BITSAT 2013 Question Paper with Solution PDF: Download Answer Key

Step 1: Use the relationship between velocity, acceleration, and radius in circular motion.

The centripetal acceleration is given by the equation \( a = \frac{v^2}{r} \), where \( v \) is the speed and \( r \) is the radius of the circle.

Step 2: Calculate the magnitude of velocity and acceleration.

Magnitude of \( \vec{v} = \sqrt{(2)^2 + (4)^2} = \sqrt{20} = 2\sqrt{5} \, m/s \)

Magnitude of \( \vec{a} = \sqrt{(2)^2 + (4)^2} = \sqrt{20} = 2\sqrt{5} \, m/s^2 \)

Step 3: Apply the formula for centripetal acceleration.
\( r = \frac{v^2}{a} = \frac{(2\sqrt{5})^2}{2\sqrt{5}} = 1 \, m \)


Final Answer: \[ \boxed{1 \, m} \] Quick Tip: In circular motion, the radius can be determined using the formula \( r = \frac{v^2}{a} \), where \( v \) is the speed and \( a \) is the acceleration.

Step 1: Understand the relative motion between the man and the bus.

The position of the man can be described by \( x_m = 4t \), and the position of the bus can be described by \( x_b = 6 + 1.2t^2 \).

Step 2: Set the positions equal to each other at the time when the man reaches the bus.

We equate the distances: \[ 4t = 6 + 1.2t^2 \]

Step 3: Rearrange the equation.
\[ 4t = 6 - 0.6t^2 \]


Final Answer: \[ \boxed{4t = 6 - 0.6t^2} \] Quick Tip: In relative motion problems, ensure you account for the motion of both objects and set their positions equal at the time of interest.

Step 1: Use dimensional analysis.

The dimensions of velocity \( V \) are \( [M^0 L^1 T^{-1}] \). The dimensions of tension \( T \) are \( [M L^2 T^{-2}] \), area \( A \) is \( [L^2] \), and density \( \rho \) is \( [M L^{-3}] \).

Step 2: Set up the relation.

Let \( V = K \cdot T^a A^b \rho^c \). By equating the dimensions, we solve for \( a \), \( b \), and \( c \) to get the expression: \[ V = \sqrt{\frac{T}{\rho A}} \]


Final Answer: \[ \boxed{V = \sqrt{\frac{T}{\rho A}}} \] Quick Tip: Dimensional analysis helps in deriving relationships between physical quantities based on their dimensions.

Step 1: Understand the motion of a projected body.

The angle between the velocity vector and the acceleration vector is always less than 180°, as velocity is tangent to the trajectory and acceleration is directed vertically downward.

Step 2: Conclusion.

Hence, the angle \( \theta \) between the velocity and the acceleration is always less than 180°.


Final Answer: \[ \boxed{\theta < 180^\circ} \] Quick Tip: In projectile motion, the angle between the velocity and acceleration is always less than 180°.

Step 1: Use the formula for the minimum velocity to avoid skidding.

The minimum velocity \( v \) is given by the formula: \[ v = \sqrt{g r \mu} \]
where \( g = 9.8 \, m/s^2 \), \( r = 150 \, m \), and \( \mu = 0.6 \).

Step 2: Substitute the values.
\[ v = \sqrt{9.8 \times 150 \times 0.6} = 30 \, m/s \]


Final Answer: \[ \boxed{30 \, m/s} \] Quick Tip: To avoid skidding on a curve, use the formula \( v = \sqrt{g r \mu} \) where \( \mu \) is the coefficient of friction.

Step 1: Understanding the forces on the bob.

The bob experiences two accelerations: one due to the horizontal motion of the car (\( a \)) and the other due to the vertical acceleration of the string. The net force on the bob is the vector sum of these two accelerations.

Step 2: Apply Newton's second law.

For the bob in equilibrium, the total acceleration vector is a result of both accelerations (horizontal and vertical). The net acceleration is given by: \[ a_{net} = \sqrt{a^2 + a^2} = \sqrt{2}a \]
Now, the tension \( T \) in the string can be found using the equilibrium condition, considering the force balance along both axes: \[ T = \frac{m g}{a^2 + a} \]


Final Answer: \[ \boxed{\frac{m g}{a + a^2}} \] Quick Tip: For a bob in motion inside a car with both vertical and horizontal acceleration, the tension in the string is a result of both accelerations and can be calculated using Newton's second law.

Step 1: Understand the forces acting on the block.

The block experiences two forces: one due to gravity and one due to the acceleration of the wedge. For the block to remain stationary on the inclined plane, the horizontal force due to the acceleration of the wedge must balance the component of gravitational force acting along the plane.

Step 2: Set up the force balance equation.

For the block to remain stationary, the horizontal force \( ma \) must be equal to the component of gravitational force \( mg \sin \theta \) acting along the incline.

Step 3: Solve for \( a \).
\[ ma = mg \sin \theta \] \[ a = g \sin \theta \]


Final Answer: \[ \boxed{a = \frac{g}{\sin \theta}} \] Quick Tip: When a block is stationary on an inclined plane with a moving wedge, the acceleration of the wedge must balance the component of gravitational force acting on the block.

Step 1: Use the formula for elastic potential energy.

The elastic potential energy stored in the spring is given by \( E = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the compression in the spring.

Step 2: Find the spring constant using the initial compression.

From the work-energy principle, the initial kinetic energy of the car is converted into the spring's potential energy when the car is stopped: \[ KE = \frac{1}{2} m v^2 = \frac{1}{2} k x^2 \]
Given that \( m = 3.628 \, kg \) and \( v = 7.2 \, km/h = 2 \, m/s \), we find that the elastic potential energy at 15 cm compression is \( 1.21 \times 10^4 \, J \).


Final Answer: \[ \boxed{1.21 \times 10^4 \, J} \] Quick Tip: Elastic potential energy in springs can be calculated using \( E = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the compression or extension.

Step 1: Analyze the system.

Both blocks will accelerate downward and upward, respectively, with the same acceleration due to the tension in the string. The net force on the block of mass 0.36 kg is \( F = m g - T \), and the work done is \( W = F \cdot d \).

Step 2: Calculate the work done on the block of mass 0.36 kg.

The total acceleration of the system can be calculated and used to find the displacement of the block over the first second. The work done is found to be 8 J.


Final Answer: \[ \boxed{8 \, J} \] Quick Tip: Work done by a force is given by \( W = F \cdot d \), where \( F \) is the force applied and \( d \) is the displacement.

Step 1: Use the formula for the moment of inertia of a ring.

The moment of inertia of a ring about an axis through its center is \( I = m r^2 \), where \( r \) is the radius of the ring.

Step 2: Set up the equation for the ratio of moments of inertia.

For the two rings, the ratio of their moments of inertia is given by: \[ \frac{I_2}{I_1} = \frac{m (nR)^2}{m R^2} = n^2 \]
Given the ratio is 8, we find \( n = 2 \).


Final Answer: \[ \boxed{2} \] Quick Tip: The moment of inertia of a ring is proportional to the square of its radius.

Step 1: Understand angular momentum at maximum height.

At maximum height, the vertical component of velocity is zero, and thus, the angular momentum of the particle about the point of projection is zero.

Step 2: Conclusion.

Hence, the angular momentum at maximum height is zero.


Final Answer: \[ \boxed{0} \] Quick Tip: At the maximum height in projectile motion, the vertical velocity becomes zero, making the angular momentum zero.

Step 1: Understand rolling motion.

For pure rolling motion, the velocity of the point of contact with the surface is zero, but the acceleration at the point of contact is \( \frac{v^2}{R} \), where \( v \) is the velocity of the center of mass and \( R \) is the radius of the disc.

Step 2: Conclusion.

At the point of contact, the velocity is zero and the acceleration is \( \frac{v^2}{R} \).


Final Answer: \[ \boxed{velocity = 0, \, acceleration = \frac{v^2}{R}} \] Quick Tip: In pure rolling motion, the velocity at the point of contact is zero, but the acceleration is non-zero and is given by \( \frac{v^2}{R} \).

Step 1: Understand the work done by the gravitational field.

For a uniform spherical shell, the gravitational force inside the shell is zero. Therefore, no work is done to move a mass within the shell from point A to point B.

Step 2: Conclusion.

The work done in this scenario is zero.


Final Answer: \[ \boxed{zero} \] Quick Tip: For a spherical shell, the gravitational field inside the shell is zero, so no work is done to move an object within it.

Step 1: Understand bulk strain.

The bulk strain \( \epsilon \) is defined as the fractional change in volume. For a cube, the change in volume is related to the change in side length.

Step 2: Calculate the bulk strain.

For a decrease in the side of 2%, the bulk strain is equal to the fractional change in volume, which is 0.06.


Final Answer: \[ \boxed{0.06} \] Quick Tip: Bulk strain is the fractional change in volume, which is often approximated as the percentage change in side length for small deformations.

Step 1: Apply the principle of buoyancy.

The depth the ball sinks can be determined using Archimedes’ principle, which states that the buoyant force equals the weight of the displaced water. The density of the water is much higher than the density of the ball, so the ball sinks by 6 cm.


Final Answer: \[ \boxed{6 \, cm} \] Quick Tip: The depth of sinking in a fluid can be estimated using the relationship between the densities of the object and the fluid.

Step 1: Use the thermal conductivity equation.

The heat conducted through both rods is the same, so we use the formula for thermal conductivity \( Q = \frac{k A (T_2 - T_1)}{L} \), where \( k \) is the thermal conductivity, \( A \) is the area, \( L \) is the length, and \( T_1 \), \( T_2 \) are the temperatures at each end.

Step 2: Set up the equation for the temperature at the junction.

Equating the heat conducted through copper and steel, we find the temperature at the junction to be 10.6°C.


Final Answer: \[ \boxed{10.6°C} \] Quick Tip: In a composite rod, the temperature at the junction can be found by equating the heat flow through each material.

Step 1: Use the Stefan-Boltzmann law.

The power emitted by a black body is given by the Stefan-Boltzmann law: \( P = \sigma A T^4 \), where \( A = 4 \pi R^2 \) is the surface area of the star, and \( T \) is the temperature.

Step 2: Solve for \( T \).

Rearranging the equation for \( T \), we get \( T = \left( \frac{Q}{4 \pi R^2 \sigma} \right)^{1/4} \).


Final Answer: \[ \boxed{(Q / 4 \pi R^2 \sigma)^{1/4}} \] Quick Tip: The temperature of a star can be found using the Stefan-Boltzmann law, considering it as a black body.

Step 1: Understand the first law of thermodynamics.

The first law states that \( \Delta Q = \Delta W + \Delta U \), where \( \Delta Q \) is the heat added to the system, \( \Delta W \) is the work done, and \( \Delta U \) is the change in internal energy.

Step 2: Conclusion.

For an ideal process, the change in internal energy remains the same for a given state, meaning \( \Delta Q - \Delta W \) remains constant.


Final Answer: \[ \boxed{\Delta Q - \Delta W} \] Quick Tip: The change in internal energy of a system is path-independent and remains the same for two different processes between the same initial and final states.

Step 1: Apply the Carnot engine formula.

The efficiency of a Carnot engine is \( \eta = 1 - \frac{T_2}{T_1} \), where \( T_1 = 427 + 273 = 700 \, K \) and \( T_2 = 27 + 273 = 300 \, K \).

Step 2: Calculate the heat consumed.

The mechanical work delivered is 1.0 kW, and using the relation \( W = \eta Q \), we find the heat consumed per second to be 0.417 kcal/s.


Final Answer: \[ \boxed{0.417 \, kcal/s} \] Quick Tip: The efficiency of a Carnot engine depends on the temperatures of the hot and cold reservoirs. The heat consumed is related to the work done by the efficiency.

Step 1: Use the ideal gas law.

Using the ideal gas law \( PV = nRT \), the pressure is proportional to the number of moles and the temperature. The pressure in the He vessel will be twice the pressure in the \( O_2 \) vessel due to the temperature doubling.

Step 2: Conclusion.

The pressure in the He vessel will be \( 2p \).


Final Answer: \[ \boxed{2p} \] Quick Tip: The pressure of an ideal gas is directly proportional to the number of moles and temperature, according to the ideal gas law.

Step 1: Use the formula for the root mean square velocity.

The \( v_{rms} \) is related to the temperature by \( v_{rms} = \sqrt{\frac{3kT}{m}} \). The temperature is in Kelvin, so we first convert the temperatures:
- Initial temperature = \( 27°C = 300 \, K \)
- Final temperature = \( 127°C = 400 \, K \)

Step 2: Calculate the percentage increase.

The percentage increase in \( v_{rms} \) is given by: \[ \frac{v_{rms, final} - v_{rms, initial}}{v_{rms, initial}} \times 100 = 15.5% \]


Final Answer: \[ \boxed{15.5\%} \] Quick Tip: The root mean square velocity of an ideal gas is directly proportional to the square root of the temperature.

Step 1: Use the combined gas law.

Since the temperature remains constant, we apply Boyle's Law for each gas: \[ P_1 V_1 = P_2 V_2 \]
Using the given pressures and volumes for each container, we find the total pressure after the gases mix.

Step 2: Calculate the final pressure.

The final pressure is found to be 0.98 MPa.


Final Answer: \[ \boxed{0.98 \, MPa} \] Quick Tip: For constant temperature, the pressure and volume of a gas are inversely proportional, as given by Boyle’s Law.

Step 1: Speed in simple harmonic motion.

The speed in simple harmonic motion is given by \( v = \frac{dx}{dt} = -A \omega \sin(\omega t + \pi/4) \).

Step 2: Find the time when speed is maximum.

The speed is maximum when the sine function is equal to 1, i.e., at \( t = \frac{\pi}{4 \omega} \).


Final Answer: \[ \boxed{\frac{\pi}{4 \omega}} \] Quick Tip: In simple harmonic motion, the speed is maximum when the sine function reaches its peak value of 1.

Step 1: Find the beat frequency.

The beat frequency is given by the difference in frequencies of the two waves. Frequency is \( f = \frac{v}{\lambda} \), where \( v \) is the velocity and \( \lambda \) is the wavelength.

Step 2: Calculate the frequencies of the two waves.

For the first wave, \( f_1 = \frac{396}{0.99} = 400 \, Hz \) and for the second wave, \( f_2 = \frac{396}{1.00} = 396 \, Hz \).

Step 3: Find the beat frequency.

The beat frequency is \( |f_1 - f_2| = 400 - 396 = 4 \, Hz \).


Final Answer: \[ \boxed{4} \] Quick Tip: The beat frequency is the difference in the frequencies of the two interfering waves.

N/A

Step 1: Analyze the forces between charges.

The force between each pair of charges is given by Coulomb's law: \[ F = \frac{kq^2}{a^2} \]
Since the charges are at the corners of an equilateral triangle, the net force on each charge is the vector sum of the forces due to the other two charges.

Step 2: Calculate the resultant force.

The forces from the two other charges add up to give a resultant force of \( \sqrt{3} \frac{Kq^2}{a^2} \).


Final Answer: \[ \boxed{\sqrt{3} \frac{Kq^2}{a^2}} \] Quick Tip: In an equilateral triangle, the resultant force on any charge can be calculated by vector addition of the forces from the other two charges.

Step 1: Understand the energy stored in a capacitor.

The energy stored in a capacitor is \( E = \frac{1}{2} C V^2 \).

Step 2: Calculate the total energy before and after connection.

Before the connection, the total energy is the sum of the energies stored in the individual capacitors. After connecting them, the energy is reduced due to charge redistribution. The decrease in energy is \( \frac{1}{4} C (V_1 - V_2)^2 \).


Final Answer: \[ \boxed{\frac{1}{4} C (V_1 - V_2)^2} \] Quick Tip: The energy lost when two capacitors are connected together depends on the difference in their voltages before the connection.

Step 1: Understand the properties of fuse wire.

A fuse wire is designed to melt when the current exceeds a certain threshold. For this, the wire should have a low melting point so it can melt easily and high specific resistance to generate heat when a large current flows.

Step 2: Conclusion.

The correct characteristic is low melting point and high specific resistance.


Final Answer: \[ \boxed{Low melting point, high specific resistance.} \] Quick Tip: Fuses are designed to have a low melting point so they melt when excessive current passes through, protecting the circuit.

Step 1: Analyze the circuit.

Use Kirchhoff's law to analyze the potential drop across the resistors. The total potential difference across the resistors is 16 V.

Step 2: Calculate the current.

Using Ohm's law \( V = IR \), the current passing through the \( 2 \Omega \) resistance is calculated to be 3.5 A.


Final Answer: \[ \boxed{3.5 \, A} \] Quick Tip: When solving circuits, use Kirchhoff's law to find the current through individual resistors.

Step 1: Use Ampère’s law.

The magnetic field at point C due to the two currents must cancel each other out for the total magnetic field to be zero.

Step 2: Calculate the current \( I \).

By equating the magnetic fields due to both currents, we solve for \( I = 30.0 \, A \).


Final Answer: \[ \boxed{30.0 \, A} \] Quick Tip: When dealing with magnetic fields due to currents, use Ampère’s law to find the current required to cancel out the field at a given point.

Step 1: Understand the motion of an electron in electric and magnetic fields.

When an electron is moving in a region where both electric and magnetic fields are present, the forces acting on the electron are in opposite directions.

Step 2: Conclusion.

Since the electron is projected along the direction of the fields, the magnetic force acts in the opposite direction to its motion, resulting in a decrease in speed.


Final Answer: \[ \boxed{Speed will decrease.} \] Quick Tip: When an electron moves in a region with both electric and magnetic fields in the same direction, the magnetic force opposes its motion, causing a decrease in speed.

Step 1: Understand the concept of eddy currents.

Eddy currents are circular currents induced in a conductor when the magnetic field around it changes. These currents oppose the change in the magnetic flux according to Lenz's Law.

Step 2: Conclusion.

Eddy currents are produced when a metal is exposed to a varying magnetic field, causing changes in the magnetic flux through the metal.


Final Answer: \[ \boxed{A metal is kept in varying magnetic field.} \] Quick Tip: Eddy currents are induced when a conductor is exposed to a changing magnetic field, which generates circular currents within the conductor.

Step 1: Use the formula for mutual inductance.

The mutual inductance \( M \) of two coaxial solenoids is given by the formula: \[ M = \frac{\mu_0 N_1 N_2 A}{l} \]
where \( N_1 \) and \( N_2 \) are the number of turns in the solenoids, \( A \) is the cross-sectional area, \( l \) is the length of the solenoids, and \( \mu_0 \) is the permeability of free space.

Step 2: Calculate the mutual inductance.

Substitute the given values to find the mutual inductance as \( 4.8 \pi \times 10^{-5} \, H \).


Final Answer: \[ \boxed{4.8 \pi \times 10^{-5} \, H} \] Quick Tip: The mutual inductance between two solenoids depends on their number of turns, cross-sectional area, and the length of the solenoids.

Step 1: Use the transformer current equation.

For an ideal transformer, the ratio of the currents in the primary and secondary is inversely proportional to the ratio of the number of turns: \[ \frac{I_1}{I_2} = \frac{N_2}{N_1} \]
where \( N_1 \) and \( N_2 \) are the number of turns in the primary and secondary coils, respectively, and \( I_1 \) and \( I_2 \) are the currents in the primary and secondary coils.

Step 2: Calculate the secondary current.

Given the turn ratio is 4:1, the current in the secondary will be \( 1 \, A \).


Final Answer: \[ \boxed{1 \, A} \] Quick Tip: In a step-up transformer, the current in the secondary is inversely proportional to the ratio of the turns in the secondary and primary coils.

Step 1: Understand the relationship between wavelength and radiation type.

The wavelength of electromagnetic radiation decreases as the energy increases. The order of wavelengths from smallest to largest is: \( \gamma \)-rays, X-rays, ultraviolet rays, microwaves.

Step 2: Conclusion.

The smallest wavelength corresponds to \( \gamma \)-rays.


Final Answer: \[ \boxed{\gamma \, rays} \] Quick Tip: The electromagnetic spectrum has a range of wavelengths. The higher the energy, the smaller the wavelength (e.g., \( \gamma \)-rays have the smallest wavelength).

Step 1: Understand the concept of refraction.

In refraction, the light changes its direction as it passes from one medium to another, which causes its velocity and wavelength to change. However, the frequency remains unchanged since it depends only on the source.

Step 2: Conclusion.

The frequency of light does not change during refraction.


Final Answer: \[ \boxed{Frequency} \] Quick Tip: In refraction, the frequency of light remains constant, but its wavelength and velocity change due to the properties of the new medium.

Step 1: Understand the lens formula.

For a combination of lenses in series, the equivalent focal length \( \frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} + \cdots \).

Step 2: Rearrange the parts of the lens.

When the lens is broken and rearranged, the effective focal length is halved.


Final Answer: \[ \boxed{\frac{f}{2}} \] Quick Tip: When combining multiple lenses in series, the equivalent focal length is the reciprocal of the sum of the reciprocals of the individual focal lengths.

Step 1: Use the formula for the fringe pattern.

The fringe pattern for Young’s double slit experiment is given by \( d \sin \theta = n \lambda \), where \( d \) is the distance between the slits, \( \lambda \) is the wavelength, and \( n \) is the order of the maximum.

Step 2: Calculate the new order of the maximum.

Since the wavelength decreases, the order of the maximum increases proportionally, leading to the 14th order maximum for \( \lambda = 5000 \, Å \).


Final Answer: \[ \boxed{14th} \] Quick Tip: In Young's double slit experiment, the order of maxima is inversely proportional to the wavelength.

Step 1: Understand the working of a transistor as a switch.

In the active region (III) of the transistor, it behaves like a switch, and in the cutoff region (I), it is off.

Step 2: Conclusion.

For a transistor to act as a switch, it needs to operate in both the cutoff (I) and saturation (III) regions.


Final Answer: \[ \boxed{both in region (I) and (III)} \] Quick Tip: In a transistor switch, the cutoff region and saturation region are used for switching operations.

Step 1: Analyze the given circuit.

The circuit is a combination of logic gates. An OR gate produces a high output when any of its inputs are high.

Step 2: Conclusion.

The given circuit functions as an OR gate.


Final Answer: \[ \boxed{OR gate} \] Quick Tip: In digital circuits, an OR gate outputs high when any of the inputs are high.

Step 1: Use the dilution equation.

The dilution equation is \( C_1 V_1 = C_2 V_2 \), where \( C_1 \) and \( V_1 \) are the concentration and volume of the concentrated acid, and \( C_2 \) and \( V_2 \) are the concentration and volume of the diluted solution.

Step 2: Calculate the amount of concentrated acid.

The volume of the concentrated acid required is calculated as follows: \[ Mass of conc. acid = \frac{C_2 V_2}{percentage concentration} \times 100 \] \[ \boxed{45.0 \, g} \] Quick Tip: To prepare a diluted solution from a concentrated one, use the dilution equation \( C_1 V_1 = C_2 V_2 \).

Step 1: Use the formula for the Bohr radius.

The radius for the \( n \)-th orbit is given by \( r_n = n^2 \times 0.53 \, Å \).

Step 2: Calculate the radius for the first excited state.

For \( n = 2 \), the radius is \( r_2 = 2^2 \times 0.53 = 2.12 \, Å \).


Final Answer: \[ \boxed{2.12 \, Å} \] Quick Tip: The radius of the Bohr orbit increases with the square of the principal quantum number \( n \).

Step 1: Understand the concept of screening effect.

The screening effect refers to the ability of inner electrons to shield outer electrons from the attractive force of the nucleus.

Step 2: Conclusion.

The screening effect of \( d \)-electrons is less than that of \( p \)-electrons, as \( p \)-electrons are more effective at shielding.


Final Answer: \[ \boxed{Less than p-electrons} \] Quick Tip: Electrons in the \( p \)-orbitals provide better shielding than \( d \)-electrons due to their higher probability of being closer to the nucleus.

Step 1: Understand the periodic trend.

Ionization energy increases across a period and decreases down a group. The highest ionization energies are found in the noble gases.

Step 2: Conclusion.

The peaks in the graph correspond to the noble gases, which have the highest first ionization energies.


Final Answer: \[ \boxed{Rare gases} \] Quick Tip: Rare gases (noble gases) have the highest first ionization energies due to their stable electron configurations.

Step 1: Understand the trend in ionic radii.

For isoelectronic ions, the ionic radii decrease as the nuclear charge increases because the same number of electrons are pulled closer to the nucleus by the increasing positive charge.

Step 2: Conclusion.

The ionic radius decreases significantly from \( O^{2-} \) to \( Al^{3+} \) due to the increasing nuclear charge.


Final Answer: \[ \boxed{A significant decrease from O^{2-} to Al^{3+}} \] Quick Tip: For isoelectronic ions, the ionic radius decreases with increasing nuclear charge due to stronger attraction of electrons by the nucleus.

Step 1: Apply Molecular Orbital Theory (MOT).

From MOT, paramagnetic species have unpaired electrons in their molecular orbitals.

Step 2: Conclusion.

Both \( B_2 \) and \( O_2 \) have unpaired electrons, making them paramagnetic.


Final Answer: \[ \boxed{B_2 and O_2} \] Quick Tip: Paramagnetic species have unpaired electrons in their molecular orbitals, which cause them to be attracted to a magnetic field.

Step 1: Use the formula for rms velocity.

The rms velocity is inversely proportional to the square root of the molar mass of the gas.

Step 2: Conclusion.

The increasing order of rms velocities is \( H_2 \), \( N_2 \), \( O_2 \), and \( HBr \).


Final Answer: \[ \boxed{H_2 \rightarrow N_2 \rightarrow O_2 \rightarrow HBr} \] Quick Tip: The rms velocity of a gas is inversely proportional to the square root of its molar mass.

Step 1: Apply the concept of enthalpy change.

In the dissociation of \( H_2 \), the heat of atomisation is half of the given value of \( \Delta H \).

Step 2: Conclusion.

The heat of atomisation of H is \( 81 \, Kcal \).


Final Answer: \[ \boxed{81 \, Kcal} \] Quick Tip: The heat of atomisation is half the heat of dissociation for diatomic molecules like \( H_2 \).

Step 1: Understand the concept of combustion.

The enthalpy of combustion in a bomb calorimeter is determined under constant volume, whereas in open conditions, the enthalpy change is different due to work done by the system.

Step 2: Conclusion.

The enthalpy of combustion differs by \( -7.483 \, kJ \).


Final Answer: \[ \boxed{-7.483 \, kJ} \] Quick Tip: The enthalpy of combustion measured in a bomb calorimeter differs from the value measured in open conditions due to the work done by the system.

Step 1: Analyze the equilibrium expression.

The equilibrium constant for the dissociation of iodine is given by: \[ K_c = \frac{[ I ]^2}{[ I_2 ]} \]
Since \( K_c \) is very small, this means that most of the iodine remains in the molecular form, and the concentration of \( I_2 \) will be greater than that of \( I \).

Step 2: Conclusion.

At equilibrium, the concentration of \( I_2 \) will be less than that of \( I \).


Final Answer: \[ \boxed{[ I_2 (g) ] < [ I (g) ]} \] Quick Tip: For a reaction with a small equilibrium constant, the reactants (in this case, \( I_2 \)) are favored, and the products (in this case, \( I \)) will be present in much smaller amounts.

Step 1: Use the relation between the equilibrium constant and temperature.

For the given reaction, the equilibrium constant \( K_c \) has a temperature dependence that involves \( RT \), where \( R \) is the gas constant and \( T \) is the temperature in Kelvin.

Step 2: Conclusion.

The value of \( K_c \) for this reaction is \( (RT)^{1/2} \).


Final Answer: \[ \boxed{(RT)^{1/2}} \] Quick Tip: For reactions involving gases, the equilibrium constant can depend on temperature, often involving powers of \( RT \).

Step 1: Assign oxidation states.

In \( Na_2 S_4 O_6 \), sodium (Na) has an oxidation state of +1, and oxygen (O) has an oxidation state of -2. Let the oxidation state of sulfur (S) be \( x \).

Step 2: Solve for the oxidation state of sulfur.

The total charge on the molecule is zero, so: \[ 2(+1) + 4x + 6(-2) = 0 \]
Solving for \( x \), we get \( x = +5 \).


Final Answer: \[ \boxed{+5} \] Quick Tip: To find the oxidation state of an element in a compound, use the rule that the sum of oxidation states in a neutral molecule must equal zero.

Step 1: Analyze the reactions.

In both reactions (with sulfuric acid and sodium hydroxide), zinc undergoes displacement reactions, producing hydrogen gas. The volume of hydrogen produced is proportional to the moles of zinc reacted.

Step 2: Conclusion.

In both cases, the same amount of zinc is used, and the volumes of hydrogen evolved are equal, so the ratio is 1:1.


Final Answer: \[ \boxed{1:1} \] Quick Tip: The amount of hydrogen gas produced in reactions with zinc is proportional to the amount of zinc reacted.

Step 1: Understand the thermal stability of hydrides.

As we move down the group, the alkali metals form increasingly unstable hydrides due to the increasing size of the metal cation and the decreasing lattice energy.

Step 2: Conclusion.

The thermal stability decreases in the order: \( CsH > RbH > KH > LiH \).


Final Answer: \[ \boxed{CsH > RbH > KH > LiH} \] Quick Tip: The thermal stability of hydrides decreases as the size of the alkali metal cation increases.

Step 1: Analyze carbocation stability.

The stability of a carbocation increases with electron-donating groups attached to it. The \( CH_3 - OCH_3^+ \) carbocation is stabilized by the electron-donating \( OCH_3 \) group.

Step 2: Conclusion.

The most stable carbocation is \( CH_3 - OCH_3^+ \).


Final Answer: \[ \boxed{CH_3 - OCH_3^+} \] Quick Tip: Carbocations are stabilized by groups that donate electron density, such as alkyl groups and \( OCH_3 \).

Step 1: Understand chirality.

A chiral compound does not have an internal plane of symmetry and has non-superimposable mirror images.

Step 2: Conclusion.

Only structures I and II are chiral compounds.


Final Answer: \[ \boxed{Only I and II are chiral compounds.} \] Quick Tip: A chiral molecule lacks a plane of symmetry and has a non-superimposable mirror image (enantiomer).

Step 1: Analyze the compound's symmetry.

The compound contains two chiral centers (the carbon atoms attached to \( Br \)), so it can have 4 enantiomers (two pairs of non-superimposable mirror images).

Step 2: Conclusion.

The number of enantiomers of the compound is 4.


Final Answer: \[ \boxed{4} \] Quick Tip: The number of enantiomers is determined by the number of chiral centers in the molecule. For \( n \) chiral centers, the number of stereoisomers is \( 2^n \).

Step 1: Understand the reaction type.

The reaction \( RCOOK \xrightarrow{Electrolytic} Br_2 \) involves the reduction of a carboxylate anion to a hydrocarbon, a reaction that yields good results.

Step 2: Conclusion.

This reaction will give a hydrocarbon product in good yield.


Final Answer: \[ \boxed{RCOOK \xrightarrow{Electrolytic} Br_2} \] Quick Tip: Electrolytic reduction of carboxylate salts often leads to the formation of hydrocarbons.

Step 1: Understand the reaction of acetylene with hypochlorous acid.

Acetylene reacts with hypochlorous acid to form dichloroacetaldehyde as the main product.

Step 2: Conclusion.

The main product of this reaction is dichloroacetaldehyde.


Final Answer: \[ \boxed{Dichloroacetaldehyde} \] Quick Tip: The reaction of acetylene with hypochlorous acid results in the addition of chlorine to the acetylene molecule, forming dichloroacetaldehyde.

Step 1: Understand the greenhouse effect.

The greenhouse effect is primarily due to the presence of gases such as \( CO_2 \), \( CH_4 \), and water vapor in the atmosphere, which absorb infrared radiation and trap heat.

Step 2: Conclusion.

The greenhouse effect is caused by gases that absorb infrared radiation.


Final Answer: \[ \boxed{Presence of gases, which in general are strong infrared absorbers, in the atmosphere.} \] Quick Tip: The greenhouse effect occurs when gases in the atmosphere trap heat by absorbing and emitting infrared radiation.

Step 1: Understand Frenkel defect.

A Frenkel defect occurs when an ion is displaced from its lattice position to an interstitial position. This type of defect does not significantly affect the overall density of the crystal.

Step 2: Conclusion.

Frenkel defect does not change the overall density of the ionic solid.


Final Answer: \[ \boxed{Does not change} \] Quick Tip: Frenkel defects involve the displacement of ions within the lattice, but they do not change the density of the solid significantly.

Step 1: Understand the concept of molarity.

Molarity is given by \( M = \frac{moles}{volume in liters} \). If equal weights of NaCl and KCl are dissolved in the same volume, the molarity will be higher for NaCl due to its lower molar mass.

Step 2: Conclusion.

The molarity of NaCl will be more than that of KCl.


Final Answer: \[ \boxed{That of NaCl will be more than that of KCl} \] Quick Tip: Molarity is inversely related to the molar mass of the solute. The lower the molar mass, the higher the molarity for a given mass.

Step 1: Use the formula for electrolysis.

The number of moles of metal deposited is \( moles = \frac{mass}{atomic weight} \). The number of moles of electrons involved is related to the charge passed.

Step 2: Calculate the oxidation state.

The oxidation state is found to be +3 based on the amount of metal deposited.


Final Answer: \[ \boxed{+3} \] Quick Tip: The amount of metal deposited in electrolysis is proportional to the number of moles of electrons and the oxidation state of the metal.

Step 1: Use the relation for electrolysis.

The amount of substance deposited at each electrode is proportional to the equivalent weight and the number of moles of electrons passed.

Step 2: Conclusion.

The ratio of iron deposited will be 3:2 based on the molar mass and oxidation states.


Final Answer: \[ \boxed{3:2} \] Quick Tip: The amount of metal deposited in electrolysis depends on the oxidation state and the number of moles of electrons passed through the electrolyte.

Step 1: Use the Arrhenius equation.

The rate constant depends on temperature as \( k = A e^{-\frac{E_a}{RT}} \).

Step 2: Calculate the rate constant at 300 K.

Using the Arrhenius equation, the rate constant at 300 K is \( 6.4 \times 10^{-3} \).


Final Answer: \[ \boxed{6.4 \times 10^{-3}} \] Quick Tip: The rate constant increases with temperature according to the Arrhenius equation.

Step 1: Understand the order of reaction at high pressure.

When a gas forms a monomolecular layer, the reaction is typically first-order with respect to the gas concentration.

Step 2: Conclusion.

The order of reaction in this case is first-order.


Final Answer: \[ \boxed{First-order} \] Quick Tip: At high pressures, the gas molecules behave like a monomolecular layer, which typically corresponds to a first-order reaction.

Step 1: Understand bond energy.

The bond energy of \( F_2 \) is lower than that of \( Cl_2 \) and \( Br_2 \) due to the small size of the fluorine atoms, which causes electron-electron repulsion.

Step 2: Conclusion.

The incorrect property is bond energy, as \( F_2 \) has a lower bond energy compared to \( Cl_2 \) and \( Br_2 \).


Final Answer: \[ \boxed{Bond energy: F_2 > Cl_2 > Br_2} \] Quick Tip: Bond energies generally decrease with increasing size of the halogen atoms, making \( F_2 \) weaker than \( Cl_2 \) and \( Br_2 \).

Step 1: Understand the reducing nature.

The reducing behavior of \( H_3 PO_4 \) is largely due to the presence of the \( -OH \) group and P-H bonds. These bonds are weak and prone to breaking, which makes the molecule a good reducing agent.

Step 2: Conclusion.

The strong reducing behavior of \( H_3 PO_4 \) is due to the presence of one \( -OH \) group and two P-H bonds.


Final Answer: \[ \boxed{Presence of one -OH group and two P-H bonds} \] Quick Tip: Reducing agents typically have bonds that are easy to break, such as \( -OH \) and P-H in the case of \( H_3 PO_4 \).

Step 1: Magnetic moment and spin-only formula.

The magnetic moment is given by the formula \( \mu_{eff} = \sqrt{n(n+2)} \), where \( n \) is the number of unpaired electrons.

Step 2: Determine the species with the same magnetic moment.

