MU-NLPC/Calc-aqua_rat huggingface dataset:
X and Y can do a piece of work in 20 days and 12 days respectively. X started the work alone and then after 4 days Y joined him till the completion of the work. How long did the work last?  Answers
A) 11 B) 10 C) 99 D) 87 E) 55

Solution: Work done by X in 4 days = (1/20 * 4) = 1/5   Remaining work = (1 - 1/5) = 4/5   (X + Y)'s 1 day work = (1/20 + 1/12) = 2/15   Now, 2/15 work is done by X and Y in 1 day.   So, 4/5 work will be done by X and Y in (15/2 * 4/5)= 
(15/2 * 4/5) = 6
6 days.   Hence, total time taken= (6 + 4)= 
(6 + 4) = 10
10 days.   Answer:10
Answer: B
If 3 workers collect 48 kg of cotton in 4 days, how many kg of cotton will 9 workers collect in 2 days?	Choose the correct answer.
A) 12 kg
B) 88 kg
C) 72 kg
D) 99 kg
E) 66 kg

Solution: (3 * 4)/48 = (9 * 2)/ x   x = 72 kg   Answer: 72 kg
Answer: C
Ravi can do a piece of work in 30 days while Prakash can do it in 40 days. In how many days will they finish it together?
Choose the correct choice from the following options: A) 17 1/7 days B) 27 1/7 days C) 23 2/7 days D) 16 4/11 days E) none of these

Solution: 1/30 + 1/40 = 7/120   120/7 = 17 1/7 days   17 1/7 daysNSWER:17 1/7 days
Answer: A
The volume of the sphere Q is (37/64)% less than the volume of sphere P and the volume of sphere R is (19/27)% less than that of sphere Q. By what is the surface area of sphere R less than the surface area of sphere P?  Select the correct option:
A) 77.77% B) 75% C) 67.5% D) 87.5% E) None of these

Solution: Explanation :   Let the volume of sphere P be 64 parts.   Therefore volume of sphere Q   =>64−(37/64)% of 64.   =>64−37=27 parts.   The volume of R is:-   =>27−(19/27)×27.   =>27−19=8 parts.   Volume ratio:   =>P:Q:R=64:27:8.   Radius ratio:   =>P:Q:R=4:3:2.   The surface area will be 16:9:5.   Surface area of R is less than the surface area of sphere P.   =>16k−4k=12k.   Now,   =>(12k/16k)×100.   =>75%.   Hence, the surface area of sphere R is less than the surface area of sphere P by 75%.   Answer : 75%
Answer: B
Eight years ago, Ajay's age was 4/3 times that of Vijay. Eight years hence, Ajay's age will be 6/5 times that of Vijay. What is the present age of Ajay?	Choose the correct answer
A) 33
B) 40
C) 99
D) 77
E) 12

Solution: Let the present ages of Ajay and Vijay be 'A' and 'V' years respectively.   A - 8 = 4/3 (V - 8) and A + 8 = 6/5 (V + 8)   3/4(A - 8) = V - 8 and 5/6(A + 8) = V + 8   V = 3/4 (A - 8) + 8 = 5/6 (A + 8) - 8   => 3/4 A - 6 + 8 = 5/6 A + 20/3 - 8   => 10 - 20/3 = 10/12 A - 9/12 A   => 10/3 = A/12 => A = 40.   Answer: 40:
Answer: B
A cistern has a leak which would empty the cistern in 20 minutes. A tap is turned on which admits 10 liters a minute into the cistern, and it is emptied in 24 minutes. How many liters does the cistern hold? Options:
A) 4800
B) 1200
C) 1680
D) 1100
E) 1300

Solution: 1/x - 1/20 = -1/24   x = 120   120 * 10= 
120 * 10 = 1_200
1200   Answer:1200
Answer: B
Jose completes a piece of work in 10 days, Raju completes the same work in 40 days. If both of them work together, then the number of days required to complete the work is	Pick one: A) 5 days B) 6 days C) 8 days D) 10 days E) 12 days

