PhysicsCBSE Class 12 · Sample paper, session 2023-24 · 2 marks · Very short answer

CBSE Class 12 Physics sample paper 2023-24: question 21(a)

From the official CBSE sample question paper for 2023-24, section B, question 21(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(a) · 2 marks · Very short answerRay Optics and Optical Instruments

Show that the least possible distance between an object and its real image in a convex lens is 4f, where f is the focal length of the lens.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

Open one hint at a time and try again before the next.

Hint 1

Let the object distance be $x$ ; then the image distance is $d - x$ .

Hint 2

Put these in the lens formula to get a quadratic in $x$ ; when does it have real roots?

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let $d$ be the distance between the object and its real image, with the object at distance $x$ from the lens. Then $u = - x$ , $v = + (d - x)$ . Lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ .
½ mark
$\frac{1}{f} = \frac{1}{d - x} + \frac{1}{x} = \frac{d}{x ( d - x )}$
½ mark
$\Rightarrow x d - x^{2} = f d \Rightarrow x^{2} - d x + f d = 0 \Rightarrow x = \frac{d \pm d ^{2} - 4 f d}{2}$
½ mark
For a real image, $x$ must be real, so $d^{2} - 4 f d \geq 0 \Rightarrow d \geq 4 f$ . Hence the least distance between object and its real image is $4 f$ .
½ mark

Answer: $d_{min} = 4 f$ (shown)

Where the marks are

Takes $u = - x$ , $v = d - x$ with lens formula (½ mark)
Gets $1/ f = d / [x (d - x)]$ (½ mark)
Forms the quadratic $x^{2} - d x + f d = 0$ and solves it (½ mark)
Uses real-root condition $d^{2} - 4 f d \geq 0$ to get $d \geq 4 f$ (½ mark)

Revise the chapter

Chapter: Ray Optics and Optical Instruments (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Ray Optics and Optical Instruments, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Physics sample paper 2023-24: question 21(a)

Hints

Step-by-step solution

Revise the chapter

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