A magnified image of real object is to be obtained on a large screen 1 m from it. This can be achieved by

Options

A. using a convex mirror of focal length less than 0.25 m
B. using a concave mirror of focal length less than 0.25 m
C. using a convex lens of focal length less than 0.25 m
D. using a concave lens of focal length less than 0.25 m

Answer

To solve the problem of obtaining a magnified image of a real object on a large screen 1 meter away, we can follow these steps:

### Step 1: Understand the Requirements
We need to obtain a magnified image of a real object on a screen that is 1 meter away from the object. This means that the image distance \( v \) should be 1 meter.

### Step 2: Identify the Optical Devices
We can use either a concave mirror or a convex lens to achieve this. Both devices can form real images, but we need to determine the conditions under which they can produce a magnified image.

### Step 3: Use the Lens/Mirror Formula
For a lens or mirror, we use the formula:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
where \( f \) is the focal length, \( u \) is the object distance, and \( v \) is the image distance.

### Step 4: Set Up the Equation
Given that \( v = 1 \, m \) (the distance to the screen), we can express the object distance \( u \) in terms of \( f \):
\[ \frac{1}{f} = \frac{1}{u} + 1 \]
Rearranging gives:
\[ \frac{1}{u} = \frac{1}{f} - 1 \]
This implies:
\[ u = \frac{f}{1 - f} \]

### Step 5: Determine Conditions for Magnification
To achieve a magnified image, the object distance \( u \) must be less than twice the focal length \( f \) (i.e., \( u < 2f \)).

### Step 6: Substitute and Solve
Substituting the expression for \( u \):
\[ \frac{f}{1 - f} < 2f \]
Cross-multiplying gives:
\[ f < 2f(1 - f) \]
This simplifies to:
\[ f < 2f - 2f^2 \]
Rearranging leads to:
\[ 2f^2 < f \quad \Rightarrow \quad 2f^2 - f < 0 \]
Factoring gives:
\[ f(2f - 1) < 0 \]
This inequality holds when \( f < 0.25 \, m \).

### Step 7: Conclusion
Thus, we conclude that to obtain a magnified image of a real object on a screen 1 meter away, we can use:
1. A concave mirror with a focal length less than 0.25 m.
2. A convex lens with a focal length less than 0.25 m.

### Final Answer
The options that can achieve this are:
- Using a concave mirror of focal length less than 0.25 m.
- Using a convex lens of focal length less than 0.25 m.

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