Differential Equations

Step 1: Translate statement into differential equation.
Rate of change proportional to population:

Step 2: Solve the differential equation.

Integrate:


Step 3: Use the condition that population never increases.
Population never increases .
Since , this implies:

Step 4: Identify correct form among options.
The form with and matches option (B).
Final Answer:

Step 1: First, we need to check if the function is continuous at . - For , the function is , where represents the greatest integer function. - For , the function is .
Step 2: Check if the function is continuous at :
- Left-hand limit at is: - Right-hand limit at is: Since the left-hand and right-hand limits are not equal, the function is not continuous at .
Step 3: Since the function is not continuous at , it cannot be differentiable there. Therefore, the correct answer is:

Step 1: Using Vieta's formulas, the sum and product of the roots of the quadratic equation are given by: Step 2: We need to find the value of the expression We combine the terms into a single fraction: Simplifying the numerator: Using the identity , we substitute: Simplifying further: Step 3: Now, for the denominator: Using the same values for and , this becomes: Step 4: Combining the numerator and denominator, we get: Therefore, the value of the expression is .

Step 1: Coordinates of points A, B, and C. The coordinates of the mid-points of the sides of the triangle are given as:
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Step 2: Calculating distances , , and . To calculate the distances between the points, we use the distance formula in 3D: -
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Step 3: Summing the squares of the distances. Step 4: Calculating the ratio. Now, we calculate the ratio: Simplifying the expression: Thus, the value of the expression is , so the correct answer is (A).

Step 1: Using the standard area formula for an ellipse.
The area of an ellipse is given by: where and .
Step 2: Calculating the area.
Thus, the correct answer is (A).