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1. A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs. 250 and Rs. 300 per day respectively. In addition, a male operator gets Rs. 15 per call he answers and a female operator gets Rs. 10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?
a) 15
b) 14
c) 12
d) 10

2. There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or to person 2; task 2 must be assigned to either person 3 or person 4. Every person is to be assigned one task. In how many ways can the assignment be done?
a) 144
b) 180
c) 192
d) 360

3. The number of employees in Obelix Menhir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of employees who are not, can possibly be:
a) 101:88
b) 87:100
c) 110:111
d) 97:84

4. For a positive integer n, let Pn denote the product of the digits of n, and Sn denote the sum of the digits of n. The number of integers between 10 and 1000 for which Pn + Sn = n is
a) 81
b) 16
c) 18
d) 9

5. There are 6 boxes numbered 1, 2... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is
a) 5
b) 21
c) 33
d) 60

6. A boat travels from point A to point B upstream and returns from point B to point A downstream. If the round trip takes the boat 5 hours and the distance between point A and point B is 120 kms and the speed of the stream is 10 km/hr, how long did the upstream journey take?
a) 2 hours 40 minutes
b) 2 hours 24 minutes
c) 3 hours 
d) 2 hours

7. Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be
a) 85.5
b) 92.5
c) 90.5
d) 87.5

8. Two liquids A and B are in the ratio 5 : 1 in container 1 and in container 2, they are in the ratio 1 : 3. In what ratio should the contents of the two containers be mixed so as to obtain a mixture of A and B in the ratio 1 : 1?
a) 2 : 3 
b) 4 : 3 
c) 3 : 2 
d) 3 : 4

9. Once I had been to the post-office to buy stamps of five rupees, two rupees and one rupee. I paid the clerk Rs 20, and since he did not have change, he gave me three more stamps of one rupee. If the number of stamps of each type that I had ordered initially was more than one, what was the total number of stamps that I bought?
a) 10 
b) 9 
c) 12
d) 8

10. Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide?
a) 1/12
b) 1/6
c) ¼
d) 1/31. A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs. 250 and Rs. 300 per day respectively. In addition, a male operator gets Rs. 15 per call he answers and a female operator gets Rs. 10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?
a) 15
b) 14
c) 12
d) 10

2. There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or to person 2; task 2 must be assigned to either person 3 or person 4. Every person is to be assigned one task. In how many ways can the assignment be done?
a) 144
b) 180
c) 192
d) 360

3. The number of employees in Obelix Menhir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of employees who are not, can possibly be:
a) 101:88
b) 87:100
c) 110:111
d) 97:84

4. For a positive integer n, let Pn denote the product of the digits of n, and Sn denote the sum of the digits of n. The number of integers between 10 and 1000 for which Pn + Sn = n is
a) 81
b) 16
c) 18
d) 9

5. There are 6 boxes numbered 1, 2... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is
a) 5
b) 21
c) 33
d) 60

6. A boat travels from point A to point B upstream and returns from point B to point A downstream. If the round trip takes the boat 5 hours and the distance between point A and point B is 120 kms and the speed of the stream is 10 km/hr, how long did the upstream journey take?
a) 2 hours 40 minutes
b) 2 hours 24 minutes
c) 3 hours 
d) 2 hours

7. Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be
a) 85.5
b) 92.5
c) 90.5
d) 87.5

8. Two liquids A and B are in the ratio 5 : 1 in container 1 and in container 2, they are in the ratio 1 : 3. In what ratio should the contents of the two containers be mixed so as to obtain a mixture of A and B in the ratio 1 : 1?
a) 2 : 3 
b) 4 : 3 
c) 3 : 2 
d) 3 : 4

9. Once I had been to the post-office to buy stamps of five rupees, two rupees and one rupee. I paid the clerk Rs 20, and since he did not have change, he gave me three more stamps of one rupee. If the number of stamps of each type that I had ordered initially was more than one, what was the total number of stamps that I bought?
a) 10 
b) 9 
c) 12
d) 8

10. Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide?
a) 1/12
b) 1/6
c) ¼
d) 1/3

Answers with solutions-

1. (d)
Let x females and y males be employed.
As the total number of calls to be answered = 1000 and males and females can handle 40 and 50 calls respectively everyday
50x + 40y = 1000
40y = 1000 – 50x
∴ y = 25 – x – x/4
As 7 < x 12, x can be 8 or 12
If x = 8, y = 15 and if x = 12, y = 10
The total cost of employing x females and y males
= 300x + 250y + (50 × 10 × x) + (40 × 10 × y)
= 800x + 650y
If x = 8 and y = 15, cost = Rs. 16150
If x = 12 and y = 10, cost = Rs. 16100
Thus cost is minimized when the number of male operators is 10.

