ISRO CSE 2014 PYQ

IPv4 multicast Ethernet MAC addresses use the prefix 01:00:5E. In the fourth octet, the most significant bit must be 0, so that octet ranges from 00 to 7F. The address 01:00:5E:FF:FF:FF has that bit set and is outside the IPv4 multicast mapping range; it is therefore not a valid IPv4 multicast MAC address in this mapping.

The first two rows are linearly independent because [9,5,2,2] is not a scalar multiple of [1,2,1,-1], so rank is at least 2. Row reduction gives a third row of zeros: R2 ← R2-9R1 gives [0,-13,-7,11], and R3 ← R3-7R1 gives [0,-13,-7,11], so the second and third rows become identical after eliminating the first column. There are exactly two independent rows; therefore rank(A)=2.

For four distinct labelled vertices, Cayley’s formula states that the number of labelled trees is $n^{n-2}$. With $n=4$, this is $4^2$ = 16. This assumes “trees” means undirected labelled trees on the named vertices A, B, C, and D.

Using the empirical relation for a moderately symmetric unimodal distribution, Mode $\approx 3\times Median - 2\times Mean$. Rearranging gives Median $\approx (Mode+2\times Mean)/3$. Substituting Mode=15 and Mean=30 gives (15+60)/3 = 25. This is an empirical approximation, not an identity valid for every dataset.

The /26 block contains 64 addresses, from 130.34.12.64 through 130.34.12.127. Splitting it into four equal subnets borrows two host bits, producing four /28 subnets: .64-.79, .80-.95, .96-.111, and .112-.127. Address 130.34.12.132 is outside the original /26 block, so it is not part of the organization’s allocation.