Gate IT 2008

Question 21

The exponent of 11 in the prime factorization of 300! is

• 27
• 28
• 29
• 30

Question 22

In how many ways can b blue balls and r red balls be distributed in n distinct boxes?

• [(n+b-1)!(n+r-1)!]/[(n-1)!b!(n-1)!r!]
• [(n+(b+r)-1)!]/[(n-1)!(n-1)!(b+r)!]
• n!/(b!r!)
• [(n+(b+r)-1)!]/[n!(b+r-1)!]

Question 23

Consider the field C of complex numbers with addition and multiplication. Which of the following form(s) a subfield of C with addition and multiplication?
(S1) the set of real numbers
(S2) {(a + ib) | a and b are rational numbers}
(S3) {a + ib | (a² + b²) ≤ 1}
(S4) {ia | a is real}

• only S1
• S1 and S3
• S2 and S3
• S1 and S2

Question 24

G is a simple undirected graph. Some vertices of G are of odd degree. Add a node v to G and make it adjacent to each odd degree vertex of G. The resultant graph is sure to be

• regular
• Complete
• Hamiltonian
• Euler

Question 25

Consider the following Hasse diagrams.

Which all of the above represent a lattice?

• (i) and (iv) only
• (ii) and (iii) only
• (iii) only
• (i), (ii) and (iv) only

Question 26

If M is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct? (S1) Each row of M can be represented as a linear combination of the other rows (S2) Each column of M can be represented as a linear combination of the other columns (S3) MX = 0 has a nontrivial solution (S4) M has an inverse

• S3 and S2
• S1 and S4
• S1 and S3
• S1, S2 and S3

Question 27

If f(x) is defined as follows, what is the minimum value of f(x) for x ∊ (0, 2] ?

• 2
• 2(1/12)
• 2(1/6)
• 2(1/2)

Question 28

If the final states and non-final states in the DFA below are interchanged, then which of the following languages over the alphabet {a,b} will be accepted by the new DFA?

• Set of all strings that do not end with ab
• Set of all strings that begin with either an a or a b
• Set of all strings that do not contain the substring ab,
• The set described by the regular expression b*aa*(ba)*b*

Question 29

Consider the following languages.

L₁ = {aⁱ b^j c^k | i = j, k ≥ 1}
L₁ = {aⁱ b^j | j = 2i, i ≥ 0}
Which of the following is true?

• L₁ is not a CFL but L₂ is
• L₁ ∩ L₂ = ∅ and L₁ is non-regular
• L₁ ∪ L₂ is not a CFL but L₂ is
• There is a 4-state PDA that accepts L₁, but there is no DPDA that accepts L₂

Question 30

Consider a CFG with the following productions. S → AA | B A → 0A | A0 | 1 B → 0B00 | 1 S is the start symbol, A and B are non-terminals and 0 and 1 are the terminals. The language generated by this grammar is

• {0ⁿ 10²ⁿ | n ≥ 1}
• {0ⁱ 10^j 10^k | i, j, k ≥ 0} ∪ {0ⁿ 10²ⁿ | n ≥ l}
• {0ⁱ 10^j | i, j ≥ 0} ∪ {0ⁿ 10²ⁿ | n ≥ l}
• The set of all strings over {0, 1} containing at least two 0\'s
• None of the above