Question

Which of the following correctly represents the image formed on the screen?

• A.
• B.
• C.
• D.
• A. For a convex mirror magnification is always negative.
• B. For all virtual images formed by a mirror magnification is positive.
• C. For a concave lens magnification is always positive.
• D. For real and inverted images, magnification is always negative.
• A. \( f \)
• B. \( 2f \)
• C. \( \frac{f}{2} \)
• D. \( \frac{f}{4} \)
• A. 10 cm
• B. 12 cm
• C. 16 cm
• D. 20 cm
• A. \( X_1 X_2 \)
• B. \( \sqrt{X_1 + X_2} \)
• C. \( \sqrt{X_1 X_2} \)
• D. \( \sqrt{\frac{X_2}{X_1}} \)

Answer 1

The Correct Option is B

The image formed by a thin lens can be obtained by applying the lens formula: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] where:
\( f \) is the focal length,
\( v \) is the image distance,
\( u \) is the object distance.
The lens has two focal points, one on each side of the lens, as the light converges or diverges depending on the type of lens (convex or concave).

Answer 2

For a concave lens, the magnification is always negative because the image formed by a concave lens is always virtual, erect, and diminished in size. Therefore, option (C) is incorrect. Other statements are true as they align with the behavior of mirrors and lenses.

Answer 3

When a convex lens is cut into two equal parts perpendicular to the principal axis, the focal length of each part is halved. The new focal length \( f' \) of each part is: \[ f' = \frac{f}{2} \] This is because the lens curvature increases when it is cut in half, effectively reducing the focal length.

Answer 4

We use the lens formula: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] where:
\( u = -20 \) cm (object distance, conventionally taken as negative),
\( v = 50 - 20 = 30 \) cm (image distance),
\( f \) is the focal length.
Substituting values: \[ \frac{1}{f} = \frac{1}{30} - \frac{1}{-20} \] \[ \frac{1}{f} = \frac{1}{30} + \frac{1}{20} \] Taking LCM of 30 and 20: \[ \frac{1}{f} = \frac{2}{60} + \frac{3}{60} = \frac{5}{60} \] \[ f = \frac{60}{5} = 12 \text{ cm} \] Thus, the focal length of the lens is 12 cm, so the correct answer is (2).

Answer 5

From the properties of a biconvex lens, the focal length \( f \) is given by the geometric mean of the distances \( X_1 \) and \( X_2 \): \[ f = \sqrt{X_1 X_2} \] This relation is derived from the lens formula and paraxial approximation when the object and image distances are measured from the focal points. Thus, the correct answer is (3) \( \sqrt{X_1 X_2} \).