Introduction:
Booth's algorithm is a multiplication algorithm used to multiply two signed binary numbers in two's complement form. It is a method that reduces the number of partial products that need to be added together to obtain the final result. The performance of Booth's algorithm can vary depending on the pattern of the multiplier.

Explanation:
Booth's algorithm operates by scanning the multiplier from right to left and performing certain operations based on the current bit and the previous bit. The algorithm uses a series of shifts, additions, and subtractions to calculate the final result.

The performance of Booth's algorithm is determined by the number of operations required to complete the multiplication. In general, the worst performance occurs when there are a large number of non-zero bits in the multiplier pattern. Let's analyze each option to determine the worst performance case:

Option A:
The multiplier pattern is "101010 .. 1010". This pattern has alternating non-zero bits, which means that there are a large number of non-zero bits in the multiplier. This results in a larger number of partial products that need to be added together, leading to a higher number of operations. Therefore, option A gives the worst performance for Booth's algorithm.

Option B:
The multiplier pattern is "100000 .. 0001". This pattern has only two non-zero bits, one at the leftmost position and one at the rightmost position. This results in a smaller number of partial products compared to option A. Therefore, option B does not give the worst performance for Booth's algorithm.

Option C:
The multiplier pattern is "111111 .. 1111". This pattern has all non-zero bits, but they are contiguous. This results in a smaller number of partial products compared to option A. Therefore, option C does not give the worst performance for Booth's algorithm.

Option D:
The multiplier pattern is "011111 .. 1110". This pattern has non-zero bits at the rightmost positions and a zero bit at the leftmost position. This results in a smaller number of partial products compared to option A. Therefore, option D does not give the worst performance for Booth's algorithm.

Conclusion:
The worst performance for Booth's algorithm occurs when there are a large number of non-zero bits in the multiplier pattern. Among the given options, option A has the largest number of non-zero bits, resulting in the worst performance.