GATE IT 2006

Question 21

When a coin is tossed, the probability of getting a Head is p,0<p<1. Let N be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that successive tosses are independent, the expected value of N is

• 1/p
• 1/(1−p)
• 1/p2
• 1/(1−p2)

Question 22

Let P, Q and R be sets let Δ denote the symmetric difference operator defined as PΔQ = (P U Q) - (P ∩ Q). Using Venn diagrams, determine which of the following is/are TRUE? PΔ (Q ∩ R) = (P Δ Q) ∩ (P Δ R) P ∩ (Q ∩ R) = (P ∩ Q) Δ (P Δ R)

• I only
• II only
• Neither I nor II
• Both I and II

Question 23

What is the cardinality of the set of integers X defined below? X = {n | 1 ≤ n ≤ 123, n is not divisible by either 2, 3 or 5}

• 28
• 33
• 37
• 44

Question 24

Consider the undirected graph G defined as follows. The vertices of G are bit strings of length n. We have an edge between vertex u and vertex v if and only if u and v differ in exactly one bit position (in other words, v can be obtained from u by flipping a single bit). The ratio of the chromatic number of G to the diameter of G is

• 1/(2^n-a)
• 1/n
• 2/n
• 3/n

Question 25

What are the eigenvalues of the matrix P given below
[Tex] \\begin{pmatrix} a & 1 & 0\\\\ 1 & a & 1 \\\\ 0 & 1 &a \\end{pmatrix} [/Tex]

• a, a -√2, a + √2
• a, a, a
• 0, a, 2a
• -a, 2a, 2a

Question 26

Match the following iterative methods for solving algebraic equations and their orders of convergence.

• I-R, II-S, III-P, IV-Q
• I-S, II-R, III-Q, IV-P
• I-S, II-Q, III-R, IV-P
• I-S, II-P, III-Q, IV-R

Question 27

The following definite integral evaluates to

• 1/2
• √10
• None

Question 28

Consider the regular grammar below
S → bS | aA | ϵ
A → aS | bA
The Myhill-Nerode equivalence classes for the language generated by the grammar are

• {w ∊ (a + b)* | #a(w) is even) and {w ∊ (a + b)* | #a(w) is odd}
• {w ∊ (a + b)* | #a(w) is even) and {w ∊ (a + b)* | #b(w) is odd}
• {w ∊ (a + b)* | #a(w) = #b(w) and {w ∊ (a + b)* | #a(w) ≠ #b(w)}
• {ϵ}, {wa | w ∊ (a + b)* and {wb | w ∊ (a + b)*}

Question 29

Which of the following statements about regular languages is NOT true?

• Every language has a regular superset
• Every language has a regular subset
• Every subset of a regular language is regular
• Every subset of a finite language is regular

Question 30

Which of the following languages is accepted by a non-deterministic pushdown automaton (PDA) but NOT by a deterministic PDA?

• {aⁿbⁿcⁿ ∣ n≥0}
• {a^lb^mcⁿ ∣ l≠m or m≠n}
• {aⁿbⁿ ∣ n≥0}
• {a^mbⁿ∣ m,n≥0}