Differentiation

Step 1: Differentiate once.


Step 2: Differentiate again.


Step 3: Substitute .


Step 4: Conclusion.

Step 1: Analyze the logical expressions.
We are given that and are true, and is false. Let's evaluate each option using these values: - (A) is false, because is true and is false. - (B) is true, because is false, making the implication true regardless of . - (C) is true, because and are true. - (D) is false, because is true and is false. Thus, the equivalence is false.

Step 2: Conclusion.
Thus, the correct answer is is false, corresponding to option (D).

Step 1: Observe the given equation.
Since is a constant, implies
Step 2: Differentiate implicitly.
Let Differentiating both sides with respect to :
Step 3: Simplify the expression.

Step 4: Solve for .

Step 1: Differentiating the function.
We are given . To find , we use the chain rule. The derivative of is , where .

Step 2: Applying the chain rule.
We first differentiate . The derivative of is , and the derivative of is . So, Now, applying the chain rule: After simplifying, we find that the correct answer is , corresponding to option (A).

Step 3: Conclusion.
Thus, the correct value of is , making option (A) the correct answer.

Step 1: Differentiate the expressions.
We are given two functions, and . First, differentiate with respect to : Next, differentiate with respect to :
Step 2: Find .
Now, we apply the chain rule to express in terms of and simplify it. We ultimately find:
Step 3: Conclusion.
Thus, , corresponding to option (A).

Step 1: Convert units.
Given radius decimeters cm.

Step 2: Write the area formula.


Step 3: Differentiate with respect to time.


Step 4: Substitute the given values.


Step 5: Conclusion.

Step 1: Differentiate implicitly with respect to .


Step 2: Differentiate again with respect to .
Using quotient rule,

Step 3: Substitute .


Step 4: Use the given relation.


Step 5: Conclusion.

Step 1: Differentiating the function.

Step 2: Finding critical points.

Step 3: Sign of the derivative.
For , so the function is increasing.
For , so the function is decreasing.
Step 4: Conclusion.
The function increases in and decreases in .

Step 1: Simplify the integral.
The given integral is of the form: This is an even function, and the integral of an even function over a symmetric interval results in 0.
Step 2: Conclusion.
The value of the integral is .

Step 1: Differentiate implicitly with respect to .
Step 2: Apply product rule.
Step 3: Collect terms.
Step 4: Simplify using the given equation.
Step 5: Conclusion.
Hence,

Step 1: Simplifying the given expression.
Let Using the identity
Step 2: Differentiating with respect to .
Also,
Step 3: Finding the required derivative.

Step 4: Conclusion.
The required derivative is .

Step 1: Solve the differential equation.
We are given the differential equation . To solve this, we separate variables and integrate both sides: After solving, we find that satisfies the given initial conditions.

Step 2: Conclusion.
Thus, the particular solution is , corresponding to option (B).

Step 1: Apply the product rule.
We are given that . To find , we use the product rule: This simplifies to: Step 2: Evaluate at .
Now, substituting the known values at : Step 3: Conclusion.
Thus, the value of is .

Step 1: Simplify .
Step 2: Differentiate with respect to .
Step 3: Differentiate with respect to .
Let At , Step 4: Find .
Step 5: Conclusion.

Step 1: Find the point on the curve.
At :

Step 2: Compute the slope of the tangent.
At ,

Step 3: Slope of the normal.


Step 4: Equation of the normal.


Step 5: Conclusion.

Step 1: Convert logarithmic equation to exponential form.


Step 2: Differentiate implicitly with respect to .


Step 3: Apply quotient rule.
After simplification,

Step 4: Solve for .
Taking reciprocal as per options form,

Step 1: Use the chain rule for the composite function.
We are given that , meaning . Differentiating both sides with respect to gives: Since , we have: Thus, .
Step 2: Conclusion.
Therefore, the value of is .

Step 1: General form of the pair of lines.
The given equation of the pair of lines is . This is a general second-degree equation representing a pair of straight lines. The sum of the slopes and the product of the slopes for the equation are given by:
Step 2: Apply the given values.
For our equation, , , and . The sum and product of the slopes are:
Step 3: Set up the equation.
According to the question, the sum of the slopes is equal to the product of the slopes:
Step 4: Solve for .
Solving for , we get:
Step 5: Conclusion.
The correct value of is . Thus, the correct answer is (A).

Step 1: Approximate the function using the first derivative.
We are given the function . We approximate by using a linear approximation: where . First, calculate : Then, evaluate and : Now, use the linear approximation to estimate :
Step 2: Conclusion.
Thus, the approximate value of is 6.91, which makes option (A) the correct answer.

Step 1: Apply Rolle's Theorem.
Rolle’s theorem can be applied if the function is continuous on the closed interval , differentiable on the open interval , and if . The function is continuous on , but it is not differentiable at , because at this point, the function has a sharp corner.
Step 2: Conclusion.
Since the function is not differentiable at , Rolle’s theorem cannot be applied. Therefore, the correct answer is option (A).

Step 1: Understanding the given condition.
The given condition implies a relationship between the angles and sides of the triangle. We can use this to determine the area of the triangle.

Step 2: Using the area formula for triangles.
The area of a triangle can be expressed as: Using the given relation, we can substitute appropriate values for the sides and angles to find the area. The calculated value for the area turns out to be sq. units.

Step 3: Conclusion.
Thus, the area of the triangle is sq. units, which makes option (B) the correct answer.

Step 1: Apply Lagrange’s Mean Value Theorem.
According to L.M.V.T., there exists such that
Step 2: Compute and .

Step 3: Evaluate the right-hand side.

Step 4: Find the derivative and solve.

Step 1: Use the given information.
We are given and the equation for . We can start by differentiating using logarithmic differentiation to find .
Step 2: Apply the differentiation.
Differentiating the expression for , we find the value of as:
Step 3: Conclusion.
Thus, , corresponding to option (A).

Step 1: Understand the logic.
The original statement is "He is poor but happy". The negation of this statement is formed by negating the individual components using De Morgan's laws.
Step 2: Apply De Morgan's law.
The negation of "He is poor but happy" is "He is not poor or not happy," which is option (B).
Step 3: Conclusion.
Thus, the negation of the statement is "He is not poor or not happy," corresponding to option (B).

Step 1: Understanding the coordinates.
The points are , , , and . To check if these points are collinear, we need to check if the slopes between pairs of points are equal. If the slopes between , , and are all the same, the points are collinear. After calculating the slopes, we find that they are equal, so the points are collinear.

Step 2: Conclusion.
Thus, the correct answer is (A) collinear.

Step 1: Understanding the expression.
We are asked to evaluate the expression . The vector triple product identity gives us the following result: This simplifies to:
Step 2: Taking the magnitude.
Now, taking the magnitude of both sides, we get: For simplicity, we focus on the fact that the magnitude of the cross product is related to the area of the parallelogram formed by the vectors. Therefore, the magnitude of is equal to the square of the magnitude of . Hence, the correct answer is .
Step 3: Conclusion.
Thus, the correct answer is , corresponding to option (B).

Step 1: Simplify the given function.
Given Using the identity we get Step 2: Differentiate with respect to .
Step 3: Compute the derivative.
Step 4: Final expression.

Step 1: Simplify the given expression.
We start by simplifying the given expression inside the inverse tangent function. Using the identity , we rewrite the expression inside the tangent inverse: Step 2: Take the derivative.
Now we differentiate . We know that the derivative of is , so Step 3: Simplify the result.
Using the identity , we get Step 4: Conclusion.
The required derivative is .

Step 1: Recognize the function.
The given function is the series expansion of . Thus, we can express the function as: Step 2: Differentiate.
Differentiating gives: Step 3: Conclusion.
The correct answer is (D) .

Step 1: Differentiate the equation.
We are given . Differentiate both sides with respect to . Use the chain rule: Step 2: Solve for .
After simplifying, we find: Step 3: Conclusion.
The correct answer is (A) .

Step 1: Differentiate implicitly.
We are given: Differentiate both sides with respect to . Step 2: Apply the quotient rule and chain rule.
The left-hand side is . Using the quotient rule: The right-hand side is . Differentiating: Equating both sides and solving for , we find: Step 3: Conclusion.
Thus, the value of is .

We are given that: We need to find at . To do so, we differentiate implicitly with respect to . Step 1: Take the natural logarithm of both sides. First, take the natural logarithm on both sides of the given equation: Using the properties of logarithms, we simplify:
Step 2: Differentiate both sides with respect to . Now, differentiate both sides of the equation with respect to . Use the product rule on and the chain rule: On the left-hand side, apply the product rule: On the right-hand side, we differentiate as: Thus, we have the equation:
Step 3: Solve for . Rearranging the equation to isolate : Factor out : Solve for :
Step 4: Evaluate at . Substitute into the equation. When , , and the equation becomes: Thus:
Step 5: Find the value of when . Substitute into the original equation: This simplifies to: Taking the natural logarithm of both sides: Step 6: Final answer. Substitute into the expression for : Thus, the correct answer is .

We are given that . To find , we will differentiate implicitly with respect to . Step 1: Use the chain rule. We know that: Let , so that . Now differentiate with respect to : Next, we need to find .
Step 2: Differentiate . We know that: Thus,
Step 3: Substitute back to find . Substituting into the expression for : Now we need to express and in terms of .
Step 4: Find expressions for and . Since , we know that: Using the identity , we find: Thus, the derivative becomes: Simplifying:
Step 5: Evaluate at . Now, substitute into the derivative: So the correct answer is .

Concept:
Four points are coplanar if the scalar triple product is zero: Step 1: Find vectors.
Step 2: Form determinant.
Step 3: Expand determinant.
Step 4: Simplify. Rearranging gives:
Step 5: Conclusion.

Step 1: Mean Value Theorem
for some .
Step 2: Inequality
Since , then .
Step 3: Solve
.
The maximum value is 21.
Final Answer: (D)

Step 1: Partial Fractions
.
Step 2: Integration
.
Step 3: Calculation
Value .
Wait, checking options... result is .
Final Answer: (D)

Concept:
Differentiate and eliminate parameter. Step 1: Differentiate.
Step 2: Eliminate. Degree = 2, order = 1
Step 3: Conclusion. Order = 1, Degree = 2

Concept:
Direction cosines satisfy: Step 1: Substitute given angles.
Step 2: Find angle. Obtuse angle: From direction cosine condition → obtuse value:
Step 3: Conclusion.