2. (a)
Task 2 can be assigned in 2 ways (either to person 3 or person 4).
Task 1 can then be assigned in 3 ways (persons 3 or 4, 5 and 6)
The remaining 4 tasks can be assigned to the remaining 4 persons in 4! = 24 ways
∴ The assignment can be done in 24 × 2 × 3 = 144 ways.

3. (d)
Consider options. As the number of employees is prime we can add the numerator and denominator of ratios directly to find the number of employees.
1. Number of employees = 101 + 88 = 189 Number of employees = 189, which is not a prime number.
∴ Option 1 is eliminated.
2. Number of employees = 87 + 100 = 187 Number of employees = 187, which is not a prime number.
∴ Option 2 is eliminated.
3. Number of employees = 110 + 111 = 221 Number of employees = 221, which is not a prime number. ∴ Option 3 is eliminated.
4. Number of employees = 85 + 98 = 183 Number of employees = 183, which is not a prime number. ∴ Option 4 is eliminated.
5. Number of employees = 97 + 84 = 181 Number of employees = 181, which is a prime number.
∴ The ratio of employees = 97:84

4. (d)
n can be a 2 digit or a 3 digit number.

Case (I)
Let n be a 2 digit number.
Let n = 10x + y, where x and y are non-negative integers,
Pn = xy and Sn = x + y
Now, Pn + Sn = n
∴ xy + x + y = 10x + y
∴ xy = 9xy = 9
There are 9 two digit numbers (19, 29, 29, ... ,99) for which y = 9

Case (II)
Let n be a 3 digit number.
Let n = 100x + 10y + z, where x, y and z are non-negative integers,
Pn = xyz and Sn = x + y + z Now, Pn + Sn = n xyz + x + y + z = 100x + 10y + z
∴ xyz = 99x + 9y
∴ z = 99/y + 9/x

From the above expression, 0 < x, y < 9
But, we cannot find any value of x and y, for which z is a single digit number.
∴ There are no 3 digit numbers which satisfy Pn + Sn = n
Hence, option d.

5. (b)
GRRRRR, RGRRRR, RRGRRR, RRRGRR, RRRRGR, RRRRRG
GGRRRR, RGGRRR, RRGGRR, RRRGGR, RRRRGG
GGGRRR, RGGGRR, RRGGGR, RRRGGG
GGGGRR, RGGGGR, RRGGGG
GGGGGR, RGGGGG
GGGGGG
Hence 21 ways.

6. (c)
The total distance covered by the boat during the round trip = 120 * 2 = 240 kms. The boat took 5 hours to undertake the round trip. Therefore, the average speed of travel = 240/5 = 48 km/hr.

Let U be the speed of the boat upstream. Let B be the speed of the boat in still water and D be the speed of the boat downstream.

Upstream speed of boat = Still water speed - speed of stream
= B - 10.

Downstream speed of boat = Still water speed + speed of stream
= B + 10.

As the distance between A and B is the same as the distance between B and A, the average speed of travel for the round trip is given by = 48 km/hr (two different speeds equal distances in each of the speeds, then the average speed is given by the formula given above)

= 48.

Solving for B, we get B2 - 48B - 100 = 0
==> B2 - 50B + 2B - 100 = 0
=> B = 50 or B = -2. As speed cannot be negative, we get the speed of the boat in still water = 50 km/hr.

Therefore, the upstream speed = 50 - 10 = 40 km/hr.
Time taken to travel upstream = = 3 hours.

7. (d)
The surface area of a sphere is proportional to the square of the radius.

8. (d)
The problem can easily be solved by alligation. In container 1, the ratio of liquid A to the total liquid is 5/(5 + 1) = 5/6. In container 2, this ratio is 1/(1 + 3) = ¼. In the final mixture, this ratio will be 1/(1 + 1) = ½. Alligating as shown, we get the required ratio as 3 : 4.

9. (b)
At least two stamps of each type were ordered initially. So Rs. 2(5 + 2 + 1) = Rs. 16 have been spent. That leaves Rs. (20 - 16) = Rs. 4. In these Rs. 4, three more stamps of one rupee were given, thus accounting for Rs. 19 in all. Since one more rupee remains, it means that one more stamps of Rs. 2 was bought initially. So the total number of stamps is 2(0f Rs. 5) + 3 (of Rs. 2) + 4(of Re. 1). Note that this is the only possible combination of stamps which is consistent with the given data.

10. (c)
Their relative speed is 15 km/h they are 20 km apart hence in 80 min they will collide. Now in 79 min distance covered by first boat is 79/12 km and distance traveled by second boat is 79/6 km so they are ¼ km apart. Hence answer option is (c).

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