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Cosine of the Angle between Vectors?
In a triangle ABC, D lies on the same plane as ABC and is the foot of the perpendicular from A to BC. Given that cos(BAC) = 3/5, find the value of cos(CDB) if BC = 1 and AB = AC = √2.
How does a capacitor store electric charge?
A 2 μF capacitor is connected to a 12 V battery. Calculate the electric charge stored on the plates of the capacitor when the switch is closed. Assume the capacitor behaves as an ideal capacitor. Also, explain why the capacitor does not allow the current to flow when the switch is closed.
Induction's Time-Domain Response?
Consider a long solenoid carrying a current of 2 A. Suddenly, the current is switched off. Describe and explain the nature of the induced electromotive force (e.m.f.) in the solenoid as a function of time, assuming the magnetic flux through the solenoid is 0.02 Wb.
Modeling Population Growth?
The rate of growth of a population is proportional to its present size, and the population of a city is currently 100,000. If the growth rate is 0.02 per year, estimate the population after 5 years using the differential equation dP/dt = kP, where P is the population and k is the growth rate.
Can a function be continuous everywhere?
Consider a function f(x) = |x| in the interval [-1, 1]. Analyze the continuity of f(x) at x = 0, and then discuss whether the function can be made continuous at this point by modifying it slightly.
Maximizing Profit with Resource Constraints?
The manager of a factory wants to maximize profit by producing two products, A and B, which require 3 and 2 hours of labor, respectively. However, the labor union has imposed a constraint of 180 hours per day for both products combined. The profit per unit of A and B is ₹ 150 and ₹ 100, respectively. How would you formulate this problem as a linear programming model to determine the optimal production levels of A and B?
Capacitance of a Parallel-Plate Capacitor?
A parallel-plate capacitor is made of two plates, each 5 cm x 10 cm in size, separated by a 2 mm thick dielectric material. The plates are 1 cm apart when empty. How will the capacitance change if the gap between the plates is reduced to 1 mm and the dielectric material is replaced with a different one of double the dielectric constant?
Vector Algebra: Orthogonal Projection?
A vector μ is orthogonal to a vector α = <2, 3, -1> if and only if the projection of α onto μ is zero. If the vector α has a magnitude of 4 units, then find the possible range of the magnitude of μ, assuming that μ is orthogonal to α.
Magnetic Field of a Current Carrying Wire?
A long, straight wire carries a current of 20 A in a direction perpendicular to the plane of the paper. Two identical loops, each having a radius of 0.1 m, are placed symmetrically on either side of the wire. One loop is made of copper (conductivity = 5.8 × 10^7 S/m) and the other of a non-ferromagnetic material (conductivity = 3.0 × 10^7 S/m).
Capacitance in a Real-World Scenario?
A parallel plate capacitor is used in a fire alarm system. The plates are 5 cm apart and have an area of 0.05 m^2. The dielectric constant of the material between the plates is 3. The capacitance of the capacitor is 2 μF. How would you modify the capacitor to increase its capacitance by 50%?
Matrix Transformation Inversion?
Consider a matrix T that represents a 90° clockwise rotation of the coordinate plane. If T is applied twice in succession, describe the resulting transformation and justify your answer.
Mass of a Thin Disk: A Simplified Approach?
A thin disk of radius 5 cm is formed by revolving the region bounded by the curves y = √(x) and y = -√(x) about the x-axis. What is the mass of this disk if its density varies as √(x) with respect to x? Assume the density function to be continuous and differentiable.
Potential Differences in a Circuit?
Suppose you are designing a portable power bank with a capacity of 20,000 mAh. The power bank uses a rechargeable battery with an internal resistance of 0.5 Ω and an external resistance of 10 Ω in series. How would you determine the maximum potential difference across the power bank's terminals during charging, and what factors would you consider to minimize this potential difference?
Induction in a Coiled Wire?
A coiled wire is placed in a magnetic field. When the magnetic field is reduced by a factor of two in a time period of 1 second, and the radius of the coil is increased by 20%, what happens to the induced emf? Explain your reasoning with calculations to justify the effect on the emf.
Tan (arcsec x + 3) - 3
In a right-angled triangle ABC, AC is the hypotenuse and ∠C = π/3. If tan(∠A) = 3, find the value of tan(arcsec(x + 3)), where x = ∠B.
Vectors in Space: A Parallelogram Law Conundrum?
Consider two vectors in 3D space: A = <2, 3, -1> and B = <4, -2, 5>. If a third vector C is such that it forms a parallelogram with A and B, what are its components if the magnitude of C is 10 units?
Lifting a Magnet: What Happens?
Imagine you're holding a magnet suspended above a copper wire coil in a circuit. When you suddenly lift the magnet, the coil begins to generate a current. Explain why this happens and discuss the direction of the induced current in the coil.
Determinants in Geometry
In a given triangle, if the determinant of the matrix formed by its vertices is non-zero, explain the implications on the triangle's shape and its possibility of being a unique shape in a 2D plane.
Determine Plane Containing 3 Points?
Suppose A, B, and C are three non-collinear points in 3D space with position vectors Σ, β, and γ respectively. Explain how you can determine whether the points A, B, and C lie on a single plane.
Modeling Population Growth?
Consider a population of rabbits that grows at a rate proportional to its size. Assuming a constant birth rate of 0.5 rabbits per rabbit per year and a carrying capacity of 1000 rabbits in the region, analyze the effect of introducing an inhibitor that reduces the birth rate by 20% when the population exceeds 500 rabbits.
Inverse Trig Functions: A Real-World Application?
A manufacturer uses a machine that operates at a speed proportional to the sine of an angle between its parts. If the machine's speed is 240 km/h when the angle is 60°, and the manufacturer wants to increase the speed to 300 km/h, what angle should be set to achieve this, assuming the relationship remains the same?
Homogeneous Co-ordinates in Action?
Consider a sphere with radius 5 units, centered at the origin. If a point (x, y, z) lies on the surface of the sphere, express the equation of the sphere in homogeneous co-ordinates. Then, use this equation to find the point on the sphere that is closest to the point (4, 3, 0).
Motion of a Particle?
A particle moves along the x-axis with its velocity given by v(t) = 2t - 5. Find the position of the particle at time t = 3 seconds, given that it starts from rest at the origin.
Minimum Power Transfer Theorem?
In an AC circuit containing a resistor and an inductor, a variable capacitor is connected in parallel with the inductor. The capacitor's capacitance is slowly reduced from a very large value to a very small value. What happens to the power consumed by the inductor in the circuit as the capacitance reduces?
Population Growth Modeling?
A city's population is growing at a rate proportional to the current population. If the initial population is 10000 and the growth rate is 2% per year, use a differential equation to model this situation and find the population after 5 years.
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