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Tetrahedron's Volume Ratio
A regular tetrahedron with edge length 6 cm and another with edge length 8 cm are similar in shape. Find the ratio of the volume of the smaller tetrahedron to the volume of the larger one.
Combination of Capacitors?
Three capacitors of capacitances 2 μF, 3 μF and 5 μF are connected in parallel across a 20 V battery. How will the total capacitance of the combination be affected if a dielectric material of dielectric constant 3 is inserted between the plates of the 2 μF capacitor?
Designing a Radar System?
A radar system uses the inverse tangent function to determine the angle of an object from a distance of 30 km, while the object's height is 20% of its distance. If the object is at a distance of 60 km, what would be the angle at which the object is detected by the radar system?
Finding the Angle of Elevation?
From the top of a building, the angle of elevation to the top of a vertical pole is observed to be 60°. If the height of the building is 10 metres, how far is the pole from the building?
Modelling Population Growth?
The population of a certain city is growing at a rate proportional to the number of people already living there. If the current population is 1 million and is increasing at a rate of 5% per annum, find the population after 5 years, given that the initial population was 500,000 20 years ago. Assume a constant growth rate and a constant initial population in the past.
Area of Triangle - A Determinant Approach?
The area of a triangle with vertices (1, 2), (3, 4), and (x, y) is 13 square units. Using determinants, show that the point (x, y) lies on the circle x^2 + y^2 - 6x - 8y = 0.
Resistance in a Wire Network?
A wire of resistance R1 is connected in series with another wire of resistance R2. The total resistance of the network is measured to be 6 ohms when a voltmeter is connected across R2. If the voltage applied is 12 volts, find the value of R2.
Finding Determinant's Significance?
Consider a surveyor using determinants to calculate the area of a plot of land. If the determinant of the matrix of distances between reference points is negative, what does it imply about the orientation of the plot? Explain with an example.
Direction Cosines and Dot Product?
If A = 2i + 3j - k and B = i - 2j + 4k, find the angle between the vectors A and B and express it in radians, given that the direction cosines of vector A are 2/√14, 3/√14, and -1/√14.
Rainwater Harvesting Model?
A small town with a population of 10,000 people plans to harvest and store rainwater for non-potable purposes. The town's rooftop area is 500,000 sq. m with an average annual rainfall of 1,000 mm. Assuming 25% of the rooftop area contributes to rainwater collection and the initial water level in the tank is 5 m. Formulate and explain the integral to calculate the total volume of rainwater collected in the tank over a year.
Inverse of a Product Matrix?
If ABC is the inverse of matrix A, and D is any invertible matrix, explain how does the relationship between A, B, C and D help in finding (ABC)−1, given that (ABC)−1 = C−1B−1A−1.
Calculating Power Consumption?
A 100 W electric bulb is used for 6 hours a day. If the cost of electricity is ₹ 3 per unit, calculate the daily cost of using this bulb and suggest ways to reduce the cost.
Functional Equivalence and Composition
In a given pair of functions, determine whether they are equivalent and provide a proof for your answer. Further, compose these functions and explain its significance in real-world scenarios.
Matrix Transformation Invariance?
Consider a matrix transformation that doubles the area of any given rectangle. If a rectangle with sides 3 and 4 is transformed, what are the dimensions of the resulting rectangle?
Induced Currents in Moving Conductors?
A wire loop is placed inside a moving magnet and connected to a sensitive ammeter. Suddenly, the magnet is removed, and the loop is rotated while moving with uniform speed. Describe the changes in the ammeter reading and explain the underlying phenomenon using the concept of electromagnetic induction.
Dodecahedron Symmetry?
A regular dodecahedron has 20 faces and 30 edges. Explain, with supporting geometric reasoning, why it is impossible to inscribe a regular tetrahedron within this dodecahedron. Consider the symmetries involved in the icosahedral group and their effect on the spatial arrangement of the dodecahedron's vertices and faces.
Comparing Resistances in Parallel
Consider a circuit with three identical resistors connected in parallel, each of resistance 2 ohms. If a voltage source of 6 volts is connected across the circuit, compare the total resistance of the circuit with the resistance of a single resistor, and explain the reasoning behind your answer.
Inverse Sine in Real-Life Scenarios?
Consider a right-angled triangle with angle θ and opposite side length y. If the length of the adjacent side is fixed at 3 units, how can you use the inverse sine function to determine the possible values of θ for different values of y?
Magnetic Field and its Effects?
A bar magnet is placed inside a hollow copper tube. Explain the observed phenomena when the tube is rotated slowly around its axis while the magnet is stationary. Assume the tube is insulated from the ground.
Designing a Circular Ruler?
The circular ruler shown here is marked with a scale of angles in degrees. If the circular ruler is folded into a straight line, a new scale of numbers is formed. Using the inverse trigonometric functions, determine the new scale of numbers on the straight line for a circular ruler with a radius of 5 cm.
A Spherical Conductor Holds Charge?
A spherical conductor of radius 5 cm has a charge of 0.5 μC distributed uniformly over its surface. Does this conductor behave as an equipotential surface everywhere inside and outside, and what does this imply about its electrostatic potential?
Finding the Diagonal of a Box?
A box has dimensions 3 cm x 4 cm x 5 cm. Show that the box can be cut into two equal halves along a plane passing through its centre and perpendicular to its length, and find the length of the diagonal of each half.
Differentiability at Discontinuity?
Consider a function f(x) = |x| at x = 0. Is f(x) differentiable at x = 0? Justify your answer with a detailed explanation of the necessary conditions for differentiability.
Inducing a Change?
In a generator, a 300-turn copper coil with a radius of 0.1 m is placed in a uniform magnetic field of 2 T. If the coil is rotated at a frequency of 50 Hz, determine the magnitude and direction of the induced EMF. Assume the plane of the coil coil is initially perpendicular to the magnetic field.
Matrix Transformation Paradox?
In a 2D plane, a matrix T is applied to a point (2, 3) resulting in a new point (5, 1). If the same matrix T is applied to a point (4, 5), the resulting point is (7, 3). What is the transformation represented by matrix T in the coordinate plane?
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