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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.
Modeling Population Growth?
A town's population is increasing at a rate proportional to the square root of the current population. If the initial population is 10000 and it takes 3 years for the population to reach 12000, use the concept of accumulation to find the total population after 5 years.
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?
Direction Ratios from a Plane and a Line?
Given a plane passing through the point (1, 1, -1) with a normal vector <2, 3, -1>, and a line passing through the point (0, 0, 2) with direction ratios <1, 2, 3>. Show that the line lies in the given plane.
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.
Modeling Population Growth?
The population of a certain region grows at a rate proportional to the product of the current population and the difference between the carrying capacity and the current population. Assuming the carrying capacity is 1000 and the initial population is 200, model this situation using a differential equation and solve it to find the population after 10 years.
Modeling Rainwater Harvesting?
A rainwater harvesting system is to be designed for a small community. The roof of the community center has a rectangular shape, measuring 20 meters by 15 meters. Assuming a uniform rainfall rate of 1 mm/hour, determine the volume of rainwater collected per hour when it rains.
Determinant of a 3x3 Matrix - Application?
Suppose you are given a 3x3 matrix representing the coefficients of three linear equations in two variables. Can you use the concept of determinants to justify that the equations have a unique solution, no solution, or infinitely many solutions?
Modeling Population Growth?
A small town has a population of 1000 people, with an initial growth rate of 2% per annum. Using the logistic growth model, derive a differential equation to represent the population growth, and explain the significance of the carrying capacity in this context.
Fountain Height Equation?
A water fountain is created by shooting water jets from a fixed point. If the height of the water jet is given by h(x) = 10x^3 - 12x^2 + 5x + 2, where x is the horizontal distance from the fountain, what equation would you derive to find the total height of the fountain when the water jets merge at a point 10 meters away from the fountain?
Cost Minimization Problem?
A company producing electronic components uses a wire of length L to manufacture a rectangular coil with a fixed perimeter. The cost of wire is directly proportional to the length of the wire. If the cost per unit length is ₹ 5, and the perimeter of the coil is 64 cm, find the dimensions of the coil that will minimize the cost. Assume the cost of wire is directly proportional to its length.
Position Vector of a Diagonal
In a triangle ABC with position vectors Φa = 2i + 3j, Φb = i - 4j, and Φc = 5i + 2j, find the position vector of the point D where AD:DB = 2:3. You may use vectors Φd or Φe to represent this position vector.
Matrix Transformation Inverse?
Given that matrix A represents a transformation that reflects points across the y-axis, and matrix B represents a transformation that rotates points by 90 degrees clockwise, determine the matrix that represents the composition of these two transformations and explain why it results in a transformation that can be inverted.
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.
Water Tank Capacity?
A water tank is in the shape of a right circular cylinder with height 10 m and base radius 4 m. Water is being pumped into the tank at a rate of 2 m^3/min. How long will it take to fill the tank to a depth of 6 m?
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?
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