WBJEE Sample Papers, Section Wise & Full Physics Test - 1 Free Online Test

Given:

Mass of the matter = 1 mg = 1 × 10^-6 kg

Speed of light (C) = 3 × 10⁸ m/s

Using the formula: E = mc²

Substitute the given values:

E = 1 × 10^-6 × (3 × 10⁸)²

E = 1 × 10^-6 × 9 × 10¹⁶

E = 9 × 10¹⁰ J

The energy equivalent of the given mass is 9 × 10¹⁰ joules.

Ammeter has it's internal resistance, due to which resistance of system inscreases I∝1/R_eq so current reading decreases that original.
In general, ammeter always reads less than the actual value because of its  Resistance.
Actual current will be Greater than 1.85 A

To find the area of cross-section of the second wire, we can use the relationship between elongation, load, cross-sectional area, and material properties. The elongation of a wire is influenced by its material properties and cross-sectional area. Given the elongations and the area of the first wire, we can set up a proportion based on the formula for elongation.

Let:

• ΔL₁ = 0.1 mm (elongation of the first wire)
• ΔL₂ = 0.05 mm (elongation of the second wire)
• A₁ = 4 mm² (area of the first wire)
• A₂ = ? (area of the second wire)

Since both wires are under the same load, we can express the ratio of their elongations as follows:

• ΔL₁ / ΔL₂ = A₂ / A₁

Substituting the known values:

• 0.1 / 0.05 = A₂ / 4

This simplifies to:

• 2 = A₂ / 4

From this, we can calculate:

• A₂ = 2 * 4
• A₂ = 8 mm²

Thus, the area of cross-section of the second wire is 8 mm².

To understand the dependency of various quantities on the orbital radius of a satellite, it is essential to analyse each option carefully.

• T/R: This ratio depends on both the period (T) and the radius (R) of the orbit. Changes in radius will affect this quantity.
• T²/R: This quantity also varies with changes in the orbital radius, as it includes both the square of the period and the radius.
• T²/R²: Similar to the above, this ratio changes with the orbital radius. Both the square of the period and the square of the radius are involved.
• T²/R³: This quantity is special because it reflects a relationship that does not change with the radius. It is constant regardless of the orbital radius.

In conclusion, among these options, only T²/R³ does not depend on the orbital radius of the satellite.

100cm=1m

a=v²/r

v= 2πr/t

v=6.28/2

v= 3.14

a= 9.8596/1

a=9.8596 m/s²

Charge on first capacitor = q₁ = C₁₁
Charge on second capacitor = q₂ = 0
When they are connected, in parallel the total charge
q = q₁ + q₂
∴ q = C₁₁
and capacitance, C = C₁ + C₂
Let V' be the common potential difference across each capacitaor, then q = CV'

To provide electromagnetic dampening.

The coil is wound on a metal frame to instantly bring it to rest. In turn, this makes it possible for the coil to very immediately reach its equilibrium position. In the metal frame, eddy currents form. The coil cannot go any further due to these eddy currents. The galvanometer is referred to as a dead beat galvanometer because the oscillation of the coil abruptly ceases.

Restoring Torque on a Suspended Bar Magnet

The restoring torque acting on a bar magnet when it is displaced from its equilibrium position in a uniform magnetic field can be explained as follows:

• The torque is proportional to the magnetic moment (μ) of the magnet.
• It also depends on the strength of the magnetic field (B).
• The angle of deflection (θ) influences the torque through the sine function.

Thus, the formula for the restoring torque (τ) is given by:

τ = μB sin(θ)

This indicates that:

• The torque increases with a larger magnetic moment.
• A stronger magnetic field also results in greater torque.
• The torque is maximised when the angle θ is 90 degrees, as sin(90°) = 1.

In summary, the restoring torque on the magnet, when displaced, is directly related to its magnetic moment, the magnetic field strength, and the sine of the angle of deflection.

The coil in the tangent galvanometer must be kept

• In the magnetic meridian: This is the direction aligned with the Earth's magnetic field.

• Perpendicular to the magnetic meridian: This orientation would not allow accurate readings.

• In any direction: This would lead to incorrect measurements.

• Need not be rotated in any direction: This implies it can remain stationary, which is not correct.

To ensure accurate measurements, the coil must be positioned in the magnetic meridian.

To determine the ratio of terminal velocities for two mercury drops with radii in the ratio of 3:4, we can use the relationship between the radius of a sphere and its terminal velocity in a viscous medium.

The terminal velocity (v) of a sphere falling through a viscous fluid is given by Stokes' law, which can be expressed as:

• v ∝ r²

Where:

• v is the terminal velocity
• r is the radius of the sphere

Given the radii of the drops are in the ratio of 3:4, we can express this as:

• Let the radius of the first drop be 3x
• Let the radius of the second drop be 4x

Now, applying Stokes' law:

• For the first drop: v₁ ∝ (3x)² = 9x²
• For the second drop: v₂ ∝ (4x)² = 16x²

To find the ratio of their terminal velocities:

• v₁ : v₂ = 9x² : 16x²
• This simplifies to 9 : 16

Hence, the ratio of their terminal velocities is 9:16.

Explanation:

Given:
- x = 7t + 4t^2
- y = 5t

Acceleration of the particle can be found by taking the second derivative of the position function with respect to time.

First, find the velocity components:
- Vx = dx/dt = d(7t + 4t^2)/dt = 7 + 8t
- Vy = dy/dt = d(5t)/dt = 5

Now, find the acceleration components:
- Ax = dVx/dt = d(7 + 8t)/dt = 8
- Ay = dVy/dt = d(5)/dt = 0

The total acceleration is given by:
- A = sqrt[(Ax)^2 + (Ay)^2] = sqrt[(8)^2 + (0)^2] = 8 m/s^2

Therefore, the acceleration of the particle at t=5s is 8 m/s^2, which corresponds to option B.

To determine the height at which the gun should be aimed, we need to consider the physics of projectile motion. Here’s a simplified breakdown of the calculation:

• The bullet is fired with an initial velocity of 1000 m/s.
• The horizontal distance to the target is 100 m.
• We assume no air resistance and use the acceleration due to gravity, g = 10 m/s².

First, we calculate the time it takes for the bullet to reach the target:

• Time (t) = Distance / Velocity = 100 m / 1000 m/s = 0.1 s.

Next, we calculate the vertical displacement (height) of the bullet during this time:

• Vertical displacement (h) can be calculated using the formula: h = 0.5 × g × t².
• Substituting the values: h = 0.5 × 10 m/s² × (0.1 s)².
• This simplifies to: h = 0.5 × 10 × 0.01 = 0.05 m.
• Converting this to centimetres: 0.05 m = 5 cm.

Therefore, the gun should be aimed at a height of 5 cm above the target.

Centripetal acceleration describes how an object moves in a circular path. It depends on the object's speed and the radius of the circle.

The formula for centripetal acceleration is:

• Acceleration increases with the square of the speed.
• A larger radius results in less acceleration.
• It is calculated using the formula: v² / r,
• where v is the speed and r is the radius.

This means that if an object's speed doubles, the centripetal acceleration will increase by a factor of four. Conversely, if the radius of the path increases, the acceleration decreases.

According to radioactive decay, 

Divide (i) by (ii), we get

When the length of a simple pendulum is increased by 1%, it affects its time period. The time period of a pendulum is given by the formula:

• The time period T is proportional to the square root of the length L: T ∝ √L.
• A small increase in length leads to a proportional increase in the time period.

Specifically:

• If the length increases by 1%, the new length becomes 1.01L.
• The new time period can be calculated as follows:
• New time period: T' = K√(1.01L) where K is a constant.
• Using the approximation for small changes, we find that:
• The increase in time period is about 0.5% when the length is increased by 1%.

Therefore, the time period increases by approximately 0.5%.

When a portion of a spring is removed, its force constant changes. In this case, one-fourth of the spring is cut away, affecting the overall stiffness of the remaining part.

To understand this, consider the following points:

• The original force constant, represented as K, indicates how stiff the spring is.
• When one-fourth of the spring is cut, three-fourths of the spring remains.
• The force constant of a spring is inversely proportional to its length. Therefore, a shorter spring will be stiffer.
• The new force constant can be calculated using the formula for the remaining spring's length. As the length decreases, the force constant increases.
• The relation shows that the new force constant of the remaining spring is given by 4K/3.

This change indicates that the stiffness of the remaining spring increases when a portion is removed, resulting in a new force constant of 4K/3.

Moment of inertia of solid cylinder about O,

Moment of inertia about point of contact P,

The continuous emission spectrum is characterised by a smooth range of frequencies without any gaps. This spectrum displays all the wavelengths of light emitted by a source, such as a hot solid or gas, allowing it to cover a wide frequency range.

In contrast, the other types of spectra can be described as follows:

• Line spectrum: Comprises distinct lines representing specific frequencies emitted by atoms or molecules. Each line corresponds to a particular energy transition.
• Band spectrum: Features a series of closely spaced lines, which appear as bands due to the transitions between vibrational levels in molecules.
• Absorption spectrum: Results from specific wavelengths being absorbed by a cooler gas, creating dark lines in the spectrum that indicate the missing frequencies.

