Step 1: Potential Energy (PE) and Kinetic Energy (KE) in SHM

The total energy in SHM is constant and is the sum of potential energy (PE) and kinetic energy (KE):

E_total = KE + PE

The potential energy at a displacement x from the mean position is:

PE = (1/2)kx²

where k is the force constant. The kinetic energy is:

KE = (1/2)k(A² - x²)

where A is the amplitude of the motion.

Step 2: Substitute x = A/n

When the particle is at (1/n) times its amplitude from the mean position, x = A/n. Substituting this into the PE and KE equations:

PE = (1/2)k(A/n)² = (1/2)kA²/n²

KE = (1/2)k(A² - (A/n)²) = (1/2)kA²(1 - 1/n²) = (1/2)kA²((n² - 1)/n²)

Step 3: Calculate the ratio KE/PE

KE/PE = [(1/2)kA²((n² - 1)/n²)] / [(1/2)kA²/n²] = n² - 1

Step 4: Final Answer

The ratio of KE to PE is n² - 1. Therefore, the correct option is (C).

Step 1: Moment of Inertia for a Solid Sphere.

The moment of inertia of a solid sphere of mass ( M ) and radius ( R ) about its diameter is:

I_{sphere (diameter)} = (2/5) M R².

Step 2: Moment of Inertia about the Given Axis (Parallel Axis Theorem).

Using the Parallel Axis Theorem, the moment of inertia about an axis parallel to the diameter and at a distance ( d ) from the center is:

I_parallel = I_center + M d².

Here: ( d = 10 , cm = 0.1 , m ) (distance from the center of each sphere to the axis).

For each sphere, the moment of inertia about the given axis is:

I_{sphere (axis)} = (2/5) M R² + M d².

Substitute ( R = 0.1 , m ) and ( d = 0.1 , m ):

I_{sphere (axis)} = (2/5) M (0.1)² + M (0.1)².

I_{sphere (axis)} = (2/5) M (0.01) + M (0.01) = (2/5) M (0.01) + (5/5) M (0.01).

I_{sphere (axis)} = (7/5) M (0.01).

Step 3: Total Moment of Inertia for Two Spheres.

Since there are two spheres and the axis passes symmetrically through the midpoint, the total moment of inertia is:

I_total = 2 × I_{sphere (axis)} = 2 × (7/5) M R².

I_total = (14/5) M R².

Step 4: Final Answer.

The total moment of inertia of the system is:

Answer: (14/5 M R²)

Step 1: Forces acting on the vehicle.

For a vehicle moving on a banked curve, the forces acting are:

• Gravitational force (mg).
• Normal reaction force (N).
• Frictional force (f).

The net force provides the necessary centripetal force for circular motion.

Step 2: Resolving forces.

The component of the normal reaction along the radius provides the centripetal force:

N sin θ = mv² / r,

where v is the speed, r is the radius of the curve, and θ is the banking angle.

Step 3: Frictionless case (Safety speed).

For the safety speed, assume friction is negligible. The normal force’s component balances the centripetal force:

v² = r g tan θ.

Solve for v:

v = √(r g tan θ).

Step 4: Final Answer.

The safety speed is:

V = √(r g tan θ).

Step 1: Understanding hydrosere succession.

Hydrosere is a type of ecological succession that starts in aquatic environments and progresses towards the climax terrestrial stage. Each stage in this progression represents a shift in the dominant vegetation and habitat conditions.

Step 2: Identifying the stages of hydrosere succession.

The typical stages in hydrosere succession are:

• Phytoplankton stage (initial stage in water)
• Submerged stage (rooted submerged plants)
• Floating stage (rooted floating plants)
• Reed swamp stage (emergent vegetation)
• Sedge meadow stage (herbaceous plants in shallow water or wet soil)
• Woodland stage (woody plants)
• Climax stage (terrestrial vegetation like forest or grassland)

Step 3: Preceding stage to sedge meadow stage.

The sedge meadow stage is characterized by sedges and other herbaceous plants. This stage is preceded by the reed swamp stage, which is dominated by emergent vegetation such as reeds and cattails.

Conclusion: The stage that comes just before the sedge meadow stage is the reed swamp stage.

Step 1: Relation between kinetic energy and frequency

The photoelectric equation is given by:

K.E = hν − ϕ

where h is Planck’s constant, ν is the frequency of incident light, and ϕ is the work function of the material.

Step 2: Frequency and wavelength relation

The frequency ν is related to the wavelength λ by:

ν = c / λ

where c is the speed of light.

Step 3: Substituting in the equation

Substituting ν = c / λ into the photoelectric equation:

K.E = (hc / λ) − ϕ

Thus, the maximum kinetic energy of the photoelectrons is inversely proportional to the wavelength λ.

Question 6:

If \( L \) is the inductance and \( R \) is the resistance, then the unit of \( \frac{L}{R} \) is:

Question 7:

A lift weighing \( 250 \, \text{kg} \) is to be lifted up at a constant velocity of \( 0.20 \, \text{m/s} \). What would be the minimum horsepower of the motor to be used?

Question 8:

A large number of bullets are fired in all directions with the same speed \( v \). What is the maximum area on the ground on which these bullets will spread?

Question 1:

Which of the following best describes sympatric speciation?

1. (A) Speciation that occurs due to geographic isolation
2. (B) Speciation that occurs in the same geographic region without physical separation
3. (C) Speciation that occurs due to migration to a new habitat
4. (D) Speciation that occurs through gradual accumulation of small changes over time

Question 2:

Which of the following causes typhoid fever?

1. (A) Salmonella typhi
2. (B) Mosquito
3. (C) Plasmodium
4. (D) Nicotine

Question 3:

Which scientists are credited with proposing the transpiration pull theory, also known as the cohesion-tension theory?

1. (A) Charles Darwin and Francis Darwin
2. (B) Stephen Hales
3. (C) Henry Dixon and John Joly
4. (D) Ernst Münch

Step 1: Units of Inductance and Resistance.

The inductance (L) is measured in henries (H), and the resistance (R) is measured in ohms (Ω).

The henry (H) is defined as:

1 H = 1 ohm-second (Ω·s).

Step 2: Derive the Unit of L/R.

The expression L/R has the unit:

Unit of L/R = (Unit of L) / (Unit of R) = henry / ohm.

Substituting 1 H = Ω·s:

L/R = (Ω·s) / Ω.

Simplify:

L/R = seconds (s).

Step 3: Final Answer.

The unit of L/R is:

Seconds (s).

Step 1: Calculate power required to lift the lift.

The force required to lift the lift is equal to its weight:

F = m × g = 250 × 9.8 = 2450 N

The power required is given by:

P = F × v = 2450 × 0.20 = 490 W

Step 2: Convert to horsepower.

Since 1 hp = 746 W, the required horsepower is:

Power in hp = 490 / 746 ≈ 0.66 hp

Step 1: Horizontal range of a projectile.

The horizontal range (R) of a projectile is given by:

R = (u² × sin 2θ) / g

where u is the initial speed, θ is the angle of projection, and g is the acceleration due to gravity.

Step 2: Maximum horizontal range.

The maximum horizontal range occurs when sin 2θ = 1, i.e., θ = 45°:

R_max = u² / g

Step 3: Area of spread.

Assuming bullets are fired in all directions, the spread forms a circle with radius R_max. The area (A) is:

A = π × (R_max)² = π × (u² / g)²

Step 1: Definition of sympatric speciation.

Sympatric speciation refers to the process by which new species evolve from a single ancestral species while inhabiting the same geographic region. Unlike allopatric speciation, there is no physical separation between populations.

Step 2: Mechanism of sympatric speciation.

Sympatric speciation often occurs due to factors such as:

• Behavioral isolation
• Ecological isolation (e.g., habitat preferences within the same region)
• Polyploidy in plants
• Genetic divergence without physical barriers

Step 3: Elimination of incorrect options.

• (A) Incorrect: Geographic isolation is a characteristic of allopatric speciation.
• (C) Incorrect: Migration to a new habitat implies geographic separation, which is not sympatric.
• (D) Incorrect: Gradual accumulation of changes is part of the speciation process but does not define sympatric speciation specifically.

Step 1: Causative organism of typhoid fever.

Typhoid fever is caused by the bacterium Salmonella typhi. This bacterium is transmitted through contaminated food and water.

Step 2: Symptoms of typhoid fever.

Symptoms include:

• High fever
• Weakness
• Abdominal pain
• Headaches
• Constipation or diarrhea

Step 3: Elimination of incorrect options.

• (B) Incorrect: Mosquitoes transmit diseases like malaria and dengue, not typhoid.
• (C) Incorrect: Plasmodium is a parasite responsible for malaria.
• (D) Incorrect: Nicotine is a chemical compound and not a causative agent of infectious diseases.

Step 1: Understanding the transpiration pull theory.

The transpiration pull theory, also known as the cohesion-tension theory, explains the movement of water from roots to leaves in tall plants. It attributes this upward movement to:

• Cohesion of water molecules (water-water attraction)
• Adhesion of water molecules to xylem walls
• Negative pressure (tension) generated due to transpiration from leaves

Step 2: Contribution of Ernst Münch.

Ernst Münch is credited with proposing the cohesion-tension theory. His work highlighted the role of transpiration in generating a pulling force and the cohesive nature of water in enabling the upward flow through the xylem vessels.

Step 3: Elimination of incorrect options.

• (A) Incorrect: Charles Darwin and Francis Darwin contributed to understanding plant tropisms but not the cohesion-tension theory.
• (B) Incorrect: Stephen Hales made significant contributions to plant physiology, especially in water uptake, but did not propose this theory.
• (C) Incorrect: Henry Dixon and John Joly are associated with early explanations of water transport but are not credited with the cohesion-tension theory.

Step 1: Understanding symbiosis and its examples.

Symbiosis refers to a close and long-term biological interaction between two different biological organisms, which can be mutualistic, commensalistic, or parasitic. The question asks for an example where the relationship is not a typical case of symbiosis.

• (A): Orchid and the tree on which it grows represent commensalism. Orchids grow on trees to access sunlight but do not harm the tree.
• (B): Cattle egret and grazing cattle represent mutualism, as cattle stir up insects for the egret to feed on.
• (C): Sea anemone and clownfish is mutualistic. Clownfish get protection from predators, and anemones get nutrients from the waste of the clownfish.
• (D): Female wasp and fig species is not a general symbiosis example but represents a specialized mutualistic relationship for reproduction (pollination).

Step 2: Conclusion.

While all options describe ecological interactions, (D) is a highly specific reproductive mutualism case, not a common example of general symbiosis. Hence, it is the correct answer.

Step 1: Understanding circulatory systems.

Arthropods, such as insects, crustaceans, and arachnids, have an open circulatory system. This means that their blood (or hemolymph) is not confined to blood vessels but instead bathes the internal organs directly in an open cavity called the hemocoel.

Step 2: Explanation of other options.

• (B): Closed circulatory system is found in vertebrates and some invertebrates (e.g., earthworms) where blood flows through a network of vessels.
• (C): No circulatory system is characteristic of small organisms like flatworms and cnidarians, which rely on diffusion.
• (D): Diffusion is essential in small organisms but not the primary mode of circulation in arthropods.

Step 3: Conclusion.

The open circulatory system is the hallmark of arthropods for transporting nutrients and gases.

Step 1: Understanding cytoplasmic streaming.

Cytoplasmic streaming, also known as cyclosis, is the movement of the cytoplasm within a cell to distribute nutrients, organelles, and other substances more effectively. This process is facilitated by the cytoskeleton, specifically actin filaments and motor proteins like myosin.

Step 2: Explanation of other options.

• (A) Endocytosis: This is the process by which cells engulf substances into vesicles.
• (B) Exocytosis: This is the process of expelling substances from the cell.
• (C) Cytokinesis: This refers to the division of the cytoplasm during cell division.

Step 3: Conclusion.

Cytoplasmic streaming is essential for intracellular transport and is especially prominent in large cells, such as plant cells.

Step 1: Analyze the assertion.

The statement “Insects are important pollinators for many flowering plants” is correct because many plants rely on insects to transfer pollen, which is essential for reproduction and seed formation.

Step 2: Analyze the reasoning.

The reasoning explains that insects visit flowers for nectar or pollen. During this process, they facilitate the transfer of pollen between flowers, enabling cross-pollination. This reasoning directly supports the assertion.

Step 3: Verify the connection between the assertion and reasoning.

The reasoning not only explains the assertion but also provides the biological basis for the statement. Hence, both the assertion and reasoning are correct, and the reasoning correctly explains the assertion.

Step 1: Overview of the Krebs cycle.

The Krebs cycle, also known as the citric acid cycle, is a crucial metabolic pathway in cellular respiration. It takes place in the mitochondrial matrix and is responsible for oxidizing acetyl-CoA to carbon dioxide while reducing NAD⁺ and FAD to NADH and FADH₂, which are used in the electron transport chain to generate ATP.

Step 2: Oxidation events in the Krebs cycle.

Oxidation occurs in the following steps of the Krebs cycle:

• Isocitrate is oxidized to α-ketoglutarate, reducing NAD⁺ to NADH.
• α-ketoglutarate is oxidized to succinyl-CoA, reducing NAD⁺ to NADH.
• Succinate is oxidized to fumarate, reducing FAD to FADH₂.
• Malate is oxidized to oxaloacetate, reducing NAD⁺ to NADH.