Solution: If A can complete a work in x days and B can complete the same work in y days, then, both   of them together can complete the work in x y/ x+ y days.   That is, the required No. of days = 10 × 40/50 = 8 days.   8 days)
Answer: C
The equation of line m is y = 4/3*x - 100. What is the smallest possible distance in the xy-plane from the point with coordinates (0, 0) to any point on line m?	Choose the correct choice from the following answers.
A) 48  B) 50  C) 60  D) 75  E) 100

Solution: This can be solve in two steps and without any complex calculation.   Given : equation of line m as y=(4/3)x -100. So the line intercept the axes at (0,-100) and (75,0).   This can be considered a right angle triangle with right angle at (0,0) . So Base=100 , Height=75 and Hypotenuse =125 (By Pythagoras triplet)   So a perpendicular from the (0,0) to hypotenuse will be the answer.   Area of triangle= 0.5*100*75=0.5*125* x   => x=60;   SO answer is 60=60
Answer: C
A retail appliance store priced a video recorder at 20 percent above the wholesale cost of $200. If a store employee applied the 15 percent employee discount to the retail price to buy the recorder, how much did the employee pay for the recorder?  Pick.
A) $204  B) $216  C) $220  D) $230  E) $240

Solution: Wholesale cost of video recorder = 200 $   Video recorder was priced at 20 percent above 200 = 240 $   % discount given by store employee = 15   Emlpoyee paid = .85 * 240= 
.85 * 240 = 204
204 $   $204nswer $204
Answer: A
The average salary/head of allthe workers in a workshop is Rs.850, if the average salary/head of 7technician is Rs.1000 and the average salary/head of the rest is Rs.780, the total no. of workers in the work-shop is ?
Choose the correct choice from the following answers.
A) 18  B) 20  C) 22  D) 24  E) 26

Solution: Let the total number of workers be y.   So sum of salary for all workers = sum of salary of 7 technician + sum of salary for other y -7 workers.   7 x 1000 + 780(y -7) = 850 y   ⇒ 7000 + 780y - 5460 = 850y   ⇒ 70y = 1540   ∴ y = 22   So total number of workers = 22   22
Answer: C
The average of 1st 3 of 4 numbers is 16 and of the last 3 are 15. If the sum of the first and the last number is 11. What is the last numbers?
Choose the most appropriate option
A) 2  B) 4  C) 6  D) 5  E) 7

Solution: A + 4 + C = 48   4 + C + D = 45   A + D = 11   A – D = 3   A + D = 11   2D = 8   D = 4   ANSWER 4
Answer: B
A certain student has received the following test grades: 94%, 83%, 71%, 82%, and 88%. If the highest test grade is increased by 5 percentage points, which of the following statements best describes the change in the mean and the median of the test grades?  Choose the correct choice from the following answers:	A) The mean and the median will remain unchanged.	B) The mean will remain unchanged but the median will increase.	C) The mean will increase but the median will remain unchanged.	D) The mean and the median will increase by the same amount.	E) The mean and the median will increase by different amounts.

Solution: Old set = {71, 82, 83, 88, 94}.   New set = {71, 82, 83, 88, 99} --> the median (middle term) will remain the same and the mean will increase.   Answer: The mean will increase but the median will remain unchanged..
Answer: C
How many different 5 digit numbers can be formed from the number 28579 by re-arranging its digits so that the odd digits occupy even position Choose the correct choice from the following answers.
A) 12
B) 5
C) 7
D) 3
E) 1

Solution: There are 3 odd numbers which have to be placed at 3 even places =3!=6 ways   There are 2 even numbers which have to be placed at 2 odd places. So 2! = 2 ways   so total number of ways of arranging all these numbers are  6 * 2= 
 6 * 2 = 12
12 ways   12nswer:12
Answer: A
By selling 32 plums in a dollar, a man loses 40%. How many plums must he sell in a dollar to gain 20% profit? Choose the correct choice.
A) 16
B) 18
C) 20
D) 25
E) 28

Solution: This is a question that is much easier than it first appears. If you work with the percents directly, you can solve it very quickly. If the man sells 32 plums for a dollar, he loses 40%. That means he is at 60% of his cost (100% - 40% = 60%). We are trying to figure out how many plums he has to sell in order to make a 20% profit, or be at 120% (100% + 20% = 120%). 120% is double 60%, meaning that we simply cut the number of plums in half to double our returns (from 60% to 120%), yielding  32/2= 
 32/2 = 16
16 plums, or answer choice 16.
Answer: A
A boy wants to go abroad by boat and return by flight. He has a choice of 6 different boat to go and 5 flight to return. In how many ways, can the boy perform his journey?	Pick
A) 20 B) 25 C) 30 D) 40 E) 50

Solution: Ans.(C)   Sol. Number of choices to go by ship = 6 Number of choices to return by airline = 5 From, fundamental principle, total number of ways of performing the journey = 6 x 5 = 30 ways
Answer: C
In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women? Answers.
A) 59	B) 60	C) 61	D) 62	E) 63

Solution: We need to select 5 men from 7 men and 2 women from 3 women.   Number of ways to do this   = 7C5 × 3C2   = 7C2 × 3C1 [∵ nCr = nC(n-r)]   =   7   ×   6   2   ×   1   ×   3   =   21   ×   3   =   63   Answer is 63
Answer: E
There are 24 stations between Ernakulam and Chennai. How many second class tickets have to be printed, so that a passenger can travel from one station to any other station?	Choose the correct choice from the following answers: A) 800 B) 820 C) 650 D) 870 E) 900

Solution: The total number of stations= 26   From 26 Stations we have to choose any two stations and the direction of travel (Ernakulam to Chennai is different from Chennai to Ernakulam) in 26P2 ways.   26P2= 26*25= 
26*25 = 650
650   ANSWER:650
Answer: C
A rectangular-shaped carpet that measures x feet by y feet is priced at $50. What is the cost of the carpet, in dollars per square yard? (1 square yard = 9 square feet) Choose the correct choice from the following choices
A) xy/360
B) 9xy/40
C) 40xy/9
D) 450xy
E) 450/(xy)

Solution: The area of the carpet in feet is xy.   The area in square yards is xy / 9.   The price per square yard is 50 / (xy/9) = 450/(xy).   The answer is 450/(xy).
Answer: E
The local recycling plant had a contract requiring anyone who obtained recycled bottles for storage for them, to bring them back to be recycled again. The plant could make one new bottle from every seven bottles returned. One week, on Monday, they got 343 bottles to recycle. Assuming that everybody brought back all the empties, how many could they eventually remake from the 343?	Select.
A) 53	B) 54	C) 55	D) 56	E) 57

Solution: In 1st level they will make 343/7= 
343/7 = 49
49 bottles.   In 2nd level they will make 49/7= 
49/7 = 7
7 bottles.   In 3rd level they will make 7/7= 
7/7 = 1
1 bottle.   Totally they can remake 49+7+1= 
49+7+1 = 57
57 bottles   ANSW57R:57
Answer: E
Four of the five parts numbered (a),(b),(c),(d) and (e) in the following equation are exactly equal. Which of the parts is not equal to the other four? The number of that part is the answer.  Choose the correct choice from the following answers
A) (a – b)2 + 4ab B) a(a + b) + b(a + b) C) (a2 – b2) + 2b(b + a) D) (a – b)b + (a + b)2b E) (a + b)2 – 2a(a – b) + a(2a – 2b)

Solution: The other parts are equal to (a + b)2.   Answer (a – b)b + (a + b)2b
Answer: D
What is the area of a rectangle whose length is twice its width and whose perimeter is equal to that of a square whose area is 1?	Answers.
A) 1/9 sq units.
B) 4/9 sq units.
C) 8/9 sq units.
D) 7/9 sq units.
E) 6/9 sq units.

Solution: Area will be 2/3 *4/3 = 8/9 sq units.   ANSWER:8/9 sq units.
Answer: C
John tossed a fair coin 6 times. What is the probability that the coin landed heads up exactly twice?  Pick:
A) 0.094  B) 0.15  C) 0.194  D) 0.25  E) 0.294

Solution: The probability is exactly same as the cion landed tail up once. That is 6*(1/2)^6 = 6/64 =0.094.   The answer is, therefore, (0.094).
Answer: A
Jack and Bill are randomly picking students to play kick-ball on the playground, one at a time (without return), from a group of ten students. Only five of the students among the twenty students are athletic. Jack picks first, then Bill, and so on.... What is the probability that Jack will get all of the athletes? (Answer in lowest terms). Pick
A) 1/3	B) 7/273	C) 1/2	D) 1/75	E) 1/252

Solution: There are 10 students of which 5/10 are athletes, so the probability of getting an athlete is P(athlete) = 1/2 on the first selected student.   Because selection is (without return), the P(athlete) has changed for following selections. Now, there are 9 students and number athletes is dependent upon whether first student was an athlete, or not... and so on.... Therefore, conditional probability! The probability that event B occurs, given that event A has already occurred is:   Formula = P(A\B) = P(A and B)/P(B).   Since, Jack picks first P(athlete) = 5/10, then Bill picks, and P(not athlete) = 5/9, next P(athlete) = 4/8, etc....   P(A\B) = (5/10)(5/9)(4/8)(4/7)(3/6)(3/5)(2/4)(2/3)(1/2) = 14400/3628800 = 1/252   Answer: 1/252
Answer: E
A 3-letter code word consists of letters B, and C. If the code includes all the two letters, how many such codes are possible? Pick
A) 72	B) 48	C) 36	D) 24	E) 6

Solution:   B-BC can be arranged in 3!/2!=3 ways;   C-BC can be arranged in 3!/2!=3 ways;   Total:  3+3= 
 3+3 = 6
6.   Answer: 6.
Answer: E
Lucy deposited $62500 in an investment fund that provided 12 percent annual return compounded quarterly. If she made no other transactions with the fund, in how much time, in months, did her investment earn a total interest of $5100?
Choose one
A) 0.5	B) 12	C) 3	D) 18	E) 6.1

Solution: A = P + I = 62500 + 5100= 
62500 + 5100 = 67_600
67600   67600= 62500(1 + 12/4*100)^(4t)   (676/625) = (103/100)^(4t)   t =1 1/2 yrs = 18 months   Answer: 18
Answer: D
The residents of Town X participated in a survey to determine the number of hours per week each resident spent watching television. The distribution of the results of the survey had a mean of 21 hours and a standard deviation of 8 hours. The number of hours that Pat, a resident of Town X, watched television last week was between 1 and 2 standard deviations below the mean. Which of the following could be the number of hours that Pat watched television last week?  Pick one:  A) 30  B) 20  C) 11  D) 14  E) 6

Solution: Given: mean=21 and SD=8   The number of hours that Pat watched television was between 1 and 2 standard deviations below the mean: 1 SD below the mean ismean-1*SD=13 and 2 SD below the mean ismean-2*SD=5, so the number of hours that Pat watched television was between 5 and 13 hours.   Answer: 11
Answer: C
If the capacity of a cylindrical tank is 1848 m3 and the diameter of its base is 14 m, then find the depth of the tank.  Choose the correct choice from the following options.
A) 10  B) 11  C) 12  D) 13  E) None of them

Solution: Let the depth of the tank be h meters. Then,   ∏ x 72 x h = 1848  h = (1848 x (7/22) x (1/49) = 12 m   Answer is 12
Answer: C
The equation q = 2y2 + 5y - 17, describes a parabola in the xy coordinate plane. If line l, with slope of 3, intersects the parabola in the upper-left quadrant at q = -5, the equation for l is
Choose the correct choice from the following answers:	A) + y + 15 = 0	B) - 3x - 11 = 0	C) + y - 16.5 = 0	D) - y - 7 = 0	E) + y + 13.5 = 0

Solution: Is option+ y - 16.5 = 0: -3q + y -6.5= 0=+ y - 16.5 = 0
Answer: C
Subtract. Simplify your answer and write it as a proper fraction or as a whole or mixed number.
4/5 - 1/5=  Choose the most appropriate option:
A) 1/5
B) 2/5
C) 3/5
D) 4/5
E) 5/5

Solution: Here's how to subtract 1/5 from 4/5:   4/5−	1/5   Since our denominators match, we can subtract the numerators.   4 − 1 = 3   So the answer is:   3/5   answer :3/5
Answer: C
What are the last two digits of 63*35*37*82*71*41?  Choose the correct choice:
A) 10  B) 30  C) 40  D) 70  E) 80

Solution: First, multiplied 82 and 35, since 2 and 5 will give a 0 in the end. Got last 2 digits 70.   no need to multiply 70 by 41 and 71, since they wont change the last 2 digits (try if you want to, but effective multiplication in case of 41 and 71 is only with 1).   Now, 70 * 3(of 63) gives : last 2 digits as 10.   lastly, 10 * 7(of 37) gives : last 2 digits 70.   ANSWER:70
Answer: D
Two trains start simultaneously from opposite ends of a 180-km route and travel toward each other on parallel tracks. Train X, traveling at a constant rate, completes the 180-km trip in 5 hours. Train Y, travelling at a constant rate, completes the 180-km trip in 4 hours. How many kilometers had Train X traveled when it met Train Y? Choose the most appropriate option
A) 72 B) 74 C) 76 D) 78 E) 80

Solution: If the two trains cover a total distance D, then Train X travels (4/9)*D while Train Y travels (5/9)*D. If the trains travel 180 km to the meeting point, then Train X travels  (4/9)*180= 
 (4/9)*180 = 80
80 km. The answer is 80.
Answer: E
What will be the cost of building a fence around a square plot with area equal to 289 sq ft, if the price per foot of building the fence is Rs. 58?	Choose the correct choice from the following:	A) Rs.3944	B) Rs.3948	C) Rs.3988	D) Rs.3949	E) Rs.3923

Solution: Let the side of the square plot be a ft.   a2 = 289 => a = 17   Length of the fence = Perimeter of the plot = 4a = 68 ft.   Cost of building the fence = 68 * 58 = Rs.3944.   Rs.3944nswer: Rs.3944
Answer: A
Find the product of the local value and absolute value of 4 in 20468?  Select:
A) 1000 B) 1100 C) 1200 D) 1340 E) 1600

Solution: Local value of 4 = 4 x 100 = 400   Place value of 4 = 4   There fore = 4 x 400 = 1600   1600
Answer: E
From a group of 4 boys and 4 girls, 4 children are to be randomly selected. What is the probability that 2 boys and 2 girls will be selected?
Choose the correct choice from the following answers:
A) 12/29 B) 14/31 C) 16/33 D) 18/35 E) 20/37

Solution: The total number of ways to choose 4 children from 8 is 8C4 = 70   The number of ways to choose 2 boys and 2 girls is 4C2*4C2 = 6*6= 
6*6 = 36
36   P(2 boys and 2 girls)= 36/70 = 18/35   The answer is 18/35.
Answer: D
From a group of 16 astronauts that includes 7 people with previous experience in space flight, a 3-person crew is to be selected so that exactly 1 person in the crew has previous experience in space flight. How many different crews of this type are possible? Choose the most appropriate option:  A) 270  B) 252  C) 272  D) 292  E) 285

