Number of cache blocks = 32KB / 32 B = 1024

=10 bits for cache index

Block offset = 32B = 5 bits

Remaining bits = Tag bits = 48 - 10 - 5=33 bits

**Number of bits in cache index:**

To determine the number of bits in the cache index, we need to consider the cache size and the block size.

Cache size: 32KB
Block size: 32B

The cache index is used to determine which cache block the data is stored in. In a direct-mapped cache, there is a one-to-one mapping between each memory block and a cache block.

The formula to calculate the cache index is:
Cache index = log2(Cache size / Block size)

Substituting the given values, we get:
Cache index = log2(32KB / 32B) = log2(1024) = 10 bits

Therefore, the cache index for this direct-mapped cache is 10 bits.

**Number of bits in cache tag:**

The cache tag is used to store the remaining bits of the memory address that are not used for the cache index. It helps identify which memory block is stored in a particular cache block.

The formula to calculate the cache tag is:
Cache tag = Total number of bits in memory address - Number of bits in cache index

Given that the CPU generates a 48-bit address and the cache index is 10 bits, we can calculate the cache tag as follows:

Cache tag = 48 bits - 10 bits = 38 bits

Therefore, the cache tag for this direct-mapped cache is 38 bits.

In summary, for a direct-mapped cache with a cache size of 32KB and a block size of 32B, the number of bits in the cache index is 10 bits and the number of bits in the cache tag is 38 bits.