List of top Questions asked in CAT- 2000

A farmer has decided to build a wire fence along one straight side of this property. For this, he planned to place several fence-posts at six metre intervals, with posts fixed at both ends of the side. After he bought the posts and wire, he found that the number of posts he had bought was five less than required. However, he discovered that the number of posts he had bought would be just sufficient if he spaced them eight metres apart. What is the length of the side of his property and how many posts did he buy?

Two full tanks, one shaped like a cylinder and the other like a cone, contain jet fuel. The cylindrical tank holds 500 litres more than the conical tank. After 200 litres of fuel has been pumped out from each tank the cylindrical tank contains twice the amount of fuel in the conical tank. How many litres of fuel did the cylindrical tank have when it was full?

Convert the number 1982 from base 10 to base 12. The result is:

For all non-negative integers \( x \) and \( y \), \( f(x, y) \) is defined as below:
\( f(0, y) = y + 1 \)
\( f(x + 1, 0) = f(x, 1) \)
\( f(x + 1, y + 1) = f(x, f(x + 1, y)) \)
Then, what is the value of \( f(1, 2) \)?

ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches E, the frog stops and stays there. Let \( a_n \) be the number of distinct paths of exactly \( n \) jumps ending in E. Then, what is the value of \( a_{2n-1} \)?

The set of all positive integers is the union of two disjoint subsets: \( \{f(1), f(2), \ldots, f(n), \ldots\} \) and \( \{g(1), g(2), \ldots, g(n), \ldots\} \), where \( f(1) < f(2) < \ldots < f(n) < \ldots \), and \( g(1) < g(2) < \ldots < g(n) < \ldots \), and \( g(n) = f(f(n)) + 1 \) for all \( n \ge 1 \). What is the value of \( g(1) \)?

There are three cities A, B and C. Each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?

In the figure AB = BC = CD = DE = EF = FG = GA, then \( \angle DAE \) is approximately

If the equation \( x^{3} - ax^{2} + bx - a = 0 \) has three real roots, then it must be the case that,

The area bounded by the three curves \( |x+y| = 1 \), \( |x| = 1 \), and \( |y| = 1 \), is equal to:

There is a vertical stack of books marked 1, 2 and 3 on Table-A, with 1 at the bottom and 3 on top. These are to be placed vertically on Table-B with 1 at the bottom and 2 on the top, by making a series of moves from one table to the other. During a move, the topmost book, or the topmost two books, or all the three, can be moved from one of the tables to the other. If there are any books on the other table, the stack being transferred should be placed, on top of the existing books, without changing the order of books in the stack that is being moved in that move. If there are no books on the other table, the stack is simply placed on the other table without disturbing the order of books in it. What is the minimum number of moves in which the above task can be accomplished?

Consider a circle with unit radius. There are 7 adjacent sectors, S1, S2, S3, ....., S7 in the circle such that their total area is (1/8)th of the area of the circle. Further, the area of the \( j^{th} \) sector is twice that of the \( (j-1)^{th} \) sector, for j = 2, ...... 7. What is the angle, in radians, subtended by the arc of S1 at the centre of the circle?

ABCD is a rhombus with the diagonals AC and BD intersecting at the origin on the x-y plane. The equation of the straight line AD is \( x + y = 1 \). What is the equation of BC?

If \( x^{2} + y^{2} = 0.1 \) and \( |x-y| = 0.2 \), then \( |x| + |y| \) is equal to:

Let N = \( 55^{3} + 17^{3} - 72^{3} \). N is divisible by:

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:
a) if \(f(x)=3f(-x)\)
b) if \(f(x)=-f(-x)\)
c) if \(f(x)=f(-x)\)
d) if \(3f(x)=6f(-x)\), for \(x\ge0\)
Mark the option that applies to the graph of this question (shown in the figure).

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:
a) if \(f(x)=3f(-x)\)
b) if \(f(x)=-f(-x)\)
c) if \(f(x)=f(-x)\)
d) if \(3f(x)=6f(-x)\), for \(x\ge0\)
Mark the option that applies to the graph of this question (shown in the figure).

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:
a) if \(f(x)=3f(-x)\)
b) if \(f(x)=-f(-x)\)
c) if \(f(x)=f(-x)\)
d) if \(3f(x)=6f(-x)\), for \(x\ge0\)
Mark the option that applies to the graph of this question (shown in the figure).

Sam has forgotten his friend's seven-digit telephone number. He remembers the following: a) the first three digits are either 635 or 674, b) the number is odd, and c) the number nine 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?

Each of the numbers \(x_1, x_2, \ldots, x_n\) \((n>4)\), is equal to 1 or \(-1\). Suppose, \(x_1x_2x_3x_4 + x_2x_3x_4x_5 + x_3x_4x_5x_6 + \cdots + x_{n-3}x_{n-2}x_{n-1}x_n + x_{n-2}x_{n-1}x_nx_1 + x_{n-1}x_nx_1x_2 + x_nx_1x_2x_3 = 0\), then:

The integers 34041 and 32506 when divided by a three-digit integer \(n\) leave the same remainder. What is \(n\)?

What is the number of distinct triangles with integral valued sides and perimeter 14?

Let x, y and z be distinct integers, that are odd and positive. Which one of the following statements cannot be true?

Let S be the set of integers x such that:

1) \( 100 \le x \le 200 \)

2) x is odd

3) x is divisible by 3 but not by 7.

How many elements does S contain?

One red flag, three white flags and two blue flags are arranged in a line such that,

A. no two adjacent flags are of the same colour

B. the flags at the two ends of the line are of different colours.

In how many different ways can the flags be arranged?