Solution: number of astronauts with previous experience = 7   Remaining =9   we need to pick exactly one from the first group   and 2 from the second group   (7C1)*(9C2)   =7*36= 
7*36 = 252
252   Answer : 252
Answer: B
A does half as much work as Band C does half as much work as A and B together. If C alone can finish the work in 80 days, then together ,all will finish the work in :	Choose the correct choice from the following options:  A) 13 1/3 days  B) 16 days  C) 15 days  D) 20 days  E) 30 days

Solution: C alone can finish the work in 80 days.   (A + B)can do it in 20 days   (A + B)s 1 days wok = 1/20.   As 1 days work : Bs 1 days Work = 1/2 : 1 = 1:2.   Aâ€™s 1 dayâ€™s work = (1/20) * (1/3) = (1/60). [Divide 1/20 in the raio 1:2] Bs 1 days work = (1/20) * (2/3) = 1/30   (A+B+c)S 1 dayâ€™s work = (1/60) + (1/30) + (1/80) = 1/16   All the three together will finish it in 16 days.   ANSWER:16 days
Answer: B
Ramu mixed 40% kerosene solution to 25% kerosene solution hence obtaining 30% kerosene solution. In what ratio were the both solutions mixed?
Choose the most appropriate option.
A) 2:9	B) 2:2	C) 2:1	D) 2:8	E) 2:5

Solution: QUESTION : A can contains a mixture of two liquids X and Y in the ratio 7:5 when 9 liters of mixture are drawn off and the can is filled with Y, the ratio of X and Y becomes 7:9. How many liters of liquid A were contained by the can initially?   12 Litres   21 litres   13 litres   22 litres   ANSWER :2:1
Answer: C
When p is divided by 9, the remainder is 3. All of the following are possible values of p, except for which of the following?	Pick:	A) 12	B) 30	C) 48	D) 56	E) 66

Solution: Four of these numbers can be expressed in the form of 9x+3, where x can be 0,1,2,3,4....   However, 56 = 9(6) + 2   The answer is 56.
Answer: D
In an examination a pupil's average marks were 63 per paper. If he had obtained 26 more marks for his Geography paper and 2 more marks for his History paper, his average per paper would have been 65. How many papers were there in the examination?
Choose the correct choice:
A) 8  B) 9  C) 10  D) 14  E) 12

Solution: Let the number of papers be x. Then,   63x + 26 + 2 = 65x   = 2x = 28   = x = 14.   ANSWER:14
Answer: D
WINK, Inc. follows a certain procedure that requires two tasks to be finished independently in order for a job to be done. On any given day, there is a 2/3 probability that task 1 will be completed on time, and a 3/5 probability that task 2 will be completed on time. On a certain day, what is the probability that task 1 will be completed on time, but task 2 will not? Select: A) 4/15 B) 3/40 C) 13/40 D) 7/20 E) 13/22

Solution: P(1 and not 2)=2/3*(1-3/5)=4/15.   4/15nswer: 4/15.
Answer: A
A little girl is numbering her collection of butterflies by placing single-digit stickers under each specimen. If the girl has 330 butterflies, how many stickers does she need? (The numbers are consecutive and the number of the first butterfly is 1).  Answers.
A) 882 B) 901 C) 1,001 D) 1,003 E) 1,346

Solution: For the first 9 butterflies we need 9 stickers.   For the next 90 we need 2 stickers each or 180 stickers.   For the next 900 butterflies we need 3 stickers each.   The first 99 butterflies correspond to 189 stickers.   Subtract 99 from the total number of butterflies (330).   This leaves 231 butterflies that use 3 stickers each. So, multiply 231 by 3 to get 693.   Add 693 to 189 stickers to get 882 stickers.   882nswer: 882
Answer: A
On a journey across Bombay, a tourist bus averages 10 km/h for 20% of the distance, 30 km/h for 60% of it and 20 km/h for the remainder. The average speed for the whole journey was  Answers
A) 10 km/h  B) 30 km/h  C) 5 km/h  D) 20 km/h  E) None of these

