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We are back with another set of CAT questions. Try to solve this set in five minutes.

Q.1 A truck traveling at 70 mph uses 30% more diesel to travel a certain distance than it does when it travels at the speed of 50 mph. If the truck can travel 19.5 kilometers on a liter of diesel at 50 mph, how far can the truck travel on 10 liters of diesel at a speed of 70 mph. (CAT 2000)

Q.2 There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required? (CAT 2003)
(A) 72 (B) 90 (C) 96 (D) 144

Q.3 A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of three cases the leading digit is 1. Then M equals (CAT 2003)
(A) 31 (B) 63 (C) 75 (D) 91

Q.4 An intelligence agency forms a code of two distinct digits selected from 0, 1, 2 … 9 such that the first digit of the code is non zero. The code handwritten on a slip, can however potentially create confusion when read upside down – For example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise? (CAT 2003)
(1) 80 (2) 78 (3) 81 (4) 69

Q.5 The digits of a three digit number A are written in the reverse order to form another three digit number B. If B > A, and B – A is perfectly divisible by 7, then which of the following is necessarily true? (CAT 2005)
(1) 100 < A < 299 (2) 106 < A < 305 (3) 112 < A < 311 (4) 118 < A < 317

Solutions:

Q.1 A truck traveling at 70 mph uses 30% more diesel to travel a certain distance than it does when it travels at the speed of 50 mph. If the truck can travel 19.5 kilometers on a liter of diesel at 50 mph, how far can the truck travel on 10 liters of diesel at a speed of 70 mph. (CAT 2000)

Solution:

Essentially, one needs to understand the relationship between usage of diesel, speed and distance. At 50 mph, it covers 19.5 kilometers in 1 liter = 195 kilometers in 10 liters. If one thing increases by x/y, the other should fall by x/(x+y) to maintain the relationship if other factors remain same.

x/y = 3/10 and x/(x+y) = 3/13
195*3/13 = 45
195 – 45 = 150

Q.2 There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required? (CAT 2003)

Solution:

There is standard way of solving this by actual counting. But a different approach is “How many connections will every town get?” A town which is part of a zone gets 6 lines from towns within that zone and 3*3 = 9 lines from towns in other zones = 15. Total number of towns is 12. 15*12 = 180 and every incoming line for town x from y town will act as incoming for town y from x town (essentially we are double counting) and hence divide by 2 = 90

Q.3 A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of three cases the leading digit is 1. Then M equals (CAT 2003)

Solution:

As the last digit in each of the cases is 1, the number when divided by 2, 3, 5 leaves a remainder of 1. Which means the number is of the form 30k + 1.
Hence, the answer is either 31 or 91. Starting with 91 and checking will give the correct answer.

Q.4 An intelligence agency forms a code of two distinct digits selected from 0, 1, 2 … 9 such that the first digit of the code is non zero. The code handwritten on a slip, can however potentially create confusion when read upside down – For example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise? (CAT 2003)

Solution:

First of all, find numbers where confusion can occur. 0, 1, 6, 8, 9. From this 0 is not possible as the first digit of the code is non-zero.
How many codes are possible with two distinct digits? 9 * 9 = 81
From remaining numbers, we can create following codes:
16, 18, 19, 61, 68, 69, 81, 86, 89, 91, 96, 98.
Out of this, 69 and 96 will not create confusion. Hence, 81 – 10 = 71 is the answer.

Q.5 The digits of a three digit number A are written in the reverse order to form another three digit number B. If B > A, and B – A is perfectly divisible by 7, then which of the following is necessarily true? (CAT 2005)

Solution:

A = 100X + 10Y + Z
B = 100Z + 10Y + X
Hence, 99Z – 99X is divisible by 7. That’s possible only when Z – X = 7
Possibilities:
Z = 9, X = 2 → maximum A = 299
Z = 8, X = 1 → minimum A = 108