A `2.0 cm` tall object is placed `15 cm` in front of concave mirrorr of focal length `10 cm`. The size and nature of the image will be

Options

A. `1.0 cm, `real
B. `4 cm, `virtual
C. `4 cm`, real
D. none of these

Explanation

To solve the problem of finding the size and nature of the image formed by a concave mirror, we will follow these steps:

### Step 1: Identify the given values
- Height of the object (h_o) = 2.0 cm
- Object distance (u) = -15 cm (negative because the object is in front of the mirror)
- Focal length (f) = -10 cm (negative for concave mirrors)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- f = focal length
- v = image distance
- u = object distance

### Step 3: Substitute the known values into the mirror formula
Substituting the values into the formula:
\[ \frac{1}{-10} = \frac{1}{v} + \frac{1}{-15} \]

### Step 4: Rearrange the equation to find v
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{-10} + \frac{1}{15} \]
Finding a common denominator (which is 30):
\[ \frac{1}{v} = \frac{-3}{30} + \frac{2}{30} = \frac{-1}{30} \]
Thus,
\[ v = -30 \text{ cm} \]

### Step 5: Determine the nature of the image
Since v is negative, the image is real and located on the same side as the object.

### Step 6: Calculate the magnification (M)
The magnification (M) is given by the formula:
\[ M = -\frac{v}{u} \]
Substituting the values:
\[ M = -\frac{-30}{-15} = -2 \]

### Step 7: Determine the size of the image (h_i)
The height of the image (h_i) can be calculated using:
\[ h_i = M \times h_o \]
Substituting the values:
\[ h_i = -2 \times 2.0 \text{ cm} = -4.0 \text{ cm} \]

### Step 8: Conclusion
The negative sign indicates that the image is inverted. Therefore, the image is:
- Size: 4.0 cm
- Nature: Real and inverted

### Final Answer
The size of the image is 4.0 cm and the nature of the image is real and inverted.

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