Geometric Progression

To solve this problem, let's first calculate the probability that both a head on the coin and a 6 on the die occur in one throw. The probability is: Next, we determine the probability of Player B winning the game. Player B wins if A does not win on the first, third, fifth, etc., throws, and then Player B wins on the subsequent throw. Hence, we need to consider the sequence of events where Player A does not win initially, followed by Player B winning.

Step 1: Calculate the probability of B winning on the second, fourth, sixth throws, etc. The probability of A not winning on the first throw is: The probability of B winning on the second throw is: Thus, the probability that B wins on the second throw is: Similarly, for B to win on the fourth throw:

Step 2: Set up the infinite series. We see a pattern here where the probability of B winning on the 2nd, 4th, 6th, etc., throws forms an infinite geometric series: The sum of this infinite geometric series is given by:

Step 3: Simplify the expression. Simplifying the above expression:

Final Answer:

We start by calculating the powers of matrix :

Next, calculate :

By observing the pattern, we deduce:

Since none of the given options match the result, the correct answer is "None of these."

Final Answer:

Step 1: Write the determinant of the coefficient matrix.

The coefficient matrix is:

The determinant of the matrix is:

Expanding along the first row, we get:

Simplifying the individual 2x2 determinants:

Now, combining like terms:

Step 2: Solve for such that .

Factorizing the cubic equation:

Next, factorizing further:

Step 3: Identify the condition for no solution.

For the system to have no solution, the determinant must equal zero, but the augmented matrix should not have the same rank. This implies that . Thus, the possible value for is .

Final Answer:

The given region is bounded by:
The circle ,
The line .

Step 1: Determine the total area of the region
The total area is the sum of two parts: 1. The area under the semicircle, and 2. The area under the curve defined by . Thus, the total area is:

Step 2: Solve the first integral (semicircular area)
The first integral represents the area of a quarter-circle, and is known:

Step 3: Solve the second integral (area under the curve)
For the second integral, we have: Let’s use the substitution , then: Substitute into the integral: Now solve:

Step 4: Combine the results
Now, combining the areas from both integrals:

Final Answer: