A very long hallway has 1000 doors numbered 1 to 1000; all doors are initially closed. One by one, 1000 people get into the hall; the first person opens each door, the second closes all door with even numbers, the third person closes door 3, opens door 6, closes door 9, opens door 12 etc. That is, the nth person changes all doors whose numbers are divisible by n. After all 1000 people have got into the hall, find out how many doors will remain open.

In the shooting of the movie, "A Beautiful Mind", Ravi, the actor, is running towards a vertical mirror with a speed of 10 m/sec on a line which is perpendicular to the plane of the mirror. The mirror is moving towards Ravi with a speed of 6 m/sec along the same line. Find the speed of Ravi’s image with respect to Ravi (the image of Ravi forms on the other side of the mirror such that the distance between the image of Ravi and the mirror is equal to the distance between Ravi and the mirror).

There are 60 workers who work for a company, out of which 25 are women. Also:
28 workers are married
26 workers are graduate
20 married workers are graduate of which 9 are men
15 men are graduate
15 men are married
How many unmarried women are graduates and how many unmarried women work in the company?

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

At what time do Ram and Shyam first meet each other?
(a) 10 a.m.
(b) 10:10 a.m.
(c) 10:20 a.m.
(d) 10:30 a.m.

At what time does Shyam overtake Ram?
(a) 10:20 a.m.
(b) 10:30 a.m.
(c) 10:40 a.m.
(d) 10:50 a.m.

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of the white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :
(a) 10
(b) 12
(c) 14
(d) 16

Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number 9 appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?
a. 10,000
b. 2,430
c. 3,402
d. 3,006

Help Distress (HD) is an NGO involved in providing assistance to people suffering from natural disasters. Currently, it has 37 volunteers. They are involved in three projects: Tsunami Relief (TR) in Tamil Nadu, Flood Relief (FR) in Maharashtra, and Earthquake Relief (ER) in Gujarat. Each volunteer working with Help Distress has to be involved in at least one relief work project.
• A Maximum number of volunteers are involved in the FR project. Among them, the number of volunteers involved in FR
project alone. is equal to the volunteers having additional involvement in the ER project.
• The number of volunteers involved in the ER project alone is double the number of volunteers involved in all the three projects.
• 17 volunteers are involved in the TR project. ,
• The number of volunteers involved in the TR project alone is one less than the number of volunteers involved in ER project
alone.
• Ten volunteers involved in the TR project are also involved in at least one more project.

Based on the information given above, the minimum number of volunteers involved in both FR and TR projects, but not in the ER project is:
(a) 1
(b) 3
(c) 4
(d) 5

Which of the following additional information would enable to find the exact number of volunteers involved in various projects?
(a) Twenty volunteers are involved in FR.
(b) Four volunteers are involved in all the three projects.
(c) Twenty three volunteers are involved in exactly one project.
(d) No need for any additional information.

After some time, the volunteers who were involved in all the three projects were asked to withdraw from one project. As a result, one of the volunteers opted out of the TR project, and one opted out of the ER project, while the remaining ones involved in all the three projects opted out of the FR project. Which of the following statements, then, necessarily follows?
(a) The lowest number of volunteers is now in TR project.
(b) More volunteers are now in FR project as compared to ER project.
(c) More volunteers are now in TR project as compared to ER project.
(d) None of the above.

After the withdrawal of volunteers, as indicated in the above question, some new volunteers joined the NGO. Each one of them was allotted only one project in a manner such that, the number of volunteers working in one project alone for each of the three projects became identical. At that point, it was also found that the number of volunteers involved in FR and ER projects was the same as the number of volunteers involved in TR and ER projects. Which of the projects now has the highest number of volunteers?
(a) ER
(b) FR
(c) TR
(d) Cannot be determined