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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.
Can you 'cut' the distance with a right-angled triangle?
In a right-angled triangle, the distance between the top of a tower and a point on the ground is 5√3 meters. If the angle between the ground and the line of sight to the top of the tower is 60°, how far is the point from the base of the tower?
Conditional Probability in Real Life
A medical research facility is conducting a study to determine the likelihood of a person developing diabetes based on their family history. It is known that 10% of people with a family history of diabetes will develop the disease, and 1% of people without a family history will develop it. If 70% of the population has a family history, what is the probability that a randomly selected person will develop diabetes?
Domain Restriction vs. Co-domain Constraint?
Consider two functions f: A → B and g: A → B with the same domain A. Function f has a domain restriction of A' ⊆ A, while function g has a co-domain constraint of B' ⊆ B. Determine the conditions under which f and g would be equal, assuming both are bijective.
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.
Domain of a Composite Function?
Consider two functions f(x) = √(x - 4) and g(x) = x^2 + 1. If h(x) = g(f(x)), find the domain of h(x), justifying your answer with proper reasoning.
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?
Line of Reflection?
Consider a line that reflects a point (3, 4) with respect to the line y = x. Find the coordinates of the reflected point and explain the relationship between the slope of the line of reflection and the reflected point.
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.
Arc Length from a Parametric Curve?
A bug is moving along a parametric curve given by x = 2cos(t) and y = 3sin(t), where t lies between 0 and π/2. If the bug moves at a rate of 1 unit per second, how much distance will it cover in the first 5 seconds?
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).
Arithmetic Progression in Permutations
In a cycle of 7 friends, A takes 3 turns to get a scoop of ice cream from a 5-person ice cream cart. Using the concept of Arithmetic Progression, how many different sequences of ice cream flavors can the friends get in their 7 turns, if the 5-person cart has 'Chocolate', 'Vanilla', 'Strawberry', 'Pistachio', and 'Butter Pecan'? Explain your reasoning.
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.
Rapid Expansion: A Queue's Growth?
A local taxi service, 'Quick Ride', offers a discount on its fare based on the number of passengers. The service charges ₹5 for the first passenger, and for each additional passenger, the fare is halved. How many passengers can a taxi with a seating capacity of 8 accommodate if the total fare collected from 8 passengers exceeds ₹800?
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