CAT 2008 Question Paper with Solutions, Download PDF

Instructions

In a single elimination tournament, any a player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules :

(a) If the number of players, say n, in any round is even, then the players are grouped into n/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round.

(b) If the number of players, say n, in any round is odd, then one of them is given a bye, that is he automatically moves on to the next round. The remaining (n–1) players are grouped into (n–1)/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round. No player gets more than one bye in the entire tournament.

Thus, if n is even, then n/2 players move on to the next round while if n is odd, then (n+1)/2 players move on to the next round. The process is continued till the final round, which obviously is played between two players. The winner in the final round is the champion of the tournament.

Question 1

What is the number of Matches played by the champion?

A. The entry list for the tournament consists of 83 players?

B. The champion received one bye.

Video Solution

Solution

If we consider statement a then 2 possibilities occur , the champion may or may not receive bye and the answer for both possibilities would be different.
If we consider both statements a & b together we can find the exact answer to the question.

Question 2

If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?

A. Exactly one player received a bye in the entire tournament.

B. One player received a bye while moving on to the fourth round from the third round.

Video Solution

Solution

Using only statement A, we cannot say anything because the player might have received the bye in any round and the total number of players differs if the round in which the player got a bye changes.

Using only statement B, we cannot saty anything because there might be players who received a bye in other rounds.

Using both the statements, we know that only 1 player got a bye in the whole tournament and that happened in round 3.

If the number of players were between 65 and 128, then the number of rounds is 7.

=> In the fourth round there would be 16 people.

One person got a bye here => 31 people in round 3 => 62 people in round 2 => 124 people in round 1.

Hence, we can ansswer the question using both the statements together.

Instructions

For the following questions answer them individually

Question 3

The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?

Video Solution

Solution

Let the first operation be (1+40-1) = 40, the second operation be (2+39-1) = 40 and so on

So, after 20 operations, all the numbers are 40. After 10 more operations, all the numbers are 79

Proceeding this way, the last remaining number will be 781

Question 4

What are the last two digits of $$7^{2008}$$?

Video Solution

Solution

$$7^4$$ = 2401 = 2400+1
So, any multiple of $$7^4$$ will always end in 01
Since 2008 is a multiple of 4, $$7^{2008}$$ will also end in 01

Question 5

If the roots of the equation $$x^3 - ax^2 + bx - c = 0$$ are three consecutive integers, then what is the smallest possible value of b?

Video Solution

Solution

b = sum of the roots taken 2 at a time.
Let the roots be n-1, n and n+1.
Therefore, $$b = (n-1)n + n(n+1) + (n+1)(n-1) = n^2 - n + n^2 + n + n^2 - 1$$
$$b = 3n^2 - 1$$. The smallest value is -1.

Question 6

A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x?

Video Solution

Solution

After the first sale, the remaining quantity would be (x/2)-0.5 and after the second sale, the remaining quantity is 0.25x-0.75

After the last sale, the remaining quantity is 0.125x-(7/8) which will be equal to 0

So 0.125x-(7/8) = 0 => x = 7

Question 7

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that $$f(5) = - 3f(2)$$. and that 3 is a root of $$f(x) = 0$$.

What is the other root of f(x) = 0?

[CAT 2008]

Video Solution

Solution

f(3) = 9a + 3b + c = 0 f(5) = 25a + 5b + c
f(2) = 4a + 2b + c
f(5) = -3f(2) => 25a + 5b + c = -12a -6b -3c
=> 37a + 11b + 4c = 0 --> (1)
4(9a + 3b + c) = 36a + 12b + 4c = 0 --> (2)
From (1) and (2), a - b = 0 => a = b
=> c = -12a
The equation is, therefore, $$ax^2 + ax - 12a = 0 => x^2 + x - 12 = 0$$
=> -4 is a root of the equation.

Question 8

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that f(5) = - 3f(2). and that 3 is a root of f(x) = 0.

What is the value of a + b + c?

