Quant Boosters  Shashank Prabhu, CAT 100 Percentiler  Set 5

Number of Questions  30
Topic  Quant Mixed Bag
Solved ? : Yes
Source : Learningroots forum

Q1) If a three digit number abc has 3 factors, how many factors does the 6digit number abcabc have?
a. 16
b. 24
c. 16 or 24
d. 20

abcabc = 1001 * abc
abc has 3 factors and so, abc is the square of a prime number.
1001=7 * 11 * 13. So, the overlap could be of 11 or 13.
Possible numbers are: 121121, 169169 or 7 * 11 * 13 * abc.
Number of factors can be 4 * 2 * 2 or 2 * 2 * 2 * 3 so 16 or 24.

Q2) At a 10,000 m race, The Flash starts first and is followed later by Bolt. The speed of Bolt is 1 m/s more than that of The Flash’s. When Bolt catches up with The Flash, The Flash increases his speed by 2 m/s, while that of Bolt remains unchanged. As a result, Bolt finishes 7 min 8 s after The Flash. If the distance had been 500 m more, then Bolt would have finished 7 min 33 s after The Flash. The time gap between the start of The Flash and Bolt is
a. 2 min
b. 2.5 min
c. 3 min
d. 1 min

500/x  500/(x+1) = 25
(x + 1  x)/x(x + 1) = 25/500 = 1/20
x(x + 1) = 20
x = 4
Bolt's speed = 4
Flash increased speed = 5
Flast actual speed = 3
25 sec = 500 m
7 min 8 sec = 8560 m
10000  8560 = 1440
1440/4 = 360
1440/3 = 480
Gap = 120 secs = 2 min

Q3) Fifteen lines are drawn in a plane such that four of them are parallel. What is the maximum number of regions into which the plane is divided?

12 lines divide into 12 * 13/2+1=79 regions. The 13th line will be parallel to the 12th one and so, can cut 11 other lines at max. So, it will add 12 more regions. Similarly, the 14th line and the 15th line will also add 12 regions each. So, total of 79+36=115.
Basically for n lines, n(n+1)/2 + 1 regions
For unbounded regions, again a bit of visualization will help.
1 line  2 unbounded regions
2 lines  4 unbounded regions
3 lines  1 bounded and 6 unbounded regions
4 lines  3 bounded and 8 unbounded regions
Similarly, for 15 lines, there would be 30 unbounded regions and 11530=85 bounded regions.

Q4) What is the number of integers greater than 10^6 that can be formed using the digits 2, 3, 0, 3, 4, 2 and 3? No other digit other than the given set of digits is allowed.

We can form 7 digit numbers only.
There are 2 cases now: either the number starts with a 0 or not. So we take the total number of cases irrespective of whether it starts with a 0 and then subtract the cases that start with a 0.
0 _ _ _ _ _ _ in 6!/(3! * 2!)=60 ways
_ _ _ _ _ _ _ in 7!/(3! * 2!)=420 ways
Numbers greater than 10^6=42060=360

Q5) All threedigit numbers, in which the ten’s digit is a natural number and is a perfect square, are formed using the digits 1 to 9. The sum of all such numbers is

_ 1 _
_ 4 _
_ 9 _
3 * 9 * 45 * 100 + 3 * 9 * 45 * 1 + 81 * 10 + 81 * 40 + 81 * 90
121500+1215+11340
134055

Q6) The Xmansion has a rectangular plot of area 900 sq.m and has to be fenced using nonmagnetic materials to ward off Magneto. Logan comes up with an idea to fortify two adjoining sides with bricks and the remaining two with a wooden fence. One metre of the wooden fence costs 10 Dollar and one metre of the brick fence costs 25 Dollar. Ted Mosby, the architect, has been given 2,000 Dollar to complete the task. Should Ted take up the contract?
a. Yes
b. No
c. Doesn't matter
d. Data insufficient

Ted would not take up the contract if even in the best case, where the material used will be minimum, he would end up making a loss. Best case is when all the sides are equal and so, the costs will be optimized. So 30 * 2 * 25 + 30 * 2 * 10 = 2100 which is still less than 2000. So, Ted won't take it up.

Q7) In how many ways can 1000 be written as a product of 3 factors?

abc=1000
a=2^a1 * 5^a2
b=2^b1 * 5^b2
c=2^c1 * 5^c2
a1+b1+c1=3... 10 solutions
a2+b2+c2=3... 10 solutions
Total solutions=100. But these will include ordered solution sets. So we need to eliminate these
1 case (10,10,10)
3 cases (1,1,1000)(2,2,250)(5,5,40) with two repeated so 9 cases in the original 100
Remaining 90 cases divided by 3! so 15 ways.
Total of 19 ways

Q8) In a mile race, Akshay can be given a start of 128 m by Bhairav. If Bhairav can give Chinmay a start of 4 m in a 100 m dash, then who out of Akshay and Chinmay will win a race of one and half miles, and what will be the final lead given by the winner to the loser? (One mile is 1,600 m.)
a. Akshay, 1/12 miles
b. Chinmay, 1/32 miles
c. Akshay, 1/24 miles
d. Chinmay, 1/16 miles

D is right.
A/B=1472/1600
B/C=100/96=1600/1536
A/C=1472/1536
So, C will win. If C covers 2400 m, A will cover 1472/1536 * 2400=25 * 92=2300 m. So, C wins by 100 m or 1/16th of a mile.

Q9) What is the value of the following expression?
[(1/ (2^2 – 1) + (1/ (4^2 – 1) + 1/ (6^2 – 1) + … 1/ (20^2 – 1)]

It will be a series in the form of
1/(1 * 3)+1/(3 * 5)...1/(19 * 21)
1/2{2/(1 * 3)+2/(3 * 5)...2/(19 * 21)}
1/2{(31)/(3 * 1)+(53)/(5 * 3)...(2119)/(21 * 19)}
1/2(11/3+1/31/5...1/191/21)
1/2(11/21)
10/21

Q10) Basanti is walking beside a railway track between Pune and Baramati at a constant speed towards Baramati. Local trains ply between the two cities at equal intervals in both directions. She encounters a train going from Pune to Baramati after every 8 minutes and a train going from Baramati to Pune after every 6 minutes. What is the time interval between the two consecutive trains going from Pune to Baramati? (Trains between Pune and Baramati in both the directions run at the same speed)