List of top Questions asked in CAT

Three small pumps and a large pump are filling a tank. Each of the three small pumps works at \( \frac{2}{3} \) the rate of the large pump. If all four pumps work at the same time, they should fill the tank in what fraction of the time that it would have taken the large pump alone?

A train approaches a tunnel AB. Inside the tunnel is a cat located at a point that is \( \frac{3}{8} \) of the distance AB measured from the entrance A. When the train whistles the cat runs. If the cat moves to the entrance of the tunnel A, the train catches the cat exactly at the entrance. If the cat moves to the exit B, the train catches the cat at exactly the exit. The speed of the train is greater than the speed of the cat by what order?

A piece of string is 40 cm long. It is cut into three pieces. The longest piece is three times as long as the middle-sized and the shortest piece is 23 cm shorter than the longest piece. Find the length of the shortest piece.

Three travellers are sitting around a fire, and are about to eat a meal. One of them has 5 small loaves of bread, the second has 3 small loaves of bread. The third has no food, but has 8 coins. He offers to pay for some bread. They agree to share the 8 loaves equally among the three travellers, and the third traveller will pay 8 coins for his share of the 8 loaves. All loaves were the same size. The second traveller (who had 3 loaves) suggests that he will be paid 3 coins, and that the first traveller be paid 5 coins. The first traveller says that he should get more than 5 coins. How much should the first traveller get?

If \( u, v, w \) and \( m \) are natural numbers such that \( u^m + v^m = w^m \), then which one of the following is true?

Three pieces of cakes of weights \( 4 \frac{1}{2} \) lb, \( 6 \frac{3}{4} \) lb and \( 7 \frac{1}{5} \) lb respectively are to be divided into parts of equal weight. Further, each part must be as heavy as possible. If one such part is served to each guest, then what is the maximum number of guests that could be entertained?

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

Six persons are playing a card game. Suresh is facing Raghubir who is to the left of and to the right of Pramod. Ajay is to the left of Dhiraj. Yogendra is to the left of Pramod. If Dhiraj exchanges his seat with Yogendra and Pramod exchanges with Raghubir, who will be sitting to the left of Dhiraj?

Amol was asked to calculate the arithmetic mean of 10 positive integers, each of which had 2 digits. By mistake, he interchanged two digits, say \( a \) and \( b \), in one of these 10 integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then \( b - a \) equals:

A car rental agency has the following terms. If a car is rented for 5 hr or less, the charge is Rs. 60 per hour or Rs. 12 per kilometre whichever is more. On the other hand, if the car is rented for more than 5 hr, the charge is Rs. 50 per hour or Rs. 7.50 per kilometre whichever is more. Akil rented a car from this agency, drove it for 30 km and ended up paying Rs. 300. For how many hours did he rent the car?

A child was asked to add first few natural numbers (i.e. \( 1 + 2 + 3 + \dots \)) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong, the child discovered he had missed one number in the sequence during addition. The number he missed was:

Suppose for any real number \( x \), \( [x] \) denotes the greatest integer less than or equal to \( x \). Let \( L(x, y) = [x] + [y] \) and \( R(x, y) = [2x] + [2y] \). Then is it impossible to find any two positive real numbers \( x \) and \( y \) for which:

Ten straight lines, no two of which are parallel and no three of which pass through any common point, are drawn on a plane. The total number of regions (including finite and infinite regions) into which the plane will be divided by the lines is:

When \( 2^{56} \) is divided by 17, the remainder would be:

The number of real roots of the equation \( \frac{A^2}{x} + \frac{B^2}{x-1} = 1 \), where \( A \) and \( B \) are real numbers not equal to zero simultaneously, is:

Neeraj has agreed to mow a lawn, which is a 20 m × 40 m rectangle. He mows it with a 1 m wide strip. If Neeraj starts at one corner and mows around the lawn toward the centre, about how many times would he go round before he has mowed half the lawn?

The owner of a local jewellery store hired three watchmen to guard his diamonds, but a thief still got in and stole some diamonds. On the way out, the thief met each watchman, one at a time. To each he gave \( \frac{1}{2} \) of the diamonds he had then, and 2 more besides. He escaped with one diamond. How many did he steal originally?

Mayank, Mirza, Little and Jaspal bought a motorbike for rupee 60. Mayank paid one-half of the sum of the amounts paid by the other boys. Mirza paid one-third of the sum of the amounts paid by the other boys. Little paid one-fourth of the sum of the amounts paid by the other boys. How much did Jaspal have to pay?

A rich merchant had collected many gold coins. He did not want anybody to know about him. One day, his wife asked, "How many gold coins do we have?" After a brief pause, he replied, "Well! If I divide the coins into two unequal numbers, then 48 times the difference between the two numbers equals the difference between the squares of the two numbers." The wife looked puzzled. Can you help the merchant’s wife by finding out how many gold coins the merchant has?

Shyam visited Ram during his brief vacation. In the mornings they both would go for yoga. In the evenings they would play tennis. To have more fun, they indulge only in one activity per day, i.e. either they went for yoga or played tennis each day. There were days when they were lazy and stayed home all day long. There were 24 mornings when they did nothing, 14 evenings when they stayed at home, and a total of 22 days when they did yoga or played tennis. For how many days Shyam stayed with Ram?

Let \( S \) denote the infinite sum \( 2 + 5x + 9x^2 + 14x^3 + 20x^4 + \dots \), where \( |x|<1 \) and the coefficient of \( x^{n-1} \) is \( \frac{1}{2} n(n + 3) \), \( n = 1, 2, \dots \). Then S equals:

If \( x^2 + 5y^2 + z^2 = 2y(2x + z) \), then which of the following statements is (are) necessarily true? A. \( x = 2y \) B. \( x = 2z \) C. \( 2x = z \)

Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is:

Only a single rail track exists between stations A and B on a railway line. One hour after the north-bound super fast train N leaves station A for station B, a south-bound passenger train S reaches station A from station B. The speed of the super fast train is twice that of a normal express train E, while the speed of a passenger train S is half that of E. On a particular day, N leaves for B from A, 20 min behind the normal schedule. In order to maintain the schedule, both N and S increased their speeds. If the super fast train doubles its speed, what should be the ratio (approximately) of the speeds of passenger train S to that of the super fast train so that the passenger train S reaches exactly at the scheduled time at A on that day?