Number Theory Previous Year Questions (CAT)  Set 0023

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

Question 1
For real numbers x, y, let
f(x, y) = Positive squareroot of (x + y), if (x + y)^{0.5} is real
= (x + y)^{2}, otherwise
g(x, y) = (x + y)^{2}, if (x + y)^{0.5} is real
= –(x + y), otherwise
Under which of the following conditions is f(x, y) necessarily greater than g(x, y)?
(1) Both x and y are less than –1
(2) Both x and y are positive
(3) Both x and y are negative
(4) y > x (CAT 2000)

Answer: Option 1

Question 2
For a real number x,
let f(x) = 1/(1 + x), if x is nonnegative
= 1+ x, if x is negative
f ^{n}(x) = f(f ^{n }^{– 1}(x)), n = 2, 3, ....
r is an integer > 2. Then, what is the value of f ^{r }^{– 1}(–r) + f ^{r}(–r) + f ^{r }^{+ 1} (–r) ?
(1) 1
(2) 0
(3) 1
(4) None of these (CAT 2000)

Answer: Option 2

Question 3
For two positive integers a and b define the function h(a,b) as the greatest common factor (G.C.F) of a, b. Let A be a set of n positive integers. G(A), the G.C.F of the elements of set A is computed by repeatedly using the function h. The minimum number of times h is required to be used to compute G is
a. n/2
b. (n – 1)
c. n
d. None of these (CAT 1999)

Question 4

Question 5

Question 6

Question 7
I have onerupee coins, 50paisa coins and 25paisa coins.
The number of coins are in the ratio 2.5 : 3 : 4
If the total amount with me is Rs. 210, find the number of onerupee coins.
a. 90
b. 85
c. 100
d. 105Â Â Â (CAT 1998 )

Question 8
In a locality, there are five small cities: A, B, C, D and E. The distances of these cities from each other are as follows.
AB = 2 km AC = 2km AD > 2 km AE > 3 km BC = 2 km
BD = 4 km BE = 3 km CD = 2 km CE = 3 km DE > 3 km
If a ration shop is to be set up within 2 km of each city, how many ration shops will be required?
a. 2
b. 3
c. 4
d. 5 (CAT 1996)

Question 9
In a locality, there are five small cities: A, B, C, D and E. The distances of these cities from each other are as follows.
AB = 2 km AC = 2km AD > 2 km AE > 3 km BC = 2 km
BD = 4 km BE = 3 km CD = 2 km CE = 3 km DE > 3 km
If a ration shop is to be set up within 3 km of each city, how many ration shops will be required?
a. 1
b. 2
c. 3
d. 4 (CAT 1996)

Question 10
A student gets an aggregate of 60% marks in five subjects in the ratio 10 : 9 : 8 : 7 : 6. If the passing marks are 50% of the maximum marks and each subject has the same maximum marks, in how many subjects did he pass the examinaiton?
a. 2
b. 3
c. 4
d. 5 Â (CAT 1997)