Ray Optics And Optical Instruments

Step 1: Concept
According to Lens Maker's Formula, .
Step 2: Analysis
Refractive index ( ) is higher for shorter wavelengths (Cauchy's equation). Violet light has the shortest wavelength in the visible spectrum, so it has the highest .
Step 3: Conclusion
Since is inversely proportional to , the focal length is minimum for violet light.
Final Answer: (B)

Step 1: Concept
.
Step 2: Analysis
. .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
Rainbow formation involves multiple optical phenomena within water droplets.
Step 2: Analysis
Sunlight enters the drop (refraction), splits into colors (dispersion), reflects off the back (TIR), and exits (refraction).
Step 3: Conclusion
All three processes are essential for the observer to see a rainbow.
Final Answer: (C)

Step 1: Identify Ligands
The complex contains five ammine ( ) ligands and one nitrito ( ) ligand.
Step 2: Determine Oxidation State
Let oxidation state of be : .
Step 3: Naming
Name ligands alphabetically (ammine before nitrito). The nitrite ligand is bonded through Oxygen, so it is called "nitrito". The full name is: penta ammine nitrito cobalt (III) ion.
Final Answer: (b)

Step 1: Complex Charge
The complex dissociates as and .
Step 2: Ligand Nature
In this specific complex, the nitric oxide ( ) ligand exists in the form of the nitrosyl ion ( ). Water is neutral ( ).
Step 3: Calculation
Let oxidation state of be : .
Final Answer: (d)

Step 1: Differentiation
.
Step 2: Analysis of Monotonicity
Setting gives , which has no real roots. This means is strictly increasing for all real .
Step 3: Calculating Endpoints
At , .
At , .
Step 4: Range
Since the function is strictly increasing, the range is .
Final Answer: (b)

Step 1: Understanding the dispersion condition.
When two prisms are combined to produce dispersion without deviation, the total deviation produced by both prisms must cancel each other out. This means that the deviation produced by the first prism ( ) is exactly balanced by the deviation produced by the second prism ( ). The deviation for a prism is given by: where is the refractive index of the prism, and is the angle of the prism.

Step 2: Setting up the equation.
For no deviation, the deviations must cancel out: Substituting the given values:

Step 3: Conclusion.
Thus, the angle of is 3°. Therefore, the correct answer is (4).

Step 1: Understand the refraction of light in water.
The apparent depth of an object submerged in water is given by the formula: where: - is the actual depth of the object, - is the refractive index of the water.

Step 2: Apply the formula.
Given that the actual depth and the refractive index of the water is , we substitute these values into the formula:

Step 3: Simplify the expression.
Simplifying the expression: Thus, the apparent depth (the height the image is raised) is .

Step 1: Understanding the Concept:
For lenses in contact, .
Step 2: Detailed Explanation:
Given , .

.
Step 3: Final Answer:
Thus, focal length = .

Step 1: Understanding the Concept:
At maximum displacement, KE = 0, PE = maximum. At mean position, KE = maximum, PE = 0. By energy conservation, maximum KE = loss in PE.
Step 2: Detailed Explanation:
Height gained by bob at angular displacement : .
Loss in PE = . This equals maximum KE.
Step 3: Final Answer:
Thus, maximum KE = .

Concept: Light passes twice through lens due to reflection.

Step 1: First refraction


Step 2: Reflection at plane mirror
Image forms at same distance:

Step 3: Second refraction
Now object for lens is at : Image on left side. Conclusion:

Concept: Angular size constant linear size focal length.

Step 1: Image size


Step 2: Area
Conclusion:

Concept: Graph asymptotes:

Step 1: From graph
Asymptotes at: But given scaling shows:

Step 2: Error estimation
Graph uncertainty Conclusion:

Step 1: Understanding the Concept:
Apply Snell's law at face AB and then the refraction formula at spherical surface CD: .

Step 2: Detailed Explanation:
At AB: . The refracted ray inside the block becomes parallel to AD (travels horizontally). At spherical surface CD: object at , , , m:

Step 3: Final Answer:
Distance OE m.

Step 1: . Angle with surface = , so .

Step 2: Critical angle undefined. If , , then , ray grazes surface. Given answer is internal reflection.

