Total number of values of `' x '` satisfying the equation, `2^(cos^2x-1)=3. 2^(cos^2x)-4` and the inequality `x^2lt=30` is: 1 (b) 2 (c) 3 (d) 4

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \( 2^{\cos^2 x - 1} = 3 \cdot 2^{\cos^2 x} - 4 \) and the inequality \( x^2 \leq 30 \), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ 2^{\cos^2 x - 1} = 3 \cdot 2^{\cos^2 x} - 4 \]
We can rewrite \( 2^{\cos^2 x - 1} \) as:
\[ \frac{2^{\cos^2 x}}{2} = 3 \cdot 2^{\cos^2 x} - 4 \]
Multiplying both sides by 2 gives:
\[ 2^{\cos^2 x} = 6 \cdot 2^{\cos^2 x} - 8 \]

### Step 2: Rearranging the Equation
Rearranging the equation, we have:
\[ 2^{\cos^2 x} - 6 \cdot 2^{\cos^2 x} + 8 = 0 \]
This simplifies to:
\[ -5 \cdot 2^{\cos^2 x} + 8 = 0 \]
or
\[ 5 \cdot 2^{\cos^2 x} = 8 \]

### Step 3: Solve for \( 2^{\cos^2 x} \)
Dividing both sides by 5:
\[ 2^{\cos^2 x} = \frac{8}{5} \]

### Step 4: Take Logarithm
Taking logarithm base 2 on both sides:
\[ \cos^2 x = \log_2\left(\frac{8}{5}\right) \]

### Step 5: Determine the Range of \( \cos^2 x \)
Since \( \cos^2 x \) must be between 0 and 1, we need to check if \( \log_2\left(\frac{8}{5}\right) \) is within this range.

Calculating \( \log_2\left(\frac{8}{5}\right) \):
\[ \log_2\left(\frac{8}{5}\right) = \log_2(8) - \log_2(5) = 3 - \log_2(5) \]
Since \( \log_2(5) \) is approximately 2.32, we find:
\[ 3 - 2.32 \approx 0.68 \]
Thus, \( \cos^2 x \approx 0.68 \), which is valid since it lies between 0 and 1.

### Step 6: Find \( x \)
Now we have:
\[ \cos^2 x = \log_2\left(\frac{8}{5}\right) \]
This gives:
\[ \cos x = \pm \sqrt{\log_2\left(\frac{8}{5}\right)} \]
The general solutions for \( x \) are:
\[ x = \pm \cos^{-1}\left(\sqrt{\log_2\left(\frac{8}{5}\right)}\right) + 2n\pi \quad \text{and} \quad x = \pm \cos^{-1}\left(-\sqrt{\log_2\left(\frac{8}{5}\right)}\right) + 2n\pi \]

### Step 7: Apply the Inequality
Next, we apply the inequality \( x^2 \leq 30 \):
\[ -\sqrt{30} \leq x \leq \sqrt{30} \]
Calculating \( \sqrt{30} \approx 5.48 \).

### Step 8: Determine Valid Values of \( x \)
The values of \( x \) that satisfy \( x = n\pi \) within the interval \( [-\sqrt{30}, \sqrt{30}] \):
- \( n = 0 \) gives \( x = 0 \)
- \( n = 1 \) gives \( x = \pi \approx 3.14 \)
- \( n = -1 \) gives \( x = -\pi \approx -3.14 \)

The values \( 0, \pi, -\pi \) are valid solutions.

### Conclusion
Thus, the total number of values of \( x \) satisfying the equation and the inequality is **3**.