Hence, oxidation occurs four times in the Krebs cycle.

Step 1: Role of water in the Krebs cycle.

Water molecules are involved at two specific points in the Krebs cycle:

• Fumarate is hydrated to malate by the enzyme fumarase, consuming one water molecule.
• Citrate is formed from oxaloacetate and acetyl-CoA, requiring water for the reaction catalyzed by citrate synthase.

Although water is consumed in the cycle, two water molecules are released as byproducts during different enzymatic reactions.

Step 1: Understanding the life cycle of Plasmodium.

When a female Anopheles mosquito bites a human, it injects the sporozoites into the bloodstream. Sporozoites are the infective stage of the Plasmodium parasite.

Step 2: Role of sporozoites.

The sporozoites travel to the liver cells, where they multiply asexually and form merozoites. This marks the beginning of the parasite’s life cycle within the human host:

Sporozoite → Liver → Merozoite (blood stage)

Step 3: Key Point.

The sporozoite phase is the entry stage of the parasite into the human body, making it the correct answer.

Step 1: Understanding the male reproductive system.

The male reproductive system is primarily responsible for the production of male gametes (sperm) and hormones like testosterone. The testes are the key structures where spermatogenesis (the production of sperm) occurs.

Step 2: Spermatogenesis in the testes.

• Spermatogenesis occurs in the seminiferous tubules within the testes.
• These tubules are lined by germ cells and Sertoli cells, which aid in the development of sperm.

Step 3: Explanation of other options.

• (A) Prostate gland: Produces seminal fluid that nourishes and protects sperm.
• (B) Seminal vesicles: Contribute to the production of seminal fluid.
• (C) Epididymis: Stores and matures sperm but does not produce it.

Hence, the correct answer is (D) Testes.

Step 1: Understanding double fertilization.

Double fertilization is a unique process in angiosperms where two fertilization events occur simultaneously.

Step 2: The two events in double fertilization.

• One sperm cell fuses with the egg cell to form the zygote, which develops into the embryo.
• The other sperm cell fuses with the two polar nuclei in the central cell to form a triploid structure that develops into the endosperm.

Step 3: Explanation of other options.

• (B): Incorrect because the endosperm is formed from the polar nuclei, not the egg cell.
• (C): Incorrect because the seed formation involves the zygote and maternal tissues, not directly from the sperm cell.
• (D): Incorrect because the polar nuclei form the endosperm, not the zygote.

Thus, the correct sequence of events is described in (A).

Solution:

The IUPAC name for acetone is derived from its molecular structure:

• Acetone has the molecular formula C₃H₆O.
• It contains a ketone functional group (C=O) located at the second carbon atom.

Hence, the correct IUPAC name is Propan-2-one.

Solution:

Glyceraldehyde is named based on its structure:

• It is a three-carbon molecule (C₃H₆O₃) with two hydroxyl groups (-OH) attached to the second and third carbons.
• The aldehyde group (-CHO) is located at the first carbon atom.

Thus, the correct IUPAC name is 2,3-dihydroxypropanal.

Step 1: Determine the order of reaction.

The problem involves percentages of completion and time, which suggests a first-order reaction. The formula for the time required to achieve a certain completion in a first-order reaction is:

t = (2.303 / k) × log([A]₀ / [A]),

where:

• t is the time,
• k is the rate constant,
• [A]₀ is the initial concentration,
• [A] is the concentration at time t.

Step 2: Calculate the rate constant k.

For 50% completion, [A]₀ / [A] = 2. Substituting t = 15 minutes:

15 = (2.303 / k) × log(2).

k = (2.303 × log(2)) / 15.

Using log(2) = 0.3010:

k = (2.303 × 0.3010) / 15 = 0.04627 min^-1.

Step 3: Calculate the time for 90% completion.

For 90% completion, [A]₀ / [A] = 10. Substituting into the formula:

t = (2.303 / k) × log(10).

Using log(10) = 1:

t = (2.303 / 0.04627) × 1 = 49.44 minutes.

Step 4: Final Answer.

The time required to complete 90% of the reaction is 49.44 minutes.

Step 1: Determine the number of unpaired electrons.

The electronic configuration of Mn²⁺ is:

[Ar] 3d⁵

This indicates 5 unpaired electrons in the 3d subshell.

Step 2: Use the formula for magnetic moment.

The magnetic moment (µ) is given by:

µ = √n(n + 2) BM,

where n is the number of unpaired electrons.

µ = √5(5 + 2) = √35 = 5.9 BM.

Final Answer:

The magnetic moment of Mn²⁺ is 5.9 BM.