Graphical Method Of Analysis

Step 1: The hour hand on a clock completes a full rotation (360°) in 12 hours. Thus, the angle turned by the hour hand in 1 hour is: Step 2: From 8:00 AM to 8:00 PM, the total time is 12 hours.
Step 3: Therefore, the hour hand will rotate by: But from 8:00 AM to 8:00 PM, the hour hand rotates half of the circle, which is 180°. Thus, the answer is .

Step 1: The equation of a plane bisecting the angle between two planes is given by the formula: where and are the two planes.
Step 2: For the given planes: - Plane 1: has normal vector . - Plane 2: has normal vector . Using the bisector formula, the plane equation is: After calculating, the result is , but this is not one of the options.

Step 1: The given equation of the circle is: To rewrite this equation in standard form, we complete the square for both and .
Step 2: For , the coefficient of is 2. Half of 2 is 1, and . Add and subtract 1 inside the equation. For , the coefficient of is 4. Half of 4 is 2, and . Add and subtract 4 inside the equation: This simplifies to: Thus, the center of the circle is and the radius is .
Step 3: The point diametrically opposite to is on the straight line joining and the center of the circle. The midpoint of and the diametrically opposite point is the center . The midpoint formula is: Solving for and : Thus, the point diametrically opposite to is .

Step 1: First, calculate the modulus of the product and the denominator.
The modulus of a complex number is .
Step 2: For , the modulus is:
Step 3: For , the modulus is:
Step 4: For , the modulus is:
Step 5: The modulus of the entire expression is the product of the moduli of the numerator divided by the modulus of the denominator:

The given equations represent two skew lines. To calculate the shortest distance between them, we will use the formula: Where: - and are points on the respective lines, - and are the direction vectors of the lines.
Step 1: Write the parametric equations for both lines. For the first line , we can set: For the second line , we can set:
Step 2: Find the vectors and . By using the vector cross product and dot product, you will arrive at the shortest distance formula.
Step 3: Calculate the value of . Using the formula, we find that the shortest distance is 2.

Step 1: Understanding binomial distribution properties.
For a binomial distribution , - Mean - Variance
Step 2: Finding and .
Given: Substituting into variance equation: Using :
Step 3: Computing .
Thus, the correct answer is (B).

Step 1: Identifying restrictions on the function.
- The function contains a square root in the denominator, meaning the expression inside the square root must be strictly positive: - Solving for : Step 2: Understanding why is excluded.
- If or , then - This makes the denominator zero, which is undefined in real numbers. - Hence, the function is not defined at .
Step 3: Identifying the correct answer.
- The domain must be strictly between -3 and 3, so the correct answer is (B) .