Absolute Maxima And Absolute Minima

Step 1: Define the variables
Let the length of the rectangle be cm and the breadth be cm (since the perimeter is ).
When the rectangle is rolled along its length, the radius and height of the cylinder formed are: Step 2: Write the volume of the cylinder
The volume of the cylinder is given by: Simplify: Step 3: Differentiate with respect to
The derivative of is: Simplify: Step 4: Set to find critical points
Step 5: Use the second derivative test to confirm maxima
The second derivative is: At : Hence, is maximum when .
Step 6: Calculate the dimensions of the rectangle
When , the length is cm, and the breadth is:

1. Reflexivity: For any , , which is divisible by 2. Hence, . Therefore, is reflexive.
2. Symmetry: If , then is divisible by 2. This implies is also divisible by 2 (since addition is commutative). Hence, . Therefore, is symmetric.
3. Transitivity: If and , then and are both divisible by 2. Adding these equations: Since is divisible by 2, is also divisible by 2. Hence, . Therefore, is transitive. Since is reflexive, symmetric, and transitive, is an equivalence relation.
4. Equivalence class : The equivalence class of , , includes all elements such that . This means is divisible by 2: Thus, must also be even. The even elements in are:
Final Answer: is an equivalence relation. The equivalence class .

Step 1: Differentiate to find critical points
The derivative is: Since attains a local maximum at , we have: Solve for : Step 2: Find critical points
Substitute into : Factorize: Further factorize: The critical points are .
Step 3: Determine the nature of critical points using
The second derivative is: Evaluate at each critical point:
Conclusion:
The function attains:

Step 1: Condition for to be onto
For to be onto, every element must have a preimage . Let : Rewriting this equation: If , we have: For , . Substituting into : Solving this: which is not valid. Hence, is excluded from the range of . Therefore: Step 2: {Check if is one-one}
For to be one-one, should imply . Let: Cross-multiply: Expanding both sides: Simplify: Rearrange terms: Thus, is one-one.

Conclusion:
The function is: Onto if and . One-one since implies .

Step 1: Set up the equation
We know that , so multiplying the matrices and should yield the identity matrix . The equation is:
Step 2: Calculate the product of and
We multiply the matrices element by element and equate the result to the identity matrix. The equations formed from the first row of the resulting matrix are: which simplifies to: Thus, , which corresponds to option (B).
Step 3: Verify the options
The value satisfies the equation, matching option (B).

Step 1: Find
To find , we compute the matrix product of with itself:
Step 2: Substitute into the equation
The equation becomes: Simplifying the left side: Now equating to , we have: Equating corresponding elements gives .
Step 3: Verify the options
The correct value of is , matching option (C).

Step 1: Find the magnitudes of the vectors
Compute the magnitudes:
Step 2: Check for right angle using dot products
Calculate , , and . If one is zero, the triangle is right-angled. For example:
Step 3: Conclude the type of triangle
Since , the triangle is right-angled.

1. Point of intersection of the two lines: Let: From the first line: From the second line: Equate for consistency: - From : - From : Substituting into the first line: Thus, the point of intersection is .
2. Direction vectors of the given lines: - First line: . - Second line: .
3. Direction of the required line: The required line is perpendicular to both and . Use the cross product: Expanding:
4. Equation of the required line: The line passing through with direction vector is:
Final Answer:

1. Find the derivative: 2. Critical points: Set : (Only is in .)
3. Evaluate at endpoints and critical point: - At : - At :
4. Conclusion: The absolute maximum value is at , and the absolute minimum value is at .
Final Answer:

1. Find the derivative:
2. Critical points: The critical points are obtained by solving : Let . The equality holds for .
3. Sign of : - For , , so : is increasing. - For , , so : is decreasing.
Intervals:

1. Let the events be: - : Selection of and as CEO. - : Company increases profits.
2. Use Bayes' theorem: The required probability is:
3. Calculate the prior probabilities: From the given ratio :
4. Calculate the total probability : Substitute the given probabilities: Simplify:
5. Calculate : Substitute:
Final Answer: The probability that the increase in profits is due to 's appointment as CEO is .

Step 1: Definition of a point of inflection.
A point of inflection is a point where the concavity of the function changes. This means changes its sign at that point.
Step 2: Conditions for inflection points.
At a point , the function has a point of inflection if: - changes its sign around . - but does not change its sign.
Step 3: Analyzing the options.
Option (C) correctly states that and does not change its sign, which is consistent with the definition of a point of inflection.
Step 4: Conclusion.
The correct answer is (C).

Step 1: Understanding the function.
The function is defined as , where . The term is always non-negative, and adding 1 shifts the range of to start from 1.
Step 2: Range of .
The minimum value of is 0, so the minimum value of is: Thus, the range of is .
Step 3: Conditions for to be onto.
For to be onto, the codomain must include the entire range of . Therefore, .
Step 4: Conclusion.
The function is onto if . {10pt}

Step 1: Definition of a transpose.
For a square matrix , the transpose is obtained by interchanging the rows and columns of .
Step 2: Subtracting from .
The matrix satisfies the property: Thus, is equal to the negative of its transpose, which is the definition of a skew symmetric matrix.
Step 3: Conclusion.
For any square matrix , is always a skew symmetric matrix. {10pt}

To determine the correct constraints, analyze the feasible region depicted in the graph:
1. Line 1: The region above this line is shaded, indicating the constraint: 2. Line 2: The region below this line is shaded, indicating the constraint: 3. Non-negativity constraints: Since the shaded region is in the first quadrant: Thus, the group of constraints representing the feasible region is:
Final Answer:

1. Find the derivative:
2. Critical points: Set : (Only is in .)
3. Evaluate at endpoints and critical point: - At : - At :
4. Conclusion: The absolute maximum value is at , and the absolute minimum value is at .
Final Answer:

1. Find the derivative:
2. Critical points: The critical points are obtained by solving : Let . The equality holds for .
3. Sign of : - For , , so : is increasing. - For , , so : is decreasing.
Intervals:

For to be continuous at , the left-hand limit (LHL), right-hand limit (RHL), and the value of must all be equal.
1. Left-hand limit (LHL): For : As :
2. Right-hand limit (RHL): For : As :
3. Value at :
4. Continuity condition: Substitute: From : From :
Final Answer:

To determine the correct constraints, analyze the feasible region depicted in the graph:
1. Line 1: The region above this line is shaded, indicating the constraint:
2. Line 2: The region below this line is shaded, indicating the constraint:
3. Non-negativity constraints: Since the shaded region is in the first quadrant: Thus, the group of constraints representing the feasible region is:
Final Answer:

Step 1: Define the events
Let be the event that the lost card is a King, and be the event that the lost card is not a King. Let be the event of drawing a King from the remaining 51 cards.
Step 2: Assign probabilities to the events

Step 3: Use Bayes' Theorem
The required probability is , which is given by: Substituting the values:
Step 4: Final result
The probability that the lost card is a King is .

The parametric form of the line passing through the point with position vector

and parallel to the given line
is: where is the point on the line and is the direction vector of the line.
Thus, the parametric equations of the line are:
To convert this to the Cartesian form, eliminate from the equations:
From the equation for
:
Substitute this into the equation for :
Simplifying:
Thus, the Cartesian form of the line is:
Step 2: {Verify the options}
This matches option (B).

Step 1: Represent the lines in symmetric form: For the first line ( ), the symmetric equation is given as: Rewriting this in parametric form: The direction ratios of are: For the second line ( ), the symmetric equation is given as: Rewriting this in parametric form: The direction ratios of are:
Step 2: Apply the condition for perpendicularity: Two lines are perpendicular if the dot product of their direction ratios is zero: Substituting the direction ratios of and : Simplify the equation:
Step 3: Verify the result: For , the direction ratios of become: The dot product with is: Thus, the lines are perpendicular.
Conclusion: The value of is .

Step 1: Rewrite the function
The function can be expressed as:
Step 2: Graph the function
The graph of is a parabola, concave downwards for and concave upwards for . (Refer to the attached graph.)
Step 3: Area computation using integration
The area of the shaded region between and is given by:
Step 4: Evaluate the integral
Thus, the total area is:
Step 5: Final result
The area bounded by the curve , the X-axis, and the ordinates and is .

Step 1: Standardize the equations of the lines
The given line is: The line parallel to and passing through is:
Step 2: Vector between the lines
Let . The direction vector .
Step 3: Find the shortest distance
The shortest distance is given by: Compute : The magnitude:

Step 1: Prove that is one-one
Let , where : Cross-multiply: Simplify: If , this leads to a contradiction. Hence, , proving is one-one.
Step 2: Prove that is onto
Let . Solve : For (since ), exists in . Thus, is onto.
Step 3: Conclude the function properties
Since is both one-one and onto, is a bijection.