Integration

Step 1: Write the equation of the line and circle.
We are given the line and the equation of the circle .
Step 2: Find the mid-point of the chord.
To find the mid-point of the chord cut off by the line and the circle, we use the formula for the mid-point of a chord: where and (the values of the circle's center).
Step 3: Conclusion.
The correct co-ordinates of the mid-point are .

Step 1: Use the given condition.
We are given that . This implies . We are asked to find the maximum value of .
Step 2: Use the trigonometric identity.
Using the identity , we get: This is a standard trigonometric expression, and its maximum value occurs when , at which point:
Step 3: Conclusion.
Thus, the maximum value of is , corresponding to option (A).

Step 1: Find the derivative of with respect to .
We are given and . To find when the tangent is parallel to the x-axis, we need to find . First, find and :
Step 2: Set the derivative of with respect to equal to 0.
The derivative is given by: To have a tangent parallel to the x-axis, . Therefore: Solving for :
Step 3: Conclusion.
Thus, the value of is .

Step 1: Using substitution.
Let , then . When and when .
Step 2: Changing the limits and integral.

Step 3: Expanding and integrating.

Step 4: Conclusion.

Step 1: Integrating the given function.

Step 2: Applying the limits.

Step 3: Solving for .
Trying :
Step 4: Conclusion.
The value of is .

Step 1: Recognizing the form of the integrand.
The integrand involves the square of a rational function in . This suggests a standard substitution or recognition of the integral's form. Step 2: Solving the integral.
After performing the necessary substitution and simplifying the expression, we find that the integral evaluates to: Step 3: Conclusion.
Thus, the correct answer is .

Step 1: Break down the integrand.
We want to perform partial fraction decomposition on Step 2: Set up partial fractions.
Step 3: Solve for the constants.
Multiply both sides by and solve for . After simplification, we get Step 4: Integrate each term.
Step 5: Combine the results.
Step 6: Conclusion.
The final answer is .

Step 1: Observe the structure of the integrand.
We notice that resembles the derivative of a product involving and .
Step 2: Try differentiation of .
Step 3: Apply the product rule.
Step 4: Factor out .
Step 5: Conclusion.
Thus,

Step 1: Simplify the given equation.
The given equation is . Rewriting the trigonometric functions, we get:
Step 2: Solve for .
Multiply through by to clear the denominators: Using the identity , substitute and solve for the values of in the interval .
Step 3: Conclusion.
The correct number of solutions is 2.

Step 1: Use sine subtraction identity.
Step 2: Rewrite the integrand.
Step 3: Split the expression.
Step 4: Integrate termwise.

Concept:
Distance of point from:

• X-axis
• Y-axis
• Z-axis

Step 1: Square and add distances.
Step 2: Use given condition.
Step 3: Distance from origin.
Step 4: Conclusion.

Concept:
Area under a curve: Step 1: Area under .
Step 2: Area under .
Step 3: Compare areas. Note: Region is between curve and X-axis, but becomes negative after , so symmetry applies. Net area:
Step 4: Conclusion.

Concept:
Implication is tautology if always true. Step 1: Simplify premise.
Step 2: For tautology. Equivalent to (option form).
Step 3: Conclusion. Answer = (A)

We are tasked with evaluating two integrals, one rational and the other a simple algebraic expression. Let's start with the first one.
Step 1: Solve the first integral We are given the integral: We can attempt to simplify the expression by breaking it into simpler parts. First, we split the expression: Thus, we have two integrals: Simplify the first term: We can use substitution for the second part. Let , so . Then the integral becomes easier to handle.
Step 2: Solve the second integral Next, we focus on the second integral:
Step 3: Evaluate the result Using algebraic simplifications and solving, we arrive at the expression: Thus, the solution to the first integral is .

Step 1: Write the given integral:


Step 2: Express in terms of sine and cosine:


Step 3: Substitute into the integral:


Step 4: Simplify:


Step 5: Use substitution:
Let
So,

Step 6: Substitute everything:

Since , substitute:



Step 7: Integrate:


Step 8: Substitute back :


Step 9: Final Answer: