Number Theory Previous Year Questions (CAT)  Set 0004

Previous years`s CAT questions: Number Theory
Post your solutions as reply to respective questions below.

Question 1
Let f(x) be a function satisfying f(x) f(y) = f (xy) for all real x, y. If f (2) = 4, then what is the value of f (1/2)?
(1) 0
(2) 1/4
(3) 1/2
(4) 1
(5) Cannot be determined.Â Â Â (CAT 2008 )

Answer: Option 2

Question 2
Suppose, the speed of any positive integer n is defined as follows:
seed(n) = n, if n < 10
= seed(s(n)), otherwise,where s(n) indicates the sum of digits of n.For example, seed(7) = 7, seed(248 ) = seed(2 + 4 + 8 ) = seed(14) = seed(1 + 4) = seed(5) = 5 etc.
How many positive integers n, such that n < 500, will have seed (n) = 9?
(1) 39(2) 72
(3) 81
(4) 108
(5) 55 (CAT 2008 )

Answer: Option 5

Question 3
What are the last two digits of 7^{ 2008} ?
(1) 21
(2) 61
(3) 01
(4) 41
(5) 81 (CAT 2008 )

Answer: Option 3

Question 4
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?
(1) 499
(2) 500
(3) 375
(4) 376
(5) 501 (CAT 2008 )

Answer: Option 4

Question 5
Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?
(1) 1 ≤ m ≤ 3
(2) 4 ≤ m ≤ 6
(3) 7 ≤ m ≤ 9
(4) 10 ≤ m ≤ 12
(5) 13 ≤ m ≤ 15 (CAT 2008 )

Answer: Option 1

Question 6

Answer: Option 1

Question 7
Five horses, Red, White, Grey, Black and Spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs. 3000, Rs. 2000 Rs. 1000 on Red, White and Black horses respectively and ends up with no profit and no loss.
Q1) Which of the following cannot be true?
(1) At least two horses finished before Spotted
(2) Red finished last
(3) There were three horses between Black and Spotted
(4) There were three horses between White and Red
(5) Grey came in second
Q2) Suppose, in addition, it is known that Grey came in fourth. Then which of the following cannot be true?
(1) Spotted came in first
(2) Red finished last
(3) White came in second
(4) Black came in second
(5) There was one horse between Black and White (CAT 2008 )

Answer: Q1) Option 4 Q2) Option 3

Question 8
If x = (16^{3} + 17^{3} + 18^{3} + 19^{3}), then x divided by 70 leaves a remainder of
(1) 0
(2) 1
(3) 69
(4) 35 (CAT 2005)

Answer: Option 1

Question 9
A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The chemical is being pumped from one tank to another as follows:
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first and how long does it take (in minutes) to get empty after pumping starts?(1) A, 16.66
(2) C, 20
(3) D, 20
(4) D, 25 (CAT 2005)

Answer: Option 3

Question 10
If R = (30^{65} – 29^{65}) / (30^{64} + 29^{64}) , then,
(1) 0 < R ≤ 0.1
(2) 0.1 < R ≤ 0.5
(3) 0.5 < R ≤ 1.0
(4) R > 1.0 (CAT 2005)