In a Young’s double slit experimental set-up, if we use white light,…

The correct answer is Option (3) → (A), (B) and (D) only

In Young’s experiment the fringe width \( \beta = \lambda D/d \) is proportional to the wavelength, so with white light each colour forms its own fringe pattern.

(A) a bright (white) fringe is obtained at the center of the screen. — Correct. At the centre the path difference is zero for every wavelength, so all colours are bright there together and the central fringe is white.

(B) fringes of different colors are observed clearly, only for the first order. — Correct. Next to the central white fringe the first-order colours separate into a coloured band; farther out the colours overlap and the fringes merge into nearly uniform white light.

(C) the \( 1^{\mathrm{st}} \) order red fringes form closer to the center of the screen than the \( 1^{\mathrm{st}} \) order violet fringes. — Incorrect. Red has the longer wavelength, so its fringe lies farther from the centre.

(D) the \( 1^{\mathrm{st}} \) order violet fringes form closer to the center of the screen than the \( 1^{\mathrm{st}} \) order red fringes. — Correct.

    \( y_{1} = \lambda D/d, \quad \lambda_{\mathrm{violet}} \lt \lambda_{\mathrm{red}} \ \Rightarrow\ y_{1,\mathrm{violet}} \lt y_{1,\mathrm{red}} \)

Note: Statement (B) is an approximation. In practice a few coloured fringes can be seen on each side of the central white fringe before the colours merge. With violet at about 400 nm and red at about 700 nm, the first-order and second-order bands do not yet overlap \( (2\times400\,\mathrm{nm} = 800\,\mathrm{nm} \gt 700\,\mathrm{nm}) \); overlap begins only from the third order \( (3\times400\,\mathrm{nm} = 1200\,\mathrm{nm} \lt 2\times700\,\mathrm{nm} = 1400\,\mathrm{nm}) \). If (B) is read strictly as an exact statement and judged incorrect, the correct statements are (A) and (D), which is Option (1). The key follows the usual description that coloured fringes are clearly seen only next to the central white fringe.