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Divergent Geometric Series?
Consider a geometric series with first term a = 1 and a common ratio r = -1/2. If the series is extended indefinitely, will the sum of the series converge or diverge?
Conditional Probability Puzzle?
Suppose a bag contains 3 red, 5 blue, and 2 green balls. If two balls are drawn one after the other without replacement, what is the probability that the second ball drawn is blue, given that the first ball drawn was not blue?
Reflections in Relations?
Consider a relation $R$ on the set of all real functions defined by $f R g$ if $f = g.$ If $P$ and $Q$ are the given functions $P(x) = 2x^2+1$ and $Q(x) = rac{2x^2+1}{x},$ show that $(P cup Q) R (P cap Q).$
Optimization of a Function?
A factory produces conical shaped hats. The cost of raw materials to produce a hat is directly proportional to the square of its radius. If the volume of the hat is 200π cubic centimeters, find the radius that minimizes the cost of production.
Maximising Revenue?
A company manufactures and sells x units of a product per day. The selling price per unit is ₹(100 - 2x) and the cost price per unit is ₹(60 + 0.5x). How many units should be produced for maximum revenue?
Minimizing Production Costs?
A fertilizer factory produces two types of fertilizers, X and Y. The cost of production for X is ₹5 per unit, and for Y is ₹8 per unit. However, producing Y requires an additional machinery rental of ₹50 per week. If the factory aims to meet a weekly demand of 120 units of X and 150 units of Y, and the total weekly production budget is ₹16000, how should the production be planned to minimize costs?
City's Water Resource Management?
The city of Noida has a constant water inflow of 1000 cubic meters per hour. It has two reservoirs, A and B, with capacities of 2,00,000 cubic meters and 3,00,000 cubic meters respectively. Using definite integrals, determine the number of hours for which the reservoirs can supply water at a rate of 800 cubic meters per hour to the residents of Noida.
Conic Sections: Ellipse or Hyperbola?
In a market research survey, the relationship between the number of respondents and their ages is modeled using a conic section. If the data points (15, 20), (30, 25), (50, 20), and (70, 10) are plotted on a coordinate plane, identify the type of conic section that best represents this data and justify your choice.
Summation of Infinite Series?
A mathematician claims that the sum of the infinite series 1/2 + 1/4 + 1/8 + ... is equal to 1. Verify the claim by evaluating the sum of the series and justifying your answer with mathematical reasoning.
Fair Coin Toss Sequence?
A fair coin is tossed repeatedly, and the sequence of heads (H) and tails (T) is recorded. If the sequence starts with HHTT, what is the probability that the next toss will be a tail?
Derivative of a function: A thought experiment?
Suppose you are designing a roller coaster ride with a track shaped like a parabola. The height of the track at any point is given by the function h(x) = -2x^2 + 5x + 1. At what point on the track is the roller coaster car moving fastest, assuming it starts from rest and gains speed uniformly?
Tanking Capacity of a Tank?
A cylindrical water tank of height 10 m and radius 4 m is to be built. If the tank's capacity is to be maximized, what should be the shape of the tank, and how would you determine its optimal dimensions using an integral?
Parabola in Focus?
A camera lens uses a parabolic mirror to focus light rays onto a single point. If the parabola has its vertex at (0, 0) and the focus is 12 cm away from the vertex, find the equation of the parabola in standard form. Assume the axis of symmetry is along the x-axis.
Optimizing Witness Cost?
The Mumbai Police Department wants to allocate its personnel for surveillance in a high-security area. There are two types of witnesses: those who can identify a suspect with high accuracy and those who can do so with moderate accuracy. A high-accuracy witness costs ₹30,000 per day, while a moderate-accuracy witness costs ₹15,000 per day. For a specific event, the Police Department has a budget of ₹1,50,000. If the probability of the suspect attending the event is 0.7, and the probability of a high-accuracy witness correctly identifying the suspect is 0.85, while that of a moderate-accuracy witness is 0.6, how many high-accuracy and moderate-accuracy witnesses should be hired to achieve the highest possible probability of identifying the suspect?
Maximizing Profit with Resource Constraints?
A company manufacturing smartphones and laptops has 800 hours of labour and 400 units of memory available. The profit from each smartphone is ₹800 and from each laptop is ₹1200. Formulate a linear programming problem to maximize the profit, considering the constraints on labour and memory.
Investigating Geometric Progressions?
Consider a sequence where each term is twice the previous term and starts at 4. Examine the nature of this sequence, its first three terms, and calculate the fifth term. Can you generalize the sequence and find a formula for its nth term?
Inverse of a Function Exists?
In a given function f(x) = 2x^2, it is observed that no two different elements of the domain have the same element in the range. Does this imply that the inverse of f(x) exists, and why?
Maximizing Profit with Constraints?
Anju's Bakery produces two types of cakes - chocolate and vanilla. The profit from each chocolate cake is ₹ 15 and from each vanilla cake is ₹ 10. However, production is limited by a 4-hour workday, where each chocolate cake requires 2 hours and each vanilla cake requires 1.5 hours. Formulate a linear programming model to determine the optimal number of chocolate and vanilla cakes to produce for maximum profit.
Inverse Trigonometric Functions: Problem of Real-World Significance?
The inverse trigonometric functions are essential in calculating the angle of elevation for a ladder placed against a building. A 20-foot ladder is placed against a building with its foot 5 feet away from the wall. If the top of the ladder is 15 feet above the ground, can you determine the angle of elevation using inverse trigonometric functions?
Solving a Real-World Problem with Inverse Trigonometric Functions?
A projectile is launched from the ground at an angle of 60 degrees above the horizontal with an initial velocity of 20 m/s. Assuming negligible air resistance, find the maximum height reached by the projectile if it lands 50 meters away from the point of launch. Use inverse trigonometric functions to solve this problem.
Maximizing Revenue: An E-commerce Dilemma?
A small e-commerce business sells a product at an initial price of $100. Due to market competition, the demand for the product is decreasing at a rate of 2% per day. To maximize revenue, the business needs to adjust the product price. Using the demand function, find the optimal price at which the revenue is maximized.
Traffic Modelling Conundrum?
The city traffic engineer observes that the rate at which people arrive at a popular bus stop is given by the function f(t) = 2t^2 + 3t + 1, where t is the time in hours. Assuming the bus arrives once every 4 hours, calculate the average number of people waiting at the bus stop, excluding the last group that arrives right before the bus, over a period of 12 hours. Assume the rate of people leaving the bus stop is negligible.
Can a Circle be a Conic Section?
A conic section is a curve obtained by cutting a cone with a plane. Consider the equation of a circle as a special case of a conic section. Is it possible to obtain a circle as a conic section by suitably adjusting the parameters of the equation of a circle?
Critical Points on a Curve?
Let f(x) = |x^2 - 4x| + 1. Show that the function is differentiable at all points except at x = 2 and x = 0, and justify your answer with proper reasoning.
Interpreting Velocity Vectors in Projectile Motion?
A particle is projected from ground level with an initial velocity of 25 m/s at an angle of 60° to the horizontal. Assuming the acceleration due to gravity is 9.8 m/s² downward, represent the velocity of the particle 2 seconds after projection in terms of its horizontal and vertical components, and analyze its effect on the trajectory of the particle.
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