CAT Previous Paper 2007 Test Online

1. How many pairs of positive integers m, n satisfy
1/m + 4/n = 1/12
where n is an odd integer less than 60

2. The proportion of male students and the proportion of vegetarian students in a school are given below. The school has a total of 800 students, 80% of whom are in the Secondary Section and rest equally divided between Class 11 and 12.
What is the percentage of vegetarian students in class 12?

3. Ten years ago, the ages of the members of a joint family of eight people added up to 231 years. Three years later, one member died at the age of 60 years and a child was born during the same year. After another same years, one more member died, again at 60, and a child was born during the same year. The current average age of this eight-member joint family is nearest to

4. Mr. David manufactures  and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost in rupees, of producing x units is 240 + bx + cx², where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66 2/3%. However an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.
What is the maximum daily profit, in rupees, that Mr. David can realize from his business?

5. Let S be the set of all pairs (i, j) where 1 ≤ j ≤ n, and n ≥ 4. Any two distinct numbers of S are called "friends" if they have one constituent of the pairs in common and "enemies" otherwise. For example, if n = 4, then S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2), (1, 3) are friends. (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.
For genera n, consider any two members of S that are friends. How many other members of S will be common friends of both these members?

6. Mr. David manufactures  and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost in rupees, of producing x units is 240 + bx + cx², where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66 2/3%. However an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.
How many units should Mr. David produce daily?

7. In a tournament, there are n teams T₁, T₂ , ...............T_n, with n > 5. Each team consists of k players, k > 3. The following pairs of teams have one player in common:
T₁ & T₂, T₂ & T₃, ....... T_n-1 & T_n, & T_n, and T_n & T₁
No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together?

8. Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.

Assume the planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.
What is the plane's cruising speed in km per hour?

9. Let S be the set of all pairs (i, j) where 1 ≤ j ≤ n, and n ≥ 4. Any two distinct numbers of S are called "friends" if they have one constituent of the pairs in common and "enemies" otherwise. For example, if n = 4, then S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. Here, (1, 2), (1, 3) are friends. (1, 2) and (2, 3) are also friends, but (1, 4) and (2, 3) are enemies.
For general n, how many enemies will each member of S have?

10. Two circles with centers P and Q cut each other at two distinct points A and B. The circles have the same radii and neither P nor Q falls within the intersection of the circles. What is the smallest range that includes all possible values of the angle APQ in degrees?

11. Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

12. Shabnam is considering three alternatives to invest her surplus cash for a week. She wishes to guarantee maximum returns on her investment. She has three options, each of which can be utilized fully or partially in conjunction with others.
Option A: Invest in a public sector bank. It promises a return of +0.10%.
Option B: Invest in mutual funds of ABC Ltd. A rise in the stock market will result in a return of +5%, while a fall will entail a return of -3%.
Option C: Invest in mutual funds of CBA Ltd. A rise in the stock market will result in a return of -2.5%, while a fall entail a return of +2%.
The maximum guaranteed to a Shabnam is:

13. A quadratic function f (x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of of f (x) at x = 10?

14. The proportion of male students and the proportion of vegetarian students in a school are given below. The school has a total of 800 students, 80% of whom are in the Secondary Section and rest equally divided between Class 11 and 12.

In Class 12, twenty five percent of the vegetarians are male. What is the difference between the number of female vegetarians and male non- vegetarians?

15. Shabnam is considering three alternatives to invest her surplus cash for a week. She wishes to guarantee maximum returns on her investment. She has three options, each of which can be utilized fully or partially in conjunction with others. Option A: Invest in a public sector bank. It promises a return of +0.10%. Option B: Invest in mutual funds of ABC Ltd. A rise in the stock market will result in a return of +5%, while a fall will entail a return of -3%. Option C: Invest in mutual funds of CBA Ltd. A rise in the stock market will result in a return of -2.5%, while a fall entail a return of +2%.
What strategy will maximize the guaranteed return to Shabnam?

16. Consider the set S = {2, 3, 4, ........, 2n + 1}, where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X - Y?

17. A function f(x) satisfied f(1) = 3600, and f(1) + f(2) + .... + f(n) = n² f(n), for all positive integers n > 1. What is the value of f(9)?

18. Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.

Assume the planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.
What is the time difference between A and B?

19. The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.1n, on the nth day of 2007 (n = 1, 2, ....... 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the nth day of 2007 (n = 1, 2, ........ 365). On which date in 2007 will the prices of these two varieties of tea be equal?

20. A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja, giving her rupees instead of paise and paise instead of rupees. After buying a toffee of 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque. Which of the following is a valid statement about the cheque amount?