Integral Calculus

Step 1: Translate the given statement into symbolic form. “Suman is brilliant and dishonest if and only if Suman is rich” Dishonest
Step 2: The negation of a statement is written as .
Step 3: Therefore, the negation of the given statement is:
Step 4: Rewriting in the form given in the options:
Step 5: Hence, the correct option is (C).

Step 1: First arrange the 10 men in a row. They can be arranged in:
Step 2: These 10 men create 11 gaps (including the two ends) where women can be seated:
Step 3: To ensure that no two women sit together, choose 6 gaps out of 11 to place the women:
Step 4: Arrange the 6 women in the selected gaps:
Step 5: Multiply all possible arrangements:
Step 6: Simplify:

Step 1: Let the given points be: and the dividing point be:
Step 2: Assume that point divides internally in the ratio .
Step 3: By the section formula:
Step 4: Substitute the values:
Step 5: From the second equation:
Step 6: Hence, the ratio is:

Step 1: Use the identity for . So,
Step 2: Split the integral.
Step 3: Observe derivative structure. Note that: Thus,
Step 4: Combine into a single derivative. This is of the form:
Step 5: Differentiate the product. So,
Step 6: Integrate. Comparing with the given result:
Step 7: Find .
Hence, the correct answer is .

The integrand is an odd function overall due to denominator dominant and odd terms, while numerator even. Symmetric limits in debate imply cancellation → value 0. Hence → (B).

Step 1: Rewrite all relations in terms of . Thus, Similarly, Step 2: Add all three equations. By symmetry: Step 3: Use the given condition . Using : Step 4: Solve the quadratic. Since the equation came from squaring, actual required value: Step 5: Final conclusion.

Step 1: Write the formula for the area of a circle. Step 2: Differentiate both sides with respect to time . Step 3: Substitute the given values. Step 4: Simplify. Step 5: Final conclusion. The rate of increase of the area of the circle is:

Step 1: Evaluate . Step 2: Evaluate . Step 3: Substitute the values. Step 4: Simplify. Since principal values are taken and magnitude is considered: But using standard exam convention: Step 5: Final conclusion.