Both \( \left[ Cr(H_2 O)_6 \right]^{3+} \) and \( \left[ Fe(H_2 O)_6 \right]^{2+} \) have the same number of unpaired electrons and thus the same magnetic moment.


Final Answer: \[ \boxed{\left[ Cr(H_2 O)_6 \right]^{3+}, \left[ Fe(H_2 O)_6 \right]^{2+}} \] Quick Tip: The magnetic moment depends on the number of unpaired electrons. Species with the same number of unpaired electrons will have the same magnetic moment.

Step 1: Understand the acid strength of halogen compounds.

The acidity of halogenated compounds increases with the electronegativity of the halogen. Fluorine, being the most electronegative, makes the compound more acidic.

Step 2: Conclusion.
\( CHF_3 \) is less acidic compared to the others because the hydrogen is less acidic due to the weaker electronegativity of fluorine compared to other halogens.


Final Answer: \[ \boxed{CHF_3} \] Quick Tip: The electronegativity of the halogen influences the acidity of the compound. Fluorine is less acidic than chlorine, bromine, or iodine.

Step 1: Understand the \( S_N2 \) mechanism.

In \( S_N2 \) reactions, the leaving group is replaced by the nucleophile in a single step. The steric hindrance plays a role in determining the rate.

Step 2: Conclusion.

The least sterically hindered alkyl bromide will have the highest rate in an \( S_N2 \) reaction, so \( CH_3 CH_2 CH_2 Br \) will have the highest relative rate.


Final Answer: \[ \boxed{CH_3 CH_2 CH_2 Br} \] Quick Tip: In \( S_N2 \) reactions, less steric hindrance around the leaving group leads to a faster reaction rate.

Step 1: Analyze dehydration reactions.

Dehydration of alcohols typically produces alkenes. \( C_6 H_6 \) (benzene) is not a product of dehydration of an alcohol.

Step 2: Conclusion.
\( C_6 H_6 \) is not a product of dehydration.


Final Answer: \[ \boxed{C_6 H_6} \] Quick Tip: Dehydration of alcohols typically results in the formation of alkenes, not benzene.

Step 1: Understand the reaction.

The reaction involves the oxidation of an aldehyde or alcohol to a carboxylic acid.

Step 2: Conclusion.

The correct product is \( CHO \).


Final Answer: \[ \boxed{CHO} \] Quick Tip: Oxidation of aldehydes can lead to the formation of carboxylic acids, depending on the reaction conditions.

Step 1: Analyze the reactivity of carbonyl compounds.

The nucleophilic addition reaction is most favorable in aldehydes, as they are more electrophilic than ketones due to the lack of alkyl groups that can donate electron density.

Step 2: Conclusion.

The aldehyde \( CH_3 CHO \) is the most reactive towards nucleophilic addition.


Final Answer: \[ \boxed{CH_3 CHO} \] Quick Tip: Aldehydes are more reactive in nucleophilic addition reactions than ketones due to less steric hindrance and electron donation.

Step 1: Understand the reaction sequence.

The given reaction is a typical series of steps involving nucleophilic substitution and hydrolysis, ultimately resulting in acetaldehyde as the final product.

Step 2: Conclusion.

The correct product is \( CH_3 CHO \).


Final Answer: \[ \boxed{CH_3 CHO} \] Quick Tip: In organic reactions, substitution and hydrolysis steps can lead to aldehyde products, depending on the reagents used.

Step 1: Analyze the function of insulin.

Insulin is a hormone produced by the pancreas that regulates glucose levels in the blood.

Step 2: Conclusion.

Insulin is classified as a hormone, not an enzyme or coenzyme.


Final Answer: \[ \boxed{A hormone} \] Quick Tip: Insulin is a hormone that helps regulate blood sugar levels in the body, which is crucial for metabolic processes.

Step 1: Understand peptide bonds.

Peptide bonds in proteins are characterized by partial double bond character due to resonance, resulting in shorter C-N bond length compared to normal C-N bonds.

Step 2: Conclusion.

The C-N bond length in proteins is smaller than usual C-N bond length.


Final Answer: \[ \boxed{C-N bond length in proteins is longer than usual C-N bond length.} \] Quick Tip: Peptide bonds in proteins exhibit partial double bond character, which makes the C-N bond shorter than in typical amines.

Step 1: Understand the reaction sequence.

The reaction involves the separation of metals and their subsequent identification. In this case, sodium and aluminium can be identified through the filtrate.

Step 2: Conclusion.

The filtrate shall give tests for sodium and aluminium.


Final Answer: \[ \boxed{Sodium and aluminium} \] Quick Tip: The chemical tests for sodium and aluminium are distinct and help in their identification in a solution.

Step 1: Use the neutralization equation.

The neutralization of sulfuric acid and sodium carbonate can be represented by \( H_2 SO_4 + Na_2 CO_3 \rightarrow Na_2 SO_4 + CO_2 + H_2 O \).

Step 2: Conclusion.

The required volume is 176.66 ml based on the equivalent amounts of acid and base.


Final Answer: \[ \boxed{176.66 \, ml} \] Quick Tip: Use the equivalence factor and the molarity of the acid and base to calculate the volume needed for neutralization.

Step 1: Use the neutralization equation.

The neutralization reaction between sodium carbonate and sulphuric acid is given by: \[ Na_2 CO_3 + H_2 SO_4 \rightarrow Na_2 SO_4 + CO_2 + H_2 O \]
The molar equivalents of sodium carbonate are equal to those of sulphuric acid. Using the normality and volume of sulphuric acid, calculate the volume of sodium carbonate solution required.

Step 2: Calculate the volume.

The required volume of sodium carbonate solution is calculated as 176.66 ml using the equivalence factor.


Final Answer: \[ \boxed{176.66 \, ml} \] Quick Tip: To calculate the volume required for neutralization, use the relation \( N_1 V_1 = N_2 V_2 \), where \( N_1 \) and \( V_1 \) are the normality and volume of one reactant, and \( N_2 \) and \( V_2 \) are those of the other.

Step 1: Use the principle of inclusion-exclusion.

We know the total number of students offering Mathematics (100), and the number of students offering Mathematics and other subjects. Using inclusion-exclusion: \[ Mathematics alone = Mathematics total - (Mathematics and Physics) - (Mathematics and Chemistry) + (Mathematics, Physics and Chemistry) \] \[ Mathematics alone = 100 - 30 - 28 + 18 = 60 \]


Final Answer: \[ \boxed{60} \] Quick Tip: To find the number of students offering only one subject, use the inclusion-exclusion principle.

Step 1: Express the equations in terms of \( x \) and \( y \).

Given the equations: \[ \cos \theta + \sin \theta = x \cos \theta \quad and \quad \sin \theta = y \cos \theta \]
Substitute \( \sin \theta = y \cos \theta \) into the first equation: \[ \cos \theta + y \cos \theta = x \cos \theta \]
Factor out \( \cos \theta \): \[ \cos \theta (1 + y) = x \cos \theta \]
Therefore, \( x = 1 + y \).

Step 2: Solve for \( x^2 + y^2 \).

Now, \( x^2 + y^2 = (1 + y)^2 + y^2 = 1 + 2y + y^2 + y^2 = 1 \).


Final Answer: \[ \boxed{1} \] Quick Tip: In trigonometric equations, express one variable in terms of others and use algebraic manipulation to solve for unknowns.

Step 1: Use the identity \( \cos \left( 90^\circ - \theta \right) = \sin \theta \).

This gives: \[ \cos 76^\circ = \sin \theta \]
Therefore, \( \theta = 76^\circ \) or \( \theta = 180^\circ - 76^\circ = 104^\circ \).

Step 2: Conclusion.

The general value of \( \theta \) is \( 76^\circ \) and \( 180^\circ - 76^\circ \).


Final Answer: \[ \boxed{76^\circ and 180^\circ - 76^\circ} \] Quick Tip: When solving trigonometric equations, consider both possible solutions for angles due to periodicity and symmetry of trigonometric functions.

Step 1: Analyze the equation.

The equation \( \frac{z + 1}{z - 1} = 4 \) represents a geometric transformation. This is a Möbius transformation, which typically maps to a circle.

Step 2: Conclusion.

The locus of the point representing \( z \) is a circle with radius \( \frac{1}{2} \).


Final Answer: \[ \boxed{a circle with radius \frac{1}{2}} \] Quick Tip: Möbius transformations can map lines and circles in the complex plane. In this case, it maps to a circle.

Step 1: Use properties of roots of unity.

Since the roots of \( x^2 - x + 1 = 0 \) are cube roots of unity, we know that \( \alpha^{100} \) and \( \beta^{100} \) will satisfy the same equation as \( \alpha \) and \( \beta \).

Step 2: Conclusion.

The equation whose roots are \( \alpha^{100} \) and \( \beta^{100} \) is \( x^2 + x + 1 = 0 \).


Final Answer: \[ \boxed{x^2 + x + 1 = 0} \] Quick Tip: The roots of unity cycle periodically. In this case, the powers of the roots repeat with a period of 3.

Step 1: Analyze the inequality.

The inequality involves absolute values, so we need to break it into different intervals based on \( |x| \).

Step 2: Conclusion.

The solution set is \( [-3, 3] \cup (-4, 4) \).


Final Answer: \[ \boxed{[-3, 3] \cup (-4, 4)} \] Quick Tip: For inequalities involving absolute values, break the expression into intervals based on the behavior of the absolute value function.

Step 1: Break down the absolute value expressions.

We will consider the different cases where each of the absolute values changes based on the values of \( x \).

Step 2: Solve for \( x \).

Solving the inequality for \( x \), we find that the solution is \( x \leq 0 \) or \( x \geq \frac{8}{3} \).


Final Answer: \[ \boxed{x \leq 0 or x \geq \frac{8}{3}} \] Quick Tip: When solving inequalities involving absolute values, consider the different cases where the expressions inside the absolute values change sign.

Step 1: Fix one boy at the round table.

Since the seating is circular, we can fix one boy in position to eliminate symmetrical arrangements. We then arrange the remaining 4 boys, which can be done in \( 4! \) ways.

Step 2: Arrange the girls.

The girls can be arranged in the gaps between the boys, which can be done in \( 5! \) ways.

Step 3: Conclusion.

Thus, the total number of ways to arrange the boys and girls is \( 5! \times 4! \).


Final Answer: \[ \boxed{5! \times 4!} \] Quick Tip: In circular arrangements, fix one element in place to remove equivalent arrangements due to rotation.

Step 1: Calculate total number of ways to choose 3 balls.

Total number of ways to choose 3 balls from 9 is \( \binom{9}{3} \).

Step 2: Subtract the cases with no black balls.

The number of ways to choose 3 balls without any black balls (from white and red balls) is \( \binom{6}{3} \).

Step 3: Conclusion.

The number of ways to choose 3 balls with at least one black ball is \( \binom{9}{3} - \binom{6}{3} = 84 \).


Final Answer: \[ \boxed{84} \] Quick Tip: When calculating the number of ways to select items with certain conditions, use the complementary counting technique by subtracting the cases that don't satisfy the condition.

Step 1: Use the binomial expansion formula.

The expansion of \( (2 + 3x)^4 \) is: \[ (2 + 3x)^4 = \sum_{k=0}^{4} \binom{4}{k} 2^{4-k} (3x)^k \]

Step 2: Find the middle term.

The middle term occurs when \( k = 2 \). Thus, the middle term is: \[ \binom{4}{2} 2^{4-2} (3x)^2 = 6 \times 4 \times 9x^2 = 216x^2 \]

Step 3: Conclusion.

The coefficient of the middle term is 216.


Final Answer: \[ \boxed{216} \] Quick Tip: In a binomial expansion, the middle term is found by selecting the appropriate value of \( k \) based on the degree of the expansion.

Step 1: Use the sum of binomial coefficients.

The sum of binomial coefficients up to \( C_n \) for \( (1 + x)^n \) is \( 2^n \). Therefore, the sum of the terms in the given series is \( (n+1) \times 2^n \).

Step 2: Conclusion.

The required sum is \( (n+1)2^n \).


Final Answer: \[ \boxed{(n+1)2^n} \] Quick Tip: The sum of the binomial coefficients for a given power of \( n \) is \( 2^n \), and the sum of partial sums increases by a factor of \( (n+1) \).

Step 1: Use the formula for the sum of squares.

The sum of the first \( n \) squares is given by: \[ S = \frac{n(n+1)(2n+1)}{6} \]
For \( n = 100 \), substitute the value to find the sum.

Step 2: Conclusion.

The sum of squares of the first 100 numbers results in \( 99 \times 2^{100} + 1 \).


Final Answer: \[ \boxed{99 \times 2^{100} + 1} \] Quick Tip: The sum of squares of the first \( n \) natural numbers follows a specific formula for efficient calculation.

Step 1: Analyze the line equation.

The line passing through \( (1,1) \) has the general equation \( x + y = 2 \), which intersects the axes at \( x = 2 \) and \( y = 2 \).

Step 2: Use the area formula.

The area of the triangle formed by the intercepts is given by: \[ A = \frac{1}{2} \times 2 \times 2 = 2 \]
Thus, the quadratic equation becomes \( x^2 - Ax + 2A = 0 \).


Final Answer: \[ \boxed{x^2 - Ax + 2A = 0} \] Quick Tip: The area of a triangle formed by intercepts on the coordinate axes is \( \frac{1}{2} \times base \times height \).

Step 1: Analyze the equation.

The given equation represents a condition for the concurrent lines, and solving it gives the points of concurrency.

Step 2: Conclusion.

The points of concurrency are \( (-1, -1) \) and \( (2, -1) \).


Final Answer: \[ \boxed{(-1, -1), (2, -1)} \] Quick Tip: For a family of concurrent lines, solving the corresponding system of equations will give the points of concurrency.

Step 1: Rewrite the equation of the circle.

The given equation of the circle is: \[ x^2 + y^2 + 20(x + y) + 20 = 0 \]
Completing the square, we get the standard form of the circle equation.

Step 2: Find the equation of the tangents.

Using the formula for tangents from the origin to the circle, we get the equation \( 2x^2 + 2y^2 + 5xy = 0 \).


Final Answer: \[ \boxed{2x^2 + 2y^2 + 5xy = 0} \] Quick Tip: To find the equation of tangents from the origin to a circle, use the general formula and complete the square for the circle equation.

Step 1: Use the geometric property of ellipses.

For an ellipse, if the angle between the line joining a point on the ellipse and the two foci is 90°, then the eccentricity \( e \) is \( \frac{1}{\sqrt{2}} \).

Step 2: Conclusion.

The eccentricity of the ellipse is \( \frac{1}{\sqrt{2}} \).


Final Answer: \[ \boxed{\frac{1}{\sqrt{2}}} \] Quick Tip: For an ellipse with a right angle between the foci and any point, the eccentricity can be found using the relation \( e = \frac{1}{\sqrt{2}} \).

Step 1: Use the condition for tangency.

For a line to be tangent to the parabola, the discriminant of the quadratic equation formed by substituting the line equation into the parabola equation must be zero.

Step 2: Conclusion.

By solving the discriminant condition, we find that \( k = -15/4 \).


Final Answer: \[ \boxed{-\frac{15}{4}} \] Quick Tip: For a line to be tangent to a curve, the discriminant of the quadratic formed by substitution must be zero.

Step 1: Use the geometric property of ellipses.

In an ellipse, if the foci and end of the minor axis form an equilateral triangle, the eccentricity is \( \frac{1}{2} \).

Step 2: Conclusion.

Thus, the eccentricity of the ellipse is \( \frac{1}{2} \).


Final Answer: \[ \boxed{\frac{1}{2}} \] Quick Tip: The eccentricity of an ellipse can be derived geometrically when the foci and the ends of the minor axis form special figures like an equilateral triangle.

Step 1: Express the functions and solve.

The given equation is \( f(x) = g(x)h(x) \), and solving for \( h(x) \) involves algebraic manipulation of the terms.

Step 2: Conclusion.

We find that \( h(x) = 3 \).


Final Answer: \[ \boxed{3} \] Quick Tip: When solving for unknown functions in equations involving products, factor the given functions and solve accordingly.

Step 1: Construct the truth table.

Construct the truth table for the given logical expression and evaluate the truth value for each combination of \( p \) and \( q \).

Step 2: Conclusion.

The truth values for the expression \( (p \land q) \rightarrow (q \lor \neg p) \) are \( TFTT \).


Final Answer: \[ \boxed{TFTT} \] Quick Tip: When constructing truth tables, systematically evaluate each part of the logical expression to determine the overall truth value.

Step 1: Use the relationship between mode, mean, and median.

The relation between mode, mean, and median is given by: \[ Mode = 3 \times Median - 2 \times Mean \]
Solving for the median, we find it to be 64.


Final Answer: \[ \boxed{64} \] Quick Tip: The relationship between mode, median, and mean can help to find the missing measure in a dataset when two of them are known.

Step 1: Use the formula for variance.

The formula for variance is: \[ Variance = \frac{1}{N} \sum f(x - \mu)^2 \]
where \( N \) is the total number of frequencies, \( f \) is the frequency, and \( \mu \) is the mean of the data.

Step 2: Calculate the mean and variance.

The variance is calculated to be 1.32 using the formula.


Final Answer: \[ \boxed{1.32} \] Quick Tip: The variance measures the spread of the data points. Use the formula for variance to find the dispersion from the mean.

Step 1: Understand the properties of the relation.

The relation \( R \) is reflexive because \( |a - a| = 0 \leq 1 \). It is symmetric because if \( aRb \), then \( |a - b| = |b - a| \).

Step 2: Conclusion.

The relation is reflexive and symmetric, but not transitive.


Final Answer: \[ \boxed{reflexive and symmetric only} \] Quick Tip: To check for equivalence, verify reflexivity, symmetry, and transitivity.

Step 1: Use the trigonometric identity.

We know that \( \sin^2(x) + \cos^2(x) = 1 \), so the greatest and least values are \( 1 \) and \( 0 \), respectively.

Step 2: Conclusion.

The greatest value is 1 and the least value is 0.


Final Answer: \[ \boxed{1 and 0} \] Quick Tip: Use trigonometric identities to simplify and find the extreme values of expressions involving sine and cosine.

Step 1: Simplify the expression.

Use trigonometric and inverse trigonometric identities to simplify the expression. First, find the value of the cosine expression, then compute the inverse cosine and divide by 2.

Step 2: Conclusion.

The value of the expression is \( 3/4 \).


Final Answer: \[ \boxed{\frac{3}{4}} \] Quick Tip: For expressions involving inverse trigonometric functions, use trigonometric identities to simplify before evaluating.

Step 1: Determine the condition for determinant to vanish.

For the determinant to vanish, the rows or columns must be linearly dependent. In this case, there is no value of \( x \) that satisfies this condition.

Step 2: Conclusion.

The determinant does not vanish for any specific value of \( x \).


Final Answer: \[ \boxed{No value of x} \] Quick Tip: A determinant vanishes when its rows or columns are linearly dependent. Check for this condition when solving problems.

Step 1: Analyze the system of equations.

For the two lines to be concurrent, the system of equations must have a common solution. This happens when \( \ell = m = n = 0 \).

Step 2: Conclusion.

The lines are concurrent when \( \ell = m = n = 0 \).


Final Answer: \[ \boxed{\ell = m = n = 0} \] Quick Tip: For concurrent lines, solve the system of equations to find the values of the parameters that make the lines meet at a single point.

Step 1: Recognize the Taylor series expansion.

The given series is the Taylor series expansion of \( e^x \), and thus \( y = e^x \).

Step 2: Conclusion.

Therefore, \( y = e^x \).


Final Answer: \[ \boxed{y} \] Quick Tip: The Taylor series expansion for \( e^x \) is \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \).

Step 1: Check for continuity.

For the function to be continuous at \( x = -5 \), the limit as \( x \) approaches \( -5 \) must equal the function value at \( x = -5 \).

Step 2: Evaluate the limit.

By simplifying the expression for the limit, we find that \( a = \frac{7}{8} \).


Final Answer: \[ \boxed{\frac{7}{8}} \] Quick Tip: To determine the value of \( a \) for continuity, solve the limit of the function as \( x \) approaches the point.

Step 1: Find the derivative of the curve.

The derivative of the curve \( y = \frac{1}{x - 3} \) is given by the formula \( y' = \frac{-1}{(x - 3)^2} \), which gives the slope of the tangent.

Step 2: Use the tangent slope condition.

For the lines to be tangent with slope 2, we equate the slope to 2 and solve for the equation of the line.


Final Answer: \[ \boxed{None of these} \] Quick Tip: To find the equation of a tangent line, first calculate the derivative of the curve and set it equal to the desired slope. Then solve for the equation of the tangent.

Step 1: Analyze the function.

We differentiate the function to find the intervals where the function is increasing. The derivative is positive in the intervals \( (0, 1) \) and \( (2, \infty) \).

Step 2: Conclusion.

Thus, the function is increasing in \( (0, 1) \cup (2, \infty) \).


Final Answer: \[ \boxed{(0, 1) \cup (2, \infty)} \] Quick Tip: To determine increasing or decreasing intervals, find the derivative and solve for where it is positive (increasing) or negative (decreasing).

Step 1: Use optimization to find the maximum value.

To maximize \( xy \), apply the method of Lagrange multipliers or directly optimize by differentiating the constraint equation.

Step 2: Conclusion.

The maximum value of \( xy \) is \( \frac{c^2}{\sqrt{2ab}} \).


Final Answer: \[ \boxed{\frac{c^2}{\sqrt{2ab}}} \] Quick Tip: When maximizing expressions subject to constraints, use methods like Lagrange multipliers or differentiate the function directly.

Step 1: Use substitution for integration.

We use appropriate trigonometric substitution to solve the integral.

Step 2: Conclusion.

The integral evaluates to \( \sec^{-1} \left( \frac{x^2 + 1}{\sqrt{2x}} \right) + c \).


Final Answer: \[ \boxed{\sec^{-1} \left( \frac{x^2 + 1}{\sqrt{2x}} \right) + c} \] Quick Tip: For integrals involving powers of \( x^2 + 1 \), use trigonometric substitution to simplify the expression.

Step 1: Apply trigonometric substitution.

Use a substitution to simplify the integrand. The integral \( \int_0^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx \) can be solved by using standard integration techniques.

Step 2: Conclusion.

After performing the integration, the result is \( \frac{\pi}{4} \).


Final Answer: \[ \boxed{\frac{\pi}{4}} \] Quick Tip: For integrals involving trigonometric functions, use substitution to simplify and find the antiderivative.

Step 1: Set up the area integral.

To find the area between the curves, set up the definite integrals for both curves and subtract the values. Use the limits from \( 0 \) to \( \pi \).

Step 2: Calculate the area.

After solving the integrals, the area is found to be \( \frac{3\sqrt{3}}{4} \).


Final Answer: \[ \boxed{\frac{3\sqrt{3}}{4}} \] Quick Tip: To find the area between two curves, subtract the integrals of the two functions over the given interval.

Step 1: Rearrange the equation.

Rearrange the differential equation to separate terms involving \( x \) and \( y \).

Step 2: Solve the differential equation.

Solve the differential equation using standard techniques for solving first-order differential equations, such as substitution and integration.

Step 3: Conclusion.

The general solution is \( \log \tan \frac{y}{2} + \log \sin x = C \).


Final Answer: \[ \boxed{\log \tan \frac{y}{2} + \log \sin x = C} \] Quick Tip: When solving differential equations, use separation of variables and integration to find the general solution.

Step 1: Separate the variables.

Rewrite the differential equation in terms of separated variables and solve.

Step 2: Solve the equation.

The solution is \( f(x) = x + c \), where \( c \) is a constant.


Final Answer: \[ \boxed{f(x) = x + c} \] Quick Tip: When solving first-order differential equations, use separation of variables to isolate \( f(x) \) and solve for it.

Step 1: Use the dot product properties.

From the given conditions, we know that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \), so \( \mathbf{a} + \mathbf{b} = -\mathbf{c} \). Taking the dot product of both sides with themselves and simplifying gives us \( \mathbf{a} \cdot \mathbf{a} = \frac{3}{2} \).

Step 2: Conclusion.

The value of \( \mathbf{a} \cdot \mathbf{a} \) is \( \frac{3}{2} \).


Final Answer: \[ \boxed{\frac{3}{2}} \] Quick Tip: For unit vectors, their dot products follow specific relationships. Use these relationships to simplify expressions involving dot products.

Step 1: Use the condition for coplanarity.

Three vectors are coplanar if their scalar triple product is zero. Calculate the scalar triple product and set it equal to zero to solve for \( a \).

Step 2: Conclusion.

After performing the calculation, we find that \( a = -4 \).


Final Answer: \[ \boxed{-4} \] Quick Tip: For three vectors to be coplanar, their scalar triple product must be zero.

Step 1: Use the section formula.

To find the point of intersection with the \( XY \)-plane, substitute \( z = 0 \) into the equation of the line joining \( A \) and \( B \).

Step 2: Conclusion.

The coordinates of the point of intersection with the \( XY \)-plane are \( \left( \frac{13}{5}, 0, 0 \right) \).


Final Answer: \[ \boxed{\left( \frac{13}{5}, 0, 0 \right)} \] Quick Tip: To find the intersection of a line with the \( XY \)-plane, set \( z = 0 \) and solve for the coordinates.

Step 1: Use the formula for the angle between two planes.

The formula for the angle between two planes is: \[ \cos \theta = \frac{|A_1 A_2 + B_1 B_2 + C_1 C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2} \sqrt{A_2^2 + B_2^2 + C_2^2}} \]
where \( A_1, B_1, C_1 \) are the coefficients of the first plane and \( A_2, B_2, C_2 \) are the coefficients of the second plane.

Step 2: Conclusion.

After substituting the values, we get \( \cos^{-1} \left( \frac{4}{21} \right) \).


Final Answer: \[ \boxed{\cos^{-1} \left( \frac{4}{21} \right)} \] Quick Tip: Use the formula for the angle between two planes to find the cosine of the angle and then use the inverse cosine to get the angle.

Step 1: Use Bayes' Theorem.

Use Bayes' Theorem to calculate the probability. Bayes' Theorem states that: \[ P(B_2 | Red) = \frac{P(Red | B_2) P(B_2)}{P(Red)} \]

Step 2: Conclusion.

The probability that the red ball came from box \( B_2 \) is \( \frac{14}{39} \).


Final Answer: \[ \boxed{\frac{14}{39}} \] Quick Tip: When dealing with conditional probability, use Bayes' Theorem to reverse the probability and find the desired result.

Step 1: Use binomial probability.

We need the probability that the second win occurs at the third test. This is a binomial probability where the first two tests must be losses, and the third must be a win.

Step 2: Conclusion.

The probability of this event is \( \frac{1}{4} \).


Final Answer: \[ \boxed{\frac{1}{4}} \] Quick Tip: For probability questions involving a sequence of events, use binomial probability and the independence of events to calculate the desired probability.

Step 1: Use trigonometry.

Use the tangent function for the angular elevations at points A, B, and C to create equations involving the height of the object.

Step 2: Conclusion.

After solving the trigonometric equations, we find that the height of the object is \( \frac{a}{2} \left( b - a \right) \).


Final Answer: \[ \boxed{\frac{a}{2} \left( b - a \right)} \] Quick Tip: When dealing with objects at different points, use trigonometric functions to relate the angles and distances to calculate the height.

Step 1: Define the constraints.

The constraints on the number of articles are based on the total cost and the earnings. The inequality \( 4x + 3y \leq 24 \) ensures that the total cost does not exceed the limit.

Step 2: Conclusion.

The linear constraints are \( x \geq 0, y \geq 0, 4x + 3y \leq 24 \).


Final Answer: \[ \boxed{x \geq 0, y \geq 0, 4x + 3y \leq 24} \] Quick Tip: For linear programming problems, define the constraints based on the given limitations on resources and profits.

Step 1: Find the meaning of "SAGACIOUS".

"SAGACIOUS" means wise or having good judgment.

Step 2: Conclusion.

The correct meaning of the word is "Wise".


Final Answer: \[ \boxed{Wise} \] Quick Tip: "Sagacious" is used to describe someone who is wise or has keen insight.

Step 1: Find the meaning of "REMEDIAL".

"REMEDIAL" means corrective, intended to improve a problem or condition.

Step 2: Conclusion.

The correct meaning of the word is "Corrective".


Final Answer: \[ \boxed{Corrective} \] Quick Tip: "Remedial" is used to refer to actions intended to fix or improve something.

Step 1: Find the meaning of "RETICENT".

"RETICENT" means inclined to keep one's thoughts and feelings to oneself, or being secretive.

Step 2: Conclusion.

The correct meaning of the word is "Secretive".


Final Answer: \[ \boxed{Secretive} \] Quick Tip: "Reticent" refers to someone who is reserved and keeps their thoughts private.

Step 1: Understand the meaning of "FIDELITY".

"FIDELITY" means loyalty or faithfulness.

Step 2: Identify the opposite.

The opposite of "FIDELITY" is "Treachery", which refers to betrayal or disloyalty.


Final Answer: \[ \boxed{Treachery} \] Quick Tip: Fidelity refers to loyalty and faithfulness, while treachery refers to betrayal.

Step 1: Understand the meaning of "INFRINGABLE".

"INFRINGABLE" refers to something that cannot be violated or broken.

Step 2: Identify the opposite.

The opposite of "INFRINGABLE" is "Breakable", meaning something that can be broken or violated.


Final Answer: \[ \boxed{Breakable} \] Quick Tip: "Infringable" refers to something that cannot be violated, while "Breakable" refers to something that can be broken.

Step 1: Understand the meaning of "PROGENY".

"PROGENY" refers to offspring or children.

Step 2: Identify the opposite.

The opposite of "PROGENY" is "Parent", as a parent is the source of offspring.


Final Answer: \[ \boxed{Parent} \] Quick Tip: "Progeny" refers to offspring or children, while "Parent" refers to the individual from whom offspring are born.

Step 1: Identify the correct word.

The correct phrase should be "draw a conclusion", as "draw" is commonly used in this context.

Step 2: Conclusion.

The correct word is "Draw".


Final Answer: \[ \boxed{Draw} \] Quick Tip: In expressions like "draw a conclusion", "draw" is the appropriate verb to use.

Step 1: Identify the correct phrase.

The correct phrase is "Looking for," which is used when someone is searching for something.

Step 2: Conclusion.

Thus, "Looking for" is the correct phrase.


Final Answer: \[ \boxed{Looking for} \] Quick Tip: Use "looking for" when referring to searching or trying to find something.

Step 1: Identify the correct phrase.

The phrase "Mind your language!" is already correct. It is a common expression used to tell someone to be careful with their words.

Step 2: Conclusion.

There is no improvement needed for the given sentence.


Final Answer: \[ \boxed{No improvement} \] Quick Tip: "Mind your language" is a standard expression and does not require modification.

Step 1: Identify the correct form.

The correct expression is "used to" when referring to a habit in the past.

Step 2: Conclusion.

Thus, the correct phrase is "used to".


Final Answer: \[ \boxed{used} \] Quick Tip: Use "used to" when talking about a past habit or repeated action.

Step 1: Identify the correct preposition.

When expressing anger directed towards someone, we use the preposition "with."

Step 2: Conclusion.

Thus, the correct phrase is "angry with."


Final Answer: \[ \boxed{with} \] Quick Tip: Use "angry with" when referring to someone who is the object of anger.

Step 1: Identify the correct preposition.

The correct preposition to use with "laugh" when referring to a person is "at," as in "laugh at someone."

Step 2: Conclusion.

Thus, the correct phrase is "laugh at the poor."


Final Answer: \[ \boxed{at} \] Quick Tip: Use "laugh at" when referring to mocking or ridiculing someone.

Step 1: Analyze the sequence.

The logical flow is: First, talk about consulting with the doctor, then admitting the fact of getting better, followed by the doctor's suggestion and belief.

Step 2: Conclusion.

The correct sequence is QRPS.


Final Answer: \[ \boxed{QRPS} \] Quick Tip: When ordering sentences, ensure the flow of ideas is logical and coherent.

Step 1: Analyze the sequence.

The logical flow is: Starting with the plan of Ram, then explaining the situation and asking for help, followed by hope for his action.

Step 2: Conclusion.

The correct sequence is QPRS.


Final Answer: \[ \boxed{QPRS} \] Quick Tip: When ordering sentences, think of the most logical progression of ideas.

Step 1: Analyze the sequence.

The logical flow is to express it as a personal issue, followed by the residents' attitude and the consequences, ending with the fact that everyone will eventually face similar issues.

Step 2: Conclusion.

The correct sequence is PRSQ.


Final Answer: \[ \boxed{PRSQ} \] Quick Tip: In sentence ordering, ensure that each part flows logically from one idea to the next.

Step 1: Understand the given code.

By observing the given pattern of the code, we can map the digits to the letters of the word.

Step 2: Conclusion.

The correct code for "38249" is "MEDOA".


Final Answer: \[ \boxed{MEDOA} \] Quick Tip: To decode a message, carefully analyze the pattern and map the corresponding letters to digits.

Step 1: Observe the pattern in the given figures.

Look at the numbers in the boxes and see if there is a mathematical relationship between the numbers.

Step 2: Conclusion.

The missing number is 7 based on the identified pattern.


Final Answer: \[ \boxed{7} \] Quick Tip: For pattern-based problems, look for consistent relationships between numbers or positions in the figures.

Step 1: Understand the relationship.

Illiterates, poor people, and unemployed people can be seen as overlapping categories, as someone can belong to more than one group.

Step 2: Conclusion.

The correct representation is overlapping circles.


Final Answer: \[ \boxed{Overlapping circles} \] Quick Tip: In Venn diagrams, overlapping circles are used to represent sets with common elements.

Step 1: Calculate the directions and distances.

Follow the steps as Sushma moves in the directions provided. After calculating the total distance and direction, she is 30m East from the original position.

Step 2: Conclusion.

Sushma is 30m East from her original position.


Final Answer: \[ \boxed{30m East} \] Quick Tip: Use a step-by-step approach to calculate the final position when the path involves multiple directions.

Step 1: Analyze the seating order.

Based on the given conditions, A is sitting behind C, and C is sitting behind E. D is in front of E, and B is behind A.

Step 2: Conclusion.

The seating order is DECAB.


Final Answer: \[ \boxed{DECAB} \] Quick Tip: To solve seating arrangement problems, break down each given condition and use logical reasoning to determine the positions.

Step 1: Find the pattern.

The numbers are following a pattern where each term is obtained by multiplying the previous number by a constant and adding a number.

Step 2: Conclusion.

The next number in the series is 65.


Final Answer: \[ \boxed{65} \] Quick Tip: Look for arithmetic or geometric patterns in number sequences to identify the next number.

Step 1: Analyze the equation.

Substitute the signs \( + \) and \( \div \) into the equation to make it correct.

Step 2: Conclusion.

After interchanging the signs, the equation becomes correct when \( + \) and \( \div \) are interchanged.


Final Answer: \[ \boxed{+ and \div} \] Quick Tip: When solving sign-interchange problems, carefully check each sign and test by substituting it into the equation.

Step 1: Understand the assertion and reason.

The assertion "India is a democratic country" is true. The reason "India has a constitution of its own" is also true, but it does not directly explain the assertion that India is democratic.

Step 2: Conclusion.

Thus, both (A) and (R) are true, but (R) does not explain why India is a democratic country.


Final Answer: \[ \boxed{Both (A) and (R) are true but (R) is not the correct explanation of (A)} \] Quick Tip: When dealing with assertion and reason questions, check if the reason logically explains the assertion.

Step 1: Identify the pattern in the figure.

Observe the pattern and look for the figure that completes the pattern logically.

Step 2: Conclusion.

The correct figure that completes the original figure is Figure B.


Final Answer: \[ \boxed{Figure B} \] Quick Tip: Look for symmetry, rotation, or reflection in the figures to identify the correct option.

Step 1: Count the squares in the figure.

Count all the small and large squares present in the grid.

Step 2: Conclusion.

The total number of squares in the figure is 26.


Final Answer: \[ \boxed{26} \] Quick Tip: When counting squares, consider different sizes of squares in the figure.