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Finding Equation of Plane?
Given points A(2, 3, 1), B(1, 2, 2), and C(3, 1, 3), find the equation of the plane passing through points A, B, and C. Hence, determine if the points lie on a unique plane.
Critical Points Matter?
A function f(x) is continuous at a point x=c, but its derivative f'(x) is not defined at the same point. Is it possible for f(x) to be differentiable at x=c, given that it is continuous at the same point?
Can a charged object be made to attract another object of the same charge?
A small plastic rod is rubbed with a cloth, resulting in the transfer of electrons. The rod then attracts a small piece of paper, but not a second identical plastic rod. Analyze the phenomenon and explain the underlying principle.
Vector Equilibrium: A Geometric Conundrum?
Two forces, represented by vectors A and B, are applied at right angles to a point on a rigid structure. If the magnitudes of A and B are 10 N and 15 N, respectively, and the resultant force is perpendicular to force B, determine the vector equation representing the resultant force on the structure.
Electrostatic Shielding of a Charged Object?
A small charged object of 10 cm radius is placed inside a hollow metal sphere of 20 cm radius. What happens to the electric field and potential outside the sphere when the object is brought near the center of the sphere, and how does it affect the capacitance of the sphere?
Can Relations and Functions Unify?
Consider a relation R = {(1, 3), (2, 5), (3, 7)} and a function f(x) = 2x + 1. Analyze if there exists a function g(x) that can unify the relation R with the given function f(x), and justify your answer with a specific example or counterexample.
Solving for Volume Ratio?
A triangular prism has a trapezoidal base with parallel sides 8 cm and 12 cm and height 6 cm. If the prism is stretched by a factor of 3 in one direction, how will it affect the volume of the prism, considering its base remains trapezoidal?
Motion of a Charged Particle?
A charged particle moves along a uniform electric field. If the electric field is increased by a factor of 4, how will the path of the particle change? Explain with a neat diagram and support your answer with relevant concepts from the chapter.
Inverse Trigonometric Functions: Real-world Relevance?
The graph of y = sin^(-1) (x) is a function that maps the domain -1 ≤ x ≤ 1 to the range -π/2 ≤ y ≤ π/2. However, this graph appears to be a reflection of the graph of y = sin(x) across the line y = x. What does this symmetry imply about the properties of the function y = sin^(-1) (x)?
How does the Electric Field affect the trajectory of charged particles?
A charged particle of mass 1 kg and charge 2 μC is projected into a uniform electric field of 4 × 10^4 N/C. If the particle is projected at an angle of 60° with the direction of the electric field, what will be its trajectory and how will it change if the charge is doubled?
Tank Filling: Instantaneous Rate of Flow?
A water tank is being filled at an initial rate of 2 litres per minute. The rate of filling decreases by 0.1 litres per minute every minute. If the tank had 150 litres of water in it initially, how long will it take to fill the tank?
How does the presence of a moving conductor affect the magnetic field strength?
A straight copper wire of length 2 m is moving with a constant velocity of 10 m/s in a direction perpendicular to a magnetic field of 0.5 T. What happens to the magnetic field strength when the wire is moving in the same direction as the magnetic field? In a different direction? Assume the wire's cross-sectional area is negligible.
Magnetic Field Lines and Matter Interaction?
A rare-earth magnet is placed near a paper clip. Describe the changes in the magnetic field lines and their interactions with the paper clip, assuming the magnet's strength remains constant.
Transforming a Sphere into an Ellipsoid?
The equation of a sphere in three-dimensional space is given by x^2 + y^2 + z^2 = 4. If we replace the x term with (x + z)^2, what is the new equation of the surface, and what is its geometric interpretation?
Capacitor in Parallel: How does it affect the Total Capacitance?
Three capacitors of 10 μF, 20 μF, and 30 μF are connected in parallel across a 200 V supply. What is the total capacitance of the combination and how does it compare with the capacitance of each individual capacitor?
Can a relation be a function without being injective?
Consider a relation R = {(2, 1), (3, 1), (4, 1), (5, 1)} on the set A = {1, 2, 3, 4, 5}. Is this relation a function? If yes, explain why; if not, provide an example of a relation that is a function without being injective.
Rainwater Harvesting System?
A town planning to set up a rainwater harvesting system to supplement its water supply has a concrete tank with a trapezoidal cross-section. The dimensions of the tank are given as 10 m and 6 m for the parallel sides, with a height of 3 m. If the tank is to be filled from a point source 2 m away from the base, find the distance from the base at which the rainwater should be poured to uniformly fill the tank.
Why does a long straight wire carry current produce a magnetic field?
A current-carrying long straight wire is placed parallel to a cylindrical conductor made of a non-ferromagnetic material. The wire carries a current of 5 A, and the cylinder has a radius of 0.5 m. Explain how the direction of the magnetic field produced by the wire can be determined using the right-hand rule in different situations – when the wire and cylinder are stationary, and when the wire is moving past the cylinder while the cylinder is stationary.
Convergence of a Series?
Consider the series ∑(x^n)/(n + 1) from n = 1 to infinity. For the interval |x| < 1, discuss the convergence of this series using the criteria of continuity and differentiability of the function f(x) = ∑(x^n)/(n + 1)
Tank Filling Problem?
A tank with a cylindrical shape can hold 5000 litres of water. Water is flowing into the tank at a rate proportional to the volume already present, with the rate of flow being 100 + 0.02V litres per second, where V is the volume in litres. If the tank is initially empty, calculate the time taken to fill it to 4800 litres.
Lifting a Lift Off?
A DC motor is used to power a lift in an amusement park, and suddenly the power supply is interrupted due to a short circuit. Explain the effect on the current flowing through the motor and the lift's motion.
Induced Voltage in Alternating Fields?
A conductor is placed in an alternating magnetic field of frequency 50 Hz. Explain, with suitable mathematical reasoning, how the induced voltage in the conductor changes with time, assuming the magnetic field's amplitude remains constant.
Magnetic Field and Lenz's Law Inversion?
A superconducting material, when cooled, exhibits perfect diamagnetism. You are given a long solenoid with its axis aligned with the radius of a superconducting ring. When the current in the solenoid is switched on, devise a method to induce a current in the superconducting ring using the principle of electromagnetic induction.
Maximizing Profit under Constraints?
A company manufactures x units of product A and y units of product B using two machines, each available for at most 120 hours. It takes 2 hours to produce one unit of product A and 3 hours to produce one unit of product B. The profit is Rs. 10 per unit of product A and Rs. 15 per unit of product B. Assuming the company operates the machines for 120 hours, find the number of units of each product to maximize the profit.
Kirchhoff's Loop Rule Conundrum?
In a circuit containing a 2 Ω resistor, a 3 V battery, and an unknown inductor, the total resistance is measured to be 5 Ω. Using only Kirchhoff's loop rule, derive an expression for the inductance of the inductor and discuss its implications on the circuit.
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