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Maximising Resource Allocation
A developer has Rs 100,000 to invest in two projects. The first project yields an annual profit of x dollars for every dollar invested, while the second project yields 10% more profit. If the total investment in both projects cannot exceed Rs 100,000, how much should be invested in the first project to maximise the total profit?
Reflecting Relations?
If f: R → R is a function defined as f(x) = 2x + 1, and R is the relation defined as {(x, y) | y = f(x)}, is R a function? Justify your answer with a suitable example or counterexample.
Visualizing Complex Roots?
Consider the quadratic equation z^2 + 4z + 4 = 0. If the roots are complex numbers, describe their graphical representation on the complex plane and explain why they represent the same point.
Focusing on Reality?
Consider a satellite in a geosynchronous orbit around the Earth. Its trajectory is a parabola, but if it were to circle a star with a much weaker gravitational pull, the shape of its orbit would need to change to maintain equilibrium. How would the shape of its orbit change, and what conic section would it represent if the gravitational pull of the star were x times weaker than that of the Earth?
Applying the Central Limit Theorem?
Suppose you are given the scores of 10 students from a small town, and the mean score is 80 with a standard deviation of 5. If you were to randomly select 5 students from this town, what is the probability that their mean score would exceed 85?
Library Bookshelf Arrangement?
In a school library, there are eight books to be arranged on a bookshelf in a row. Five of the books are fiction and three are non-fiction. The fiction books are to be placed together, and the non-fiction books are to be placed together. In how many distinct ways can these eight books be arranged?
Visualizing Complex Roots?
Consider the quadratic equation x^2 + 4x + 5 = 0. If α and β are the roots of this equation, show that the point A(α, β) lies on the line y = x, and hence, find the coordinates of A.
Modulus and Argument of Complex Numbers?
In the complex plane, if the modulus of the complex number z increases by 10 units, and its argument changes from 30° to 150°, what is the corresponding change in the value of z?
What's the Best Predictor?
A school administrator wants to identify the factor that best correlates with academic performance in Grade 11. Two variables are being considered: the duration of homework (in hours) and the number of hours spent on extracurricular activities. What statistical measure would you use to determine the predictor of academic performance?
Parabolic Dish Design?
A parabolic dish is designed to collect and focus sunlight onto a central point. If the dish has a depth of 1.5 meters and a focal length of 2 meters, calculate the diameter of the dish's cross-section, assuming it is a perfect parabola.
Towers of Reflection?
The town of Shivpur has two identical towers, one on each side of the river. From a boat in the middle of the river, the angles of elevation to the top of each tower are 30° and 60° respectively. Calculate the ratio of the heights of the two towers.
Rational Inequality using Binomial Theorem?
Consider the expression $(1 + x)^5$ and the inequality $0 < (1 + x)^5 < 100$. Use the binomial theorem to determine the range of values for $x$ that satisfy this inequality.
Line Intersection Puzzle?
In a triangle XYZ, line XY is perpendicular to line ZY. Point P lies on XY and point Q lies on ZY, such that PQ is parallel to XY. How does the position of point P affect the length of PQ?
Function Composition Dilemma?
Suppose we have two functions, f(x) = 2x + 1 and g(x) = x^2 - 4. Can we find a value of x such that f(g(x)) equals 2x + 1?
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.
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.
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.
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?
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.
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