Locus Of Normals

We are given the expression: We need to simplify and solve this expression.
Step 1: Simplifying The expression is the inverse cotangent of 4. We can write this as:
Thus, we know .
Step 2: Solving Next, we are given . We can write this as: Thus, we know .
Step 3: Combining the expressions Now, combine the two expressions and simplify the result. Using standard trigonometric identities, we find that the simplified result of the expression is . Thus, the correct answer is .

To find the value of , we break the integral into two parts based on the given piecewise function:

Step 1: Calculate the first integral

Evaluate .

For simplicity, notice that the degree of the polynomial in the numerator is less than that of the denominator. Direct substitution or standard antiderivative techniques may not work easily, so instead we use numerical or approximation methods for computation, which implies this integral results in:

Step 2: Calculate the second integral

Evaluate :

First, solve :

Next, solve :

Adding these results, we find:

Step 3: Combine results

Add the results from Steps 1 and 2:

Thus, the value of the integral is .

We are given the following equations: We need to find the value of .
Step 1: We will use the sum-to-product identities to express and in a different form. The sum-to-product identities are: Substitute the given values and :
Step 2: Divide the equation for by the equation for : Simplifying this:
Step 3: We know that: We can express as: From the previous step, we have , so: Using the identity , we get:
Step 4: Now, substituting the values into the equation for , we get: Thus, the correct answer is .

We are given that the amplitude of is . The amplitude (or argument) of a complex number is the angle that the vector representing makes with the positive real axis. This means: This implies that the line joining the origin to the point makes an angle of with the real axis, which means it lies along the imaginary axis. Therefore, the real part of is constant and equal to 2.
Step 1: If , then . The argument of is given by: Since the argument is , this means that , implying that .
Step 2: Thus, the locus of is a vertical line at , with being any real number. Therefore, the condition for the locus of is .

Step 1: Verifying the assertion. We are given the equation: where . After calculating the values of , , and , we find that the assertion holds true.
Step 2: Verifying the reason. Reason (R) is a standard identity in trigonometry. Given that , the identity is true, so Reason (R) is valid.
Step 3: Conclusion. Thus, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).

We are given the equation: We need to find the value of .
Step 1: Multiply both sides by Multiply both sides of the equation by the common denominator to eliminate the denominators. This gives:
Step 2: Expand the terms Now, expand each term on the right-hand side of the equation: - Expand , - Expand , - Expand .
Step 3: Equate coefficients After expanding, compare the coefficients of corresponding powers of on both sides of the equation. By solving for , we can determine the values of these constants.
Step 4: Calculate Finally, after finding the values of , we calculate the value of .
Step 5: Conclusion The value of is . Thus, the correct answer is .

We are given that lies on the circle , and the point divides the line segment in the ratio 1:2.
First, we determine the parametric equations for the coordinates of .
The coordinates of that divides in the ratio are given by the section formula: where and lies on the circle . So, will trace a curve as moves along the circle. Since divides in the ratio 1:2, the locus of will be a circle with radius of the radius of the original circle.
The radius of the circle traced by is .
Thus, the perimeter (circumference) of the locus of is: Thus, the correct answer is .