Solution: explanation   Let the average speed be x km/h.   and Total distance = y km. Then,   0.2⁄10y + 0.6⁄30y + 0.2⁄20y = y⁄x   ⇒ x = 1⁄0.05 = 20 km/h   Answer 20 km/h
Answer: D
If a tire rotates at 400 revolutions per minute when the car is traveling 24km/h, what is the circumference of the tire?	Answers.
A) 1 meters B) 9 meters C) 8 meters D) 5 meters E) 3 meters

Solution: 400 rev / minute = 400 * 60 rev / 60 minutes   = 24,000 rev / hour   24,000 * C = 24,000 m : C is the circumference   C = 1 meters   correct answer 1 meters
Answer: A
What is the remainder when 3^33 is divided by 5?
Pick: A) 0 B) 1 C) 2 D) 3 E) 4

Solution: The units digit of powers of 3 follow a repeating cycle of four: {3, 9, 7, 1}   33 has the form 4k+1, so the units digit of 3^33 is 3.   The remainder when dividing by 5 is 3.   The answer is 3.
Answer: D
Three numbers are in the ratio 3:5:7. The largest number value is 42. Find difference between Smallest & largest number is? Choose the correct choice from the following options
A) difference=24  B) 26  C) 28  D) 30  E) 32

Solution: == 3:5:7   Total parts = 15   = The largest number value is 42   = The largest number is = 7   = Then 7 parts -----> 42 ( 7 * 6 = 42 )   = smallest number = 3 & Largest number = 7   = Difference between smallest number & largest number is = 7 - 3= 
7 - 3 = 4
4= Then 4 parts -----> 24 (4 * 6 = 24)   difference=24
Answer: A
A man in a train notices that he can count 21 telephone posts in one minute.If they are known to be 50 meters apart , then at what speed is the train travelling ?  Pick:
A) 55 km/hr
B) 57 km/hr
C) 60 km/hr
D) 63 km/hr
E) None of these

Solution: Solution   Number of gaps between 21 telephone posts = 20.   Distance travelled in 1 minute =(50x20)m = 1000 m = 1 km.   ∴ Speed = 60 km/hr.   Answer 60 km/hr
Answer: C
Prove the theorem, due to Kraitchik, asserting that 13|2^70+3^70 Choose the correct answer.
A) 1^70+3^70 == 0 (mod 13), or 1|2^70+3^70 B) 2 C) 1 D) 2^70+3^70 == 0 (mod 13), or 13|2^70+3^70 E) 1

Solution: By Fermat's theorem we have 2^12 == 1 (mod 13), hence 2^60 == 1 (mod 13), and since 2^5 == 6 (mod 13), which implies 2^10 == -3 (mod 13), we get 2^70 == -3 (mod 13). On the other hand, 3^3 == 1 (mod 13), hence 3^69 == 1 (mod 13) and 3^70 == 3 (mod 13). Therefore 2^70+3^70 == 0 (mod 13), or 13|2^70+3^70, which was to be proved.   So Answer is 2^70+3^70 == 0 (mod 13), or 13|2^70+3^70 2^70+3^70 == 0 (mod 13), or 13|2^70+3^70
Answer: D
If selling price is doubled, the profit triples. Find the profit percent  Choose the correct choice
A) 128%	B) 100%	C) 126%	D) 276%	E) 262%

Solution: Explanation:   Let C.P. be Rs. x and S.P. be Rx. y.   Then, 3(y - x) = (2y - x) => y = 2x   Profit = (y - x) = Rs. (2x - x) = Rs. x   Profit % = x/x * 100 = 100%   .Answer:100%
Answer: B
10 one question was x*y^2*z<0 then  Choose the correct choice from the following:
A) x<0
B) xyz<0
C) xz<0
D) xy<0
E) None of these

Solution: if we take y^2 then it will always greater than zero so only one option z left i.e. xz   ANSWER:xz<0
Answer: C