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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.
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?
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.
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.
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?
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.
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.
Modelling Real-World Situations?
A particle moves along the curve y = 3x^2 - 4x + 5. If it starts at (1, 2), use integration to find the total distance travelled by the particle in the first two seconds of its motion, assuming it moves in the positive y-direction.
Maximizing Profit: A Constrained Problem?
The XYZ Corporation produces two products, A and B, which require 3 hours and 2 hours of labor respectively. The profit per unit of A is ₹ 100 and B is ₹ 150. If the available labor is limited to 120 hours and 50 units of product A and 30 units of product B are already in production, formulate and solve the linear programming problem to maximize the total profit.
Determinant of a 3x3 Matrix
Consider a 3x3 matrix A where each element is the square of the corresponding position in the matrix generated by the first 3 natural numbers. Show that the determinant of matrix A is zero.
Vector Position and Reflection?
From a point A(-5, 4) in the Cartesian plane, a point B is reflected over the point C(1, -3) to obtain point B'. If the vector ℓB = ℓ(5, -6) represents vector AB, find the vector ℓB' representing vector AB' and explain why B' doesn't coincide with B + 2ℓB.
Inverse Trigonometric Functions: A Real-Life Application?
A camera's viewfinder uses trigonometric functions to calculate the angle of view. If the inverse trigonometric function is used to find the angle of elevation, how would you use it to determine the height of a building from a given distance, assuming the angle of elevation is 60 degrees?
Maximising Rainwater Harvesting
A water conservation society has built a cylindrical tank with a height of 6 meters and a radius of 4 meters. Assuming the tank is filled with rainwater to its brim, determine the depth at which the rate of increase of the volume of water is the same as the rate of increase of the surface area of the water. Justify your answer using the concept of related rates.
Population Growth Paradox?
Tommy's village has an initial population of 1000 people. It is growing at a rate proportional to the square root of the current population. If the rate of growth is 10√P people per year, where P is the population at any given time, then find the population after 10 years and explain why the population growth seems paradoxical.
Inverse Trigonometry: A Real-World Application?
A carpenter uses a level to ensure a perfectly horizontal surface. If the level's bubble is displaced by 12°, and the level's distance from the ground is 4.5 meters, how can you use inverse trigonometric functions to determine the distance between the level and the point directly below it?
Can Vectors be Added?
Given two vectors a and b, can you think of a situation where the addition of vectors a and b would not result in a new vector? Describe a scenario where this might occur and explain why this is possible in vector algebra.
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