A beam of light converges at a point P. Now a convex lens of focal length 30 cm placed in the path of the convergent beam 12 cm from P. The point at which the beam converges now is

Explanation

To solve the problem, we need to find the new point of convergence of the light beam after it passes through a convex lens placed 12 cm from point P. The focal length of the lens is given as 30 cm.

### Step-by-step Solution:

1. **Identify the Given Values:**
- Focal length of the lens, \( F = +30 \) cm (positive because it is a convex lens).
- Distance from the lens to point P, \( d = 12 \) cm (this distance is measured towards the lens).

2. **Determine the Object Distance (U):**
- Since the lens is placed 12 cm from point P and the light converges at point P, we can consider the object distance \( U \) as negative (since the object is on the same side as the incoming light):
\[ U = -12 \text{ cm} \]

3. **Use the Lens Formula:**
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]

4. **Substitute the Values into the Lens Formula:**
Substituting \( f = 30 \) cm and \( u = -12 \) cm:
\[ \frac{1}{v} = \frac{1}{30} + \frac{1}{-12} \]

5. **Calculate the Right-Hand Side:**
To add the fractions, find a common denominator:
\[ \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{-12} = \frac{-5}{60} \]
Therefore:
\[ \frac{1}{v} = \frac{2}{60} - \frac{5}{60} = \frac{-3}{60} = -\frac{1}{20} \]

6. **Solve for V:**
Taking the reciprocal gives:
\[ v = -20 \text{ cm} \]

7. **Interpret the Result:**
The negative value of \( v \) indicates that the image is formed on the same side as the object (to the left of the lens). The distance from point P is:
\[ \text{Distance from P} = 12 \text{ cm} - 20 \text{ cm} = -8 \text{ cm} \]
This means the image is 8 cm to the left of point P.

### Final Answer:
The new point at which the beam converges is 8 cm to the left of point P.