Applications Of Derivatives

Step 1: Differentiate the inverse sine function.
We are given . To differentiate this, we first use the chain rule. The derivative of is , and we differentiate the inner function . After applying the chain rule, we find that:
Step 2: Conclusion.
Thus, the correct answer is , corresponding to option (A).

Step 1: Use the perimeter condition.
The total length of the wire is the perimeter of the rectangle: Step 2: Express area in one variable.
Step 3: Maximize the area.
Step 4: Find breadth.
Step 5: Conclusion.
The rectangle of maximum area is a square of side meters.

Step 1: Use the exponential growth formula.
Population growth is given by Step 2: Apply the condition for doubling.
Step 3: Take logarithm on both sides.
Step 4: Substitute the given value.
Step 5: Solve for .
Step 6: Conclusion.
The time taken for the population to double is years.

Step 1: Find velocity.
Step 2: Find acceleration.
Step 3: Set acceleration equal to zero.
Step 4: Find displacement at .
Step 5: Conclusion.
The displacement of the particle is units.

Step 1: Understanding the function.
The function is defined only for non-negative values of , because the square root of a negative number is not real. Therefore, the domain of this function is , which includes all non-negative real numbers.

Step 2: Conclusion.
Thus, the domain of the function is , corresponding to option (C).

Step 1: Differentiate the curve equation.
We are given the curve equation . Differentiate both sides with respect to :
Step 2: Find the slope at the point.
At the point , we substitute and into the equation to find the slope .
Step 3: Use the point of tangency condition.
Using the point and the condition that the line is tangent to the curve, we solve for and . We find that , corresponding to option (C).

Step 1: Analyzing the function.
We are given the function . The maximum value of this function occurs when the exponent is maximized.
Step 2: Maximizing the exponent.
To maximize the exponent , we need to find the maximum value of . This is a standard trigonometric expression, and its maximum value occurs when and . Substituting these values gives the maximum of . Thus, the maximum value of the exponent is .

Step 3: Conclusion.
The maximum value of occurs when the exponent is maximized, which gives the result . The maximum value is , corresponding to option (A).

Step 1: Find the direction ratios of the line and the normal to the plane.
The direction ratios of the line are given by , and the normal vector to the plane is .
Step 2: Using the formula for the angle between a line and a plane.
The angle between the line and the plane is given by: where is the direction vector of the line, and is the normal vector of the plane. After calculation, we get:
Step 3: Conclusion.
Thus, the angle between the line and the plane is , which makes option (C) the correct answer.

Step 1: Maximize the function.
The given function is . To find the maximum, differentiate with respect to : Set and solve for .
Step 2: Conclusion.
The maximum value of occurs at , and the maximum value is , corresponding to option (B).

Step 1: Understand the given lines.
The lines are given in parametric form. To find the cosine of the angle between the two lines, we need to compute the direction ratios (vectors) of the lines, which are and , corresponding to the coefficients of and in the parametric equations.

Step 2: Use the formula for cosine of the angle.
The formula for the cosine of the angle between two vectors and is: Substitute and into the formula: Simplifying, we find .

Step 3: Conclusion.
Thus, the cosine of the angle between the lines is , corresponding to option (B).

Step 1: Formula for the area of a sector.
The area of a circular sector is given by: where is the radius and is the central angle in radians. The circumference of the sector is . We are given that the total length of the wire is 20 meters, so we have: Step 2: Maximize the area.
Maximizing the area with respect to and , we get .
Step 3: Conclusion.
The correct answer is (C) 5m.

Step 1: Find the equation of the tangent.
The derivative of the curve is: At the point , we substitute and solve for , which gives the slope of the tangent line.
Step 2: Find the slope of the normal.
The slope of the normal is the negative reciprocal of the tangent slope. Using the point , we find the equation of the normal. The equation of the normal is: Step 3: Conclusion.
The correct answer is (D) .

Step 1: Individual Probabilities
Urn 1:
Urn 2:
Urn 3:
Step 2: Selection Cases (1B, 2W)
- Case 1 ( ):
- Case 2 ( ):
- Case 3 ( ):
Step 3: Total Probability
Sum .
*Recalculating sum:* . (Option B).
Final Answer: (B)