Among these spectra, the continuous emission spectrum exhibits the highest frequency range, gradually decreasing to the lower frequency spectra like the line and band spectra.

In physics, when objects roll down an inclined plane, their shape and mass distribution affect their acceleration and speed.

In this scenario, we have three objects: a solid sphere, a disc, and a solid cylinder. All three objects have the same mass and are made from the same material, which ensures that their rolling behaviour is determined mainly by their shapes.

• Solid Sphere: The solid sphere has a lower moment of inertia compared to the other two shapes. This means it rotates more efficiently as it rolls down the incline.
• Disc: The disc has a higher moment of inertia than the sphere but lower than that of the solid cylinder. It will roll down the incline at a moderate pace.
• Solid Cylinder: The solid cylinder has the highest moment of inertia, causing it to roll down the incline more slowly than both the sphere and the disc.

Thus, the order in which these objects reach the bottom of the inclined plane is primarily influenced by their shape and mass distribution. Since the solid sphere has the most efficient rolling motion, it will reach the bottom first.

In forward biasing of a junction diode, the width of the depletion layer changes significantly. When a diode is forward-biased, the following occurs:

• The depletion layer narrows, allowing more charge carriers to flow through.
• This results in an increase in current across the diode.
• The decrease in width enhances the diode's conductivity.

In summary, forward biasing effectively reduces the size of the depletion layer, facilitating the flow of current. This principle is crucial for the operation of diodes in various electronic applications.

In a p- type semiconductor, the majority charge carriers are the holes whereas in the n- type semiconductor, the majority charge carriers are the free electrons.

Thermal capacity is the amount of heat required to raise the temperature of a body by 1°C. It is an important concept in thermodynamics and relates to how materials respond to heat. Below are key points about thermal capacity:

• Definition: Thermal capacity refers to the total heat energy needed to change the temperature of an object.
• Units: It is usually measured in joules per degree Celsius (J/°C).
• Factors: The thermal capacity of a material depends on its mass and specific heat, which is the heat required to raise the temperature of a unit mass by 1°C.
• Applications: Understanding thermal capacity is essential in various fields, including engineering, meteorology, and environmental science.

In summary, thermal capacity is a fundamental property that indicates how much heat a material can store and its ability to change temperature.

To find the area of a metal sheet, we multiply its length and breadth.

• Length: 3.124 m
• Breadth: 3.002 m

Calculating the area:

• Area = Length × Breadth
• Area = 3.124 m × 3.002 m
• Area = 9.373248 m²

To express this value with four significant figures, we round it:

• The first four significant figures of 9.373248 are 9.378.
• Thus, the area is 9.378 m².

The Doppler effect describes how the frequency of sound or light waves changes depending on the relative motion between the source and the observer.

Key differences in how the Doppler effect applies to sound and light include:

• Symmetry: The Doppler effect for sound is symmetrical, meaning that the frequency changes are similar regardless of whether the source is moving towards or away from the observer.
• Asymmetry in light: In contrast, the Doppler effect for light is asymmetrical. This is due to the finite speed of light and the way it impacts our perception based on relative motion.
• Speed of propagation: Sound requires a medium (like air) to travel, whereas light can move through a vacuum, leading to different effects based on movement.
• Relative speed: For sound, the changes in frequency are more pronounced at lower speeds compared to light, where even high speeds result in relatively small shifts.

In summary, while both sound and light exhibit the Doppler effect, they do so in fundamentally different ways, highlighting their unique characteristics.

To find the unknown resistance X, we can use the properties of a meter bridge. Initially, the balance point is at 55 cm, indicating a specific relationship between X and the known resistance R.

• The balance point formula for a meter bridge is given by: X / R = (L1) / (L2), where L1 is the distance from the left end (55 cm) and L2 is the remaining distance (100 cm - 55 cm = 45 cm).
• From the initial setup, we have: X / R = 55 / 45.
• When the resistance R is increased by 5 Ω, the new balance point shifts to 60 cm. The new relationship is: X / (R + 5) = 60 / 40.

Now we have two equations:

• 1. X = (R × 55) / 45
• 2. X = (R + 5) × (60 / 40)

By equating both expressions for X, we can solve for R:

• From the first equation, we find: X = 1.222 R
• From the second equation, we get: X = 1.5(R + 5)

Setting these equal gives:

• 1.222 R = 1.5(R + 5)

Solving this equation:

• 1.222 R = 1.5R + 7.5
• 0.278 R = 7.5
• R = 26.98 Ω (approximately 27 Ω)

Now substituting R back into the expression for X:

• X = 1.222 × 27 = 33 Ω (approximately).

However, in the given options, the closest value for X is 27.5 Ω.