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Determinant of a Matrix Transform

Consider a 2x2 matrix A that represents the scaling transformation of a figure. If the determinant of A is 3, and the original figure has an area of 10 square units, what is the area of the transformed figure?

Optimizing a Function?

A company producing concrete products wants to minimize its transportation cost by locating a new factory at a point on the x-axis. The cost of transportation is directly proportional to the distance from the factory to a fixed point, given by the function C(x) = (x^2 + 2x + 2) / (x^2 - 4x + 5), where x is the distance in kilometers. Determine the exact location of the factory that minimizes the transportation cost.

Rainfall Volume Calculation?

A dam's reservoir has a rectangular cross-section with a length of 100 m and a variable width, represented by the function w(x) = 5x + 2, where x is the distance from the left end in meters. If the water depth is 3 meters, calculate the volume of water in the reservoir between the points where x = 0 and x = 20 m.

Maximizing Profit?

Rahul, a small entrepreneur, wants to manufacture a limited quantity of a product to maximize his profit. He has a fixed cost of ₹1000 and a variable cost of ₹15 per unit. If he sells each unit at ₹30, formulate and solve the linear inequality to find the maximum quantity he should produce to achieve the highest profit.

Population Growth Model?

The rate of change of a population's size is proportional to the current population. Using this information, derive a differential equation to model the population growth, assuming the initial population is 1000 and the growth rate constant is 0.02.

Optimal Speed Limit?

A car travels from City A to City B at an average speed of 60 km/h. If the driver accelerates uniformly from rest to reach a top speed of 80 km/h in 5 minutes, find the speed limit on the road such that the total travel time is minimized.

Modeling Population Growth

A certain species of fish grows at a rate proportional to its current population. If the initial population is 100 and the growth rate is 25% per year, use the differential equation to determine the population after 5 years. Consider the implications of this growth on the ecosystem.

Favorable Outcomes in a Game?

Rahul and Aarav are playing a board game where they can earn rewards based on the roll of a fair six-sided die. Rahul is offered a deal: if the die lands on an even number, he gets a reward of Rs. 20, but if it lands on an odd number, he loses Rs. 15. Aarav, on the other hand, is offered a deal that guarantees him a reward of Rs. 20 if the die lands on an even number, with no loss in case of an odd number. Assuming both deals are equally likely to happen, which one should Rahul prefer, and why?

Analyzing Function Composition?

Consider two functions f(x) = 2x^2 and g(x) = √x. If h(x) is a composite function of f and g, write an expression for h(x) and explain how h(x) can be used to describe the process of finding the square root of a perfect square number.

Fair Play or Not?

In a game show, a contestant is asked to choose one of three doors, behind one of which is a car. The host then opens one of the other two doors, revealing a goat behind it. Assuming that the contestant switches doors, what is the probability that the car is behind their initial choice versus the door they switch to?

Direction Ratios of Resultant Vector?

If A = 2i - 3j + 4k and B = -3i + 2j - 5k are two vectors, find the direction ratios of the resultant vector when A is multiplied by 2 and added to B.

Optimization of a Production Process?

A factory produces metal rods of uniform length and diameter. The cost of raw materials and production is directly proportional to the surface area of the rod. If the factory produces rods of length 1 meter, find the optimal diameter that minimizes the production cost. Assume the cost is given by C = k(2πrh), where k is a constant, r is the radius, and h is the height of the rod.

Modelling Population Growth?

The population of a certain city is growing at a rate proportional to the current population. If the initial population is 1,00,000 and the growth rate is 5% per year, use the appropriate definite integral to find the population after 10 years.

If the sum of zeros of a quadratic polynomial is 5 6 and product of zeros is , then the polynomial is: x− 2 5x + 6 a) b) x + 2 5x + 6 c) d) x− 2 6x + 5 x + 2 6x− 5

Life Expectancy Conundrum?

According to a recent study, the life expectancy of Indians has increased by 5 years in the last decade. However, this increase is not uniform across all age groups. Analyze the implications of this trend on the age distribution of the Indian population.