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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?
How Many Ways to Arrange?
A bookshelf has 5 shelves, and each shelf can hold 3 books. If the same books are to be placed on each shelf, how many ways can you arrange the books on the bookshelf, considering the order of books on each shelf matters?
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.
Fairness of Loaded Dice?
Two identical dice are loaded in such a way that the probability of getting 1 on one die is 0.4, and on the other die, it is 0.5. What can you infer about the fairness of these two loaded dice in comparison to a standard die?
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.
Velocity of a Spring?
A spring is stretched by 2 cm when a force of 2 N is applied to it. Using Hooke's Law (F = kx), obtain the velocity of the spring at the instant the force applied is increased from 2 N to 3 N. Assume the spring is stretched by 2 cm at time t = 0 s.
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.
Interpreting Correlation Coefficient?
The correlation coefficient between the heights of fathers and their sons is found to be 0.8. Does this imply a causal relationship between the two variables, and why or why not?
A Billionth Portion of Infinity?
Suppose you have a tea cup containing an infinite amount of water. You start pouring out water into another cup, dividing each portion into two equal parts. If you do this an infinite number of times, what would be the amount of water left in the original cup? Assume you start with a cup that's half full, and each portion is poured out completely.
Can a function be one-to-many and onto?
Consider a function f: R → R defined by f(x) = 2x + 1. Is it possible for this function to be both one-to-many (injective) and onto (surjective)? Explain your reasoning with examples or counterexamples.
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