1. You arrive at a snack bar and you can’t decide whether to order a lime juice or a lassi. You decide to throw a fair 6-sided die to make the choice, as follows.
• If you throw 2 or 6 you order a lime juice.
• If you throw a 4, you order a lassi.
• Otherwise, you throw the die again and follow the same algorithm.
What is the probability that you end up ordering a lime juice?
(a) 1/3 (b) 1/2 (c) 2/3 (d) 3/4
2. How many times is the comparison i ≥ n performed in the following program?
int i=85, n=5;
main() {
while (i >= n) {
i=i-1;
n=n+1;
}
}
(a) 40 (b) 41 (c) 42 (d) 43
3. How many times is the comparison i ≥ n performed in the following program?
int i=85, n=5;
main() {
while (i >= n) {
i=i-1;
n=n+1;
}
}
(a) 40 (b) 41 (c) 42 (d) 43
4. The school athletics coach has to choose 4 students for the relay team. He calculates that there are 3876 ways of choosing the team if the order in which the runners are placed is not considered. How many ways are there of choosing the team if the order of the runners is to be taken into account?
(a) Between 12,000 and 25,000
(c) Between 75,000 and 99,999
(b) Between 30,000 and 60,000
(d) More than 100,000
5.In a connected undirected graph, the distance between two vertices is the number of edges in the shortest path between them. Suppose we denote by P the following property: there exists a vertex that is a neighbour of all other vertices. Consider the following statements:
(i) If P is false, then there is a pair of vertices such that the distance between them is at least 4.
(ii) If P is true, then the distance between any pair of vertices is at most 2.
What can you say about these statements?
(a) Only (i) is true.
(c) Both (i) and (ii) are true.
(b) Only (ii) is true.
(d) Neither (i) nor (ii) is true.
6. Consider a weighted undirected graph G with positive edge weights. Let (u, v) be an edge in the graph. It is known that the shortest path from a vertex s to u has weight 53 and the shortest path from s to v has weight 65. Which of the statements is always true?
(a) Weight of (u, v) ≤ 12.
(b) Weight of (u, v) = 12.
(c) Weight of (u, v) ≥ 12.
(d) Nothing can be said about the weight of (u, v).
7. Varsha lives alone and dislikes cooking, so she goes out for dinner every evening. She has two favourite restaurants, Dosa Paradise and Kababs Unlimited, to which she travels by local train. The train to Dosa Paradise runs every 10 minutes, at 0, 10, 20, 30, 40 and 50 minutes past the hour. The train to Kababs Unlimited runs every 20 minutes, at 8, 28 and 48 minutes past the hour. She reaches the station at a random time between 7:15 pm and 8:15 pm and chooses between the two restaurants based on the next available train. What is the probability that she ends up eating in Kababs Unlimited?
(a) 1/5 (b) 1/3 (c) 2/5 (d) 1/2
8. ScamTel has won a state government contract to connect 17 cities by high-speed fibre optic links. Each link will connect a pair of cities so that the entire network is connected—there is a path from each city to every other city. The contract requires the network to remain connected if any single link fails. What is the minimum number of links that ScamTel needs to set up?
(a) 34 (b) 32 (c) 17 (d) 16
9. An FM radio channel has a repository of 10 songs. Each day, the channel plays 3 distinct songs that are chosen randomly from the repository.
Mary decides to tune in to the radio channel on the weekend after her exams. What is the probability that no song gets repeated during these 2 days?
(a) C(10,3)^2 / C(10,6)
(b) C(10,6)/C(10,3)^2
(c) C(10,3)*C(7,3)/C(10,3)^2
(d) C(10,3)*C(7,3)/C(10,6)
10. Let G be an arbitrary graph on n vertices with 4n − 16 edges. Consider the following statements:
I There is a vertex of degree smaller than 8 in G.
II There is a vertex such that there are less than 16 vertices at distance exactly 2 from it.
Which of the following is true:
(a) I only
(c) Both I and II
(b) II only
(d) Neither I nor II