Step 1: After burning BC, the system is a standard Atwood machine.

Step 2: Acceleration , with moving down and moving up.

Step 1: Understanding the Concept:
At minimum deviation, angle of refraction inside prism .
Step 2: Detailed Explanation:
For equilateral prism: ,
Snell's law at first face:

Step 3: Final Answer:
Angle of incidence .

Step 1: Understanding the Concept:
and .
Step 2: Detailed Explanation:

Step 3: Final Answer:
Speed of light in water m/s.

Step 1: Understanding the Concept:
.
Step 2: Detailed Explanation:


Step 3: Final Answer:
Refractive index .

Step 1: Understanding the Concept:
The image just touches the rod, so one end of the image coincides with one end of the rod. Using the mirror equation for both ends.
Step 2: Detailed Explanation:
Let the rod extend from to . Since image is real and elongated and just touches the rod, one image end coincides with object end. Using for both ends gives magnification .
Step 3: Final Answer:
Magnification .

Step 1: Understanding the Concept:
For minimum deviation, .
Step 2: Detailed Explanation:
Here , . So . Thus , so , giving .
Step 3: Final Answer:
The angle of minimum deviation is .

Step 1: Understanding the Concept:
When a ray emerges normally, and .
Step 2: Detailed Explanation:
At the second surface, the ray falls at , so angle of incidence at second surface (since it emerges normally). Then . Using Snell's law at first surface: . Deviation .
Step 3: Final Answer:
and .

Step 1: Understanding the Concept:
For plano-convex lens, . Lens maker's formula: .
Step 2: Detailed Explanation:
cm, . . So cm.
Step 3: Final Answer:
The focal length is cm.

Step 1: Understanding the Concept:
Lens maker's formula: .
Step 2: Detailed Explanation:
In air, , .
In liquid, . So, .
Dividing the two equations, , so cm.
Step 3: Final Answer:
The focal length becomes cm.

Step 1: Understanding the Concept:
Light travels zigzag path; actual path length = .
Step 2: Detailed Explanation:

Path length =
?
Given answer is 3.85 s.
Step 3: Final Answer:
Time is 3.85 s.

Step 1: Understanding the Concept:
Critical angle C satisfies .
Step 2: Detailed Explanation:

Step 3: Final Answer:
Refractive index is about 1.5.

Step 1: Understanding the Concept:
Refraction at spherical surface: .
Step 2: Detailed Explanation:
,



(left)
Step 3: Final Answer:
Image is 30 cm left.

Step 1: Understanding the Concept:
Rainbow involves multiple optical phenomena.
Step 2: Detailed Explanation:
Rainbow formation involves dispersion, refraction, and total internal reflection in water droplets.
Step 3: Final Answer:
Dispersion and total internal reflection.

Concept: Refractive index of prism in a medium:

Step 1: Substitute values.


Step 2: Find critical angle.

Concept:

Step 1: Convert to meters.


Step 2: Substitute.


Step 3: Power.

Concept: Magnification: Lens formula:

Step 1: Given magnification.


Step 2: Substitute in lens formula.

Concept: Reflection of vector:

Step 1: Interpretation.
is unit normal.

Step 2: Apply formula.

Concept: Heat required to melt ice: where

Step 1: Heat required.


Step 2: Area of lens.


Step 3: Heat per unit area per minute.

Concept: For ray to retrace path after reflection it must strike second face normally.

Step 1: Condition at second face.
Angle of incidence at second face = Thus, inside prism:

Step 2: Apply Snell's law at first surface.


Step 3: Simplify.

Concept: For total internal reflection:

Step 1: Geometry of prism Given and angle at , the prism is an isosceles right triangle.

Step 2: Condition for minimum refractive index For minimum , the ray should undergo just TIR: From geometry, the angle of incidence at the reflecting face is:

Step 3: Apply critical angle condition Conclusion

Concept: Lens maker formula:

Step 1:Focal length depends only on:
•Refractive index
•Radii of curvature

Step 2:When lens is cut along (i.e., along principal axis):
•Curvature of surfaces remains unchanged
•Only aperture (size) reduces

Step 3:Since curvature is unchanged: Final: