`theta` is said to be well behaved if it lies in interval `[0,(pi)/(2)].` They are intelligent if they make domain of `f +g and g` equal. The vlaue of `theta` for which `h (theta)` is defined are handosome. Let
`f (x)= sqrt(thetax ^(2) -2 (theta^(2) -3) x-12theta,) g (x)=ln (x^(2) -49),`
`h (theta) ln [int_(0)^(theta) 4 cos ^(2)t dt - theta ^(2)],` where `theta` is in radians.
Complete set of alues of `theta` which are intelligent is :

Options

A. `[(6)/(7) , (7)/(2)]`
B. `(0, (pi)/(3)]`
C. `[(1)/(4), (6)/(7)]`
D. `[(1)/(2), (pi)/(2)]`

Answer

To solve the problem, we need to find the complete set of values of \( \theta \) which are "intelligent". This requires us to analyze the domains of the functions \( f(x) \) and \( g(x) \) and ensure that their domains are equal.

### Step 1: Determine the domain of \( f(x) \)

The function \( f(x) \) is defined as:
\[ f(x) = \sqrt{\theta x^2 - 2(\theta^2 - 3)x - 12\theta} \]
For \( f(x) \) to be defined, the expression inside the square root must be non-negative:
\[ \theta x^2 - 2(\theta^2 - 3)x - 12\theta \geq 0 \]
This is a quadratic inequality in \( x \). To find the roots, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = \theta \), \( b = -2(\theta^2 - 3) \), and \( c = -12\theta \).

Calculating the discriminant:
\[ D = b^2 - 4ac = [2(\theta^2 - 3)]^2 - 4(\theta)(-12\theta) \]
\[ = 4(\theta^2 - 3)^2 + 48\theta^2 \]
\[ = 4(\theta^4 - 6\theta^2 + 9 + 12\theta^2) = 4(\theta^4 + 6\theta^2 + 9) \]
This discriminant is always positive for \( \theta \) in the interval \( [0, \frac{\pi}{2}] \).

Now, applying the quadratic formula:
\[ x = \frac{2(\theta^2 - 3) \pm 2\sqrt{(\theta^2 + 3)^2}}{2\theta} = \frac{(\theta^2 - 3) \pm (\theta^2 + 3)}{\theta} \]
This gives us two roots:
1. \( x_1 = \frac{2\theta^2}{\theta} = 2\theta \)
2. \( x_2 = \frac{-6}{\theta} \)

The quadratic opens upwards (since \( a = \theta > 0 \)), so \( f(x) \) is defined outside the interval \( \left(-\frac{6}{\theta}, 2\theta\right) \).

### Step 2: Determine the domain of \( g(x) \)

The function \( g(x) \) is defined as:
\[ g(x) = \ln(x^2 - 49 \]
For \( g(x) \) to be defined, we need:
\[ x^2 - 49 > 0 \implies x < -7 \text{ or } x > 7 \]
Thus, the domain of \( g(x) \) is \( (-\infty, -7) \cup (7, \infty) \).

### Step 3: Find the intersection of the domains

For the domains of \( f(x) \) and \( g(x) \) to be equal, we need:
\[ (-\infty, -7) \cup (7, \infty) = \text{Domain of } f(x) \]
This means we need to satisfy the conditions:
1. \( -\frac{6}{\theta} < -7 \)
2. \( 2\theta > 7 \)

From the first condition:
\[ -\frac{6}{\theta} < -7 \implies \theta > \frac{6}{7} \]

From the second condition:
\[ 2\theta > 7 \implies \theta > \frac{7}{2} \]

### Step 4: Combine the conditions

The conditions we have are:
1. \( \theta > \frac{6}{7} \)
2. \( \theta < \frac{7}{2} \)

The intersection of these conditions gives us:
\[ \frac{6}{7} < \theta < \frac{7}{2} \]

### Conclusion

The complete set of values of \( \theta \) which are intelligent is:
\[ \left(\frac{6}{7}, \frac{7}{2}\right) \]