CAT Previous Paper 2005 Test Online

1. If x ≥ and y > 1, then the value of the expression log_x (x/y) + log_y (x/y) can never be

2. 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 does Shyam overtake Ram?

3. A chemical plant has four tanks (A, B, C and D), each containing 1000 liters of a chemical. The chemical is being pumped from one tank to another as follows:
From A to B @ 20 liters/minute
From C to A @ 90 liters/minute
From A to D @ 10 liters/minute
From C to D @ 50 liters/minute
From B to C @ 100 liters/minute
From D to B @ 110 liters/minute
Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping starts?

4. If R = 30⁶⁵ - 29⁶⁵/30⁶⁴ + 29⁶⁴, then

5. For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?
x² - y² = 0
(x - y)² + y² = 1

6. Two identical circles intersect so that their centers, and the points at which they intersect, from the square of side 1 cm. The area in sq. cm of the portion that is common to the circle is

7. For a positive integer n, let p_n denote the product of the digits of n, and s_n denote the sum of the digit of n. The number of integers between 10 and 1000 for which be p_n + s_n = n is:

8. Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is

9. Let S be the set of five-digit numbers formed by the digits 1, 2, 3, 4 and 5 using each digit exactly once such that exactly two odd positions are occupied by odd digits. What is the sum of the digits in the rightmost position of the number in S?

10. Consider a triangle drawn on the X - Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

11. 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?

12. If x = (16³ + 17³ + 18³ + 19³), then x divided by 70 leaves a remainder of

13. The rightmost non-zero digit of the number 30²⁷²⁰ is:

14. A rectangular floor is fully covered with square tiles of identical size. The tiles of the edges are white and the tiles in the interior are red. The number of white tiles in the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is:

15. In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls, and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is:

16. Let g(x) be a function such that g(x + 1) + g(x - 1) = g(x) for every real x. Then for what value of pis the relation g(x + p) = g(x) necessarily true for every real x?

17. A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs. 250 and Rs. 300 per day respectively. In addition, a male operator gets Rs. 15 per call he answers and a female operator gets Rs. 10 per call she answers. To minimize the total cost, how many total operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?

18. In the X - Y plane, the area of the region bounded by the graph  of |x + y| + |x - y| = 4 is:

19. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

20. Four points A, B, C and D lie on a straight line in the X - Y plane, such that AB = BC = CD, and the length of AB is 1 meter. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one meter of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is: