation \( \sin^4 x + a \sin^2 x + 4 = 0 \) for the values of \( a \) such that there is at least one real root, we can follow these steps:

### Step 1: Substitute \( \sin^2 x \)
Let \( t = \sin^2 x \). Since \( \sin^2 x \) varies between 0 and 1, we have \( t \in [0, 1] \). The equation then becomes:
\[ t^2 + at + 4 = 0 \]

### Step 2: Analyze the Quadratic Equation
For the quadratic equation \( t^2 + at + 4 = 0 \) to have at least one real root, the discriminant must be non-negative. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = a^2 - 4 \cdot 1 \cdot 4 = a^2 - 16 \]
We require:
\[ D \geq 0 \implies a^2 - 16 \geq 0 \]

### Step 3: Solve the Inequality
Solving the inequality \( a^2 - 16 \geq 0 \):
\[ a^2 \geq 16 \]
Taking the square root of both sides, we find:
\[ |a| \geq 4 \]
This gives us two intervals:
\[ a \leq -4 \quad \text{or} \quad a \geq 4 \]

### Step 4: Combine the Results
Thus, the complete set of values for \( a \) such that the equation has at least one real root is:
\[ (-\infty, -4] \cup [4, \infty) \]

### Final Answer
The correct option is:
**(B) \( (-\infty, -4] \cup [4, \infty) \)**