If the equation `x^2 +4x sintheta + tantheta=0\ (0 < theta < pi/2)` has repeated roots, then `theta` equals (i)`pi/12` (ii)`pi/6` (iii)`pi/12 or (5pi)/12` (iv) `pi/6 or pi/12`

Answer

To solve the problem, we start with the quadratic equation given:

\[ x^2 + 4x \sin \theta + \tan \theta = 0 \]

This equation has repeated roots if the discriminant \( D \) is equal to zero. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by:

\[ D = b^2 - 4ac \]

Here, we identify \( a = 1 \), \( b = 4 \sin \theta \), and \( c = \tan \theta \).

### Step 1: Calculate the Discriminant
We set the discriminant to zero for repeated roots:

\[ D = (4 \sin \theta)^2 - 4 \cdot 1 \cdot \tan \theta = 0 \]

Calculating \( D \):

\[ D = 16 \sin^2 \theta - 4 \tan \theta = 0 \]

### Step 2: Substitute \( \tan \theta \)
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substituting this into the equation gives:

\[ 16 \sin^2 \theta - 4 \left(\frac{\sin \theta}{\cos \theta}\right) = 0 \]

### Step 3: Simplify the Equation
Multiply through by \( \cos \theta \) to eliminate the fraction:

\[ 16 \sin^2 \theta \cos \theta - 4 \sin \theta = 0 \]

### Step 4: Factor the Equation
Factor out \( 4 \sin \theta \):

\[ 4 \sin \theta (4 \sin \theta \cos \theta - 1) = 0 \]

This gives us two cases to consider:

1. \( 4 \sin \theta = 0 \)
2. \( 4 \sin \theta \cos \theta - 1 = 0 \)

### Step 5: Solve the First Case
For the first case \( 4 \sin \theta = 0 \), we find:

\[ \sin \theta = 0 \]

However, since \( 0 < \theta < \frac{\pi}{2} \), this case does not apply.

### Step 6: Solve the Second Case
For the second case:

\[ 4 \sin \theta \cos \theta = 1 \]

Using the double angle identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):

\[ 2 \sin 2\theta = 1 \implies \sin 2\theta = \frac{1}{2} \]

### Step 7: Find Values of \( 2\theta \)
The solutions for \( \sin 2\theta = \frac{1}{2} \) are:

\[ 2\theta = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad 2\theta = \frac{5\pi}{6} + 2n\pi \]

For \( n = 0 \):

1. \( 2\theta = \frac{\pi}{6} \) gives \( \theta = \frac{\pi}{12} \)
2. \( 2\theta = \frac{5\pi}{6} \) gives \( \theta = \frac{5\pi}{12} \)

### Step 8: Check the Validity of Solutions
Both values \( \frac{\pi}{12} \) and \( \frac{5\pi}{12} \) fall within the range \( 0 < \theta < \frac{\pi}{2} \).

### Final Answer
Thus, the values of \( \theta \) for which the equation has repeated roots are:

\[ \theta = \frac{\pi}{12} \quad \text{or} \quad \theta = \frac{5\pi}{12} \]

The correct option is (iii) \( \frac{\pi}{12} \) or \( \frac{5\pi}{12} \).

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