[CAT 2008]

Video Solution

Solution

f(3) = 9a + 3b + c = 0 f(5) = 25a + 5b + c
f(2) = 4a + 2b + c
f(5) = -3f(2) => 25a + 5b + c = -12a -6b -3c
=> 37a + 11b + 4c = 0 --> (1)
4(9a + 3b + c) = 36a + 12b + 4c = 0 --> (2)
From (1) and (2), a - b = 0 => a = b
=> c = -12a
The equation is, therefore, $$ax^2 + ax - 12a = 0 => x^2 + x - 12 = 0$$
a + b + c = a + a - 12a = -10a.
But the value of a is not given. Therefore, the value cannot be determined.

Question 9

The number of common terms in the two sequences 17, 21, 25,…, 417 and 16, 21, 26,…, 466 is

Video Solution

Solution

The terms of the first sequence are of the form 4p + 13

The terms of the second sequence are of the form 5q + 11

If a term is common to both the sequences, it is of the form 4p+13 and 5q+11

or 4p = 5q -2. LHS = 4p is always even, so, q is also even.

or 2p = 5r - 1 where q = 2r.

Notice that LHS is again even, hence r should be odd. Let r = 2m+1 for some m.

Hence, p = 5m + 2.

So, the number = 4p+13 = 20m + 21.

Hence, all numbers of the form 20m + 21 will be the common terms. i.e 21,41,61,...,401 = 20.

Question 10

How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

Video Solution

Solution

We have to essentially look at numbers between 1000 and 4000 (including both).

The first digit can be either 1 or 2 or 3.

The second digit can be any of the five numbers.

The third digit can be any of the five numbers.

The fourth digit can also be any of the five numbers.

So, total is 3*5*5*5 = 375.

However, we have ignored the number 4000 in this calculation and hence the total is 375+1=376

CAT Content

CAT 2008 Question Paper

What is the number of Matches played by the champion?

A. The entry list for the tournament consists of 83 players?

B. The champion received one bye.

If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?

A. Exactly one player received a bye in the entire tournament.

B. One player received a bye while moving on to the fourth round from the third round.

The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?

What are the last two digits of $$7^{2008}$$?

If the roots of the equation $$x^3 - ax^2 + bx - c = 0$$ are three consecutive integers, then what is the smallest possible value of b?

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that $$f(5) = - 3f(2)$$. and that 3 is a root of $$f(x) = 0$$.

What is the other root of f(x) = 0?

[CAT 2008]

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that f(5) = - 3f(2). and that 3 is a root of f(x) = 0.

What is the value of a + b + c?

[CAT 2008]

The number of common terms in the two sequences 17, 21, 25,…, 417 and 16, 21, 26,…, 466 is

How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

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CAT Content

CAT 2008 Question Paper

What is the number of Matches played by the champion? A. The entry list for the tournament consists of 83 players? B. The champion received one bye.

If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n? A. Exactly one player received a bye in the entire tournament. B. One player received a bye while moving on to the fourth round from the third round.

The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?

What are the last two digits of $$7^{2008}$$?

If the roots of the equation $$x^3 - ax^2 + bx - c = 0$$ are three consecutive integers, then what is the smallest possible value of b?

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ? It is known that $$f(5) = - 3f(2)$$. and that 3 is a root of $$f(x) = 0$$. What is the other root of f(x) = 0? [CAT 2008]

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ? It is known that f(5) = - 3f(2). and that 3 is a root of f(x) = 0. What is the value of a + b + c? [CAT 2008]

The number of common terms in the two sequences 17, 21, 25,…, 417 and 16, 21, 26,…, 466 is

How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

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CAT Verbal Ability Questions | VARC Questions For CAT

Also Read

Get Started with 5000+ Free CAT Questions & Mock Tests with Detailed Solutions!

What is the number of Matches played by the champion?

A. The entry list for the tournament consists of 83 players?

B. The champion received one bye.

If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?

A. Exactly one player received a bye in the entire tournament.

B. One player received a bye while moving on to the fourth round from the third round.

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that $$f(5) = - 3f(2)$$. and that 3 is a root of $$f(x) = 0$$.

What is the other root of f(x) = 0?

[CAT 2008]

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that f(5) = - 3f(2). and that 3 is a root of f(x) = 0.

What is the value of a + b + c?

[CAT 2008]