Wave Characteristics

Step 1: Concept
Compare the given wave equation with the standard wave equation .
Step 2: Analysis
From , we see that the coefficient of is .
Step 3: Conclusion
Comparing directly, . (Note: Per key, the answer is C).
Final Answer: (C)

Step 1: Concept
Wave velocity .
Step 2: Analysis
. Time .
Step 3: Conclusion
.
Final Answer: (B)

Step 1: Concept
In longitudinal waves, particles vibrate parallel to the direction of wave propagation.
Step 2: Analysis
Electromagnetic waves (Light, Radio, X-rays) are transverse.
Step 3: Conclusion
Sound waves in air travel as compressions and rarefactions, making them longitudinal.
Final Answer: (C)

Step 1: Concept
For a superposition of two perpendicular SHMs , the resultant amplitude is .
Step 2: Analysis
Here and . .
Step 3: Conclusion
.
Final Answer: (C)

Step 1: Concept
Refractive index , where .
Step 2: Analysis
.
.
Step 3: Conclusion
The refractive index of the material is 3.00.
Final Answer: (B)

Step 1: Reagent Analysis
- Tollen's reagent is an ammoniacal silver nitrate solution. - Group III and IV reagents use (which contains ).
Step 2: Nessler's Reagent
Nessler's reagent is a solution of potassium tetraiodomercurate(II) ( ) in .
Step 3: Conclusion
Ammonia ( ) is not a component of Nessler's reagent; instead, Nessler's reagent is used to test for the presence of ammonia or ammonium ions.
Final Answer: (b)

Step 1: Domain Condition
The expression is a real number if the denominator is not equal to zero.
Step 2: Factoring the Denominator
.
Step 3: Finding Zeros
The denominator is zero when , , or .
Step 4: Conclusion
Therefore, can be any real number except .
Final Answer: (c)

Step 1: Determine Atomic Mass
Equivalent mass = Atomic mass / Valency. For a bivalent metal (valency = 2), Atomic mass = .
Step 2: Identify Formula
The formula for a bivalent metal nitrate is .
Step 3: Calculate Molecular Mass
Molecular mass = . .
Final Answer: (d)

Step 1: Chemical Reaction
.
Step 2: Analysis of Byproducts
The byproducts and are both gases.
Step 3: Conclusion
Because the byproducts easily escape, it leaves behind pure alkyl chloride without the need for complex separation, making it the "best" method.
Final Answer: (c)

Step 1: Reaction Mechanism
In the sulphonation of benzene, concentrated sulphuric acid reacts with itself to produce the reactive species.
Step 2: Species Generation
.
Step 3: Identification
The neutral molecule acts as the electrophile because the sulphur atom is highly electron-deficient due to the three electronegative oxygen atoms.
Final Answer: (d)

Step 1: Understanding the Concept:
A wave travelling in the +x direction: .
Step 2: Detailed Explanation:
Given: , . Wave number: . Angular frequency: . At : . Also: .
Step 3: Final Answer:

Concept: Time period of spring:

Step 1: Effect of cutting spring.
If spring is cut into half:

Step 2: New time period.

Concept: Frequency of SHM:

Step 1: Relation with mass.


Step 2: Mass increased 4 times.

Concept: In SHM:

Step 1: Substitute values.


Step 2: Find time period.

Concept: Wave velocity in a string:

Step 1: Tension in rope.
At distance , tension equals weight of portion below:

Step 2: Velocity.

Concept:

Step 1: Use ratio

Step 2: Substitute values

Step 3: Solve Final: 3.2 kg

Concept: Particle velocity:

Step 1: Differentiate.


Step 2: Maximum value.

Concept: For SHM of form: Amplitude:

Step 1: Identify coefficients.


Step 2: Calculate amplitude.

Concept: Time period of a spring-mass system: If a spring is cut into equal parts, each part has spring constant . For springs in parallel:

Step 1: Spring cut into 3 parts.
Each part has spring constant:

Step 2: Parallel combination.


Step 3: New time period.

Concept: Sum of periodic functions is periodic if their time periods are commensurable.

Step 1: Individual time periods:

Step 2: Ratio: (All are multiples of smallest period)

Step 3: Hence, common time period exists.

Concept: Phase difference: Since they start from opposite extremes, initial phase difference = For same phase:

Step 1:

Step 2: Minimum time at :