## QBM004

M161. How many cuboids exist for which the volume is less than 100 units3 and the integer side lengths are in an arithmetic sequence?
M162. Prove that 6n + 8n is divisible by 7 iff n is odd.
M163. How many solutions does the equation |x| + 2|y| = 100 have?
M164. A truck driving east at 50 mph passes a certain mile marker. A motorcyclist also driving east passes that same mile marker 45 minutes later. If the motorcyclist is driving 65 mph, how long will it take for the motorcyclist to pass the truck?
M165. Can you prove that (2n)! is divisible by 22n âˆ’ 1?
M166. The velocity v of a body moving in a straight line t seconds after starting from rest is v = 4t3 - 12 t2 meters/second
a. How many seconds after starting does its acceleration become zero?
b. Mary began walking home from school, heading south at a rate of 4 mph. Sharon left school at the same time heading north at 6 mph.
c. How long will it take for them to be 3 miles apart?
M167. The sum of the infinite series 1- (1/2) + (1/4) - (1/8) + ... is
M168. Two airplanes leave an airport simultaneously, one heading east; the other, west. The eastbound plane travels at 140 mph and the westbound plane travels at 160 mph. How long will it take for the planes to be 750 miles apart?
M169. Sachin can paint a bat in 45 minutes; Rahul, in 30 minutes. If Sachin
begins painting his bat and Rahul joins him to help after 15 minutes, how
long will it take both to finish the job?
M170.Farmer Brown told Bob and Sue that they could pick apples from his tree, but that neither of them could take more than 20. They worked for a while, and then Bob asked Sue, "Have you picked your limit yet?"
Sue replied, "Not yet. But if I had twice as many as I have now, plus half as many as I have now, I would have my limit." How many did Sue have?
M171.Q. The probability that Nike will get Cricket team contract is 2/3 and the probability that Nike will get Hockey team contract is 5/9. If the probability of getting at least one contract is 4/5, what is the probability of getting both the contract?
a. 19/45
b. 13/45
c. 12/35
d. 11/23
END

## QBM026

Q1. Find all sets of four real numbers x_{1}, x_{2}, x_{3}, x_{4} such that the sum of any one and the product of the other three is equal to 2.
Q2. During each hour of the day there is a time when the minute hand and the hour hand of the clock coincide. When does this happen between 3:00 and 4:00?
Q3. We will call a number that consists of the same sequence of digits repeated twice, such as 11 or 12391239 a doublestring number. Find the smallest doublestring number which is a square.
Q4. Two students play a game based on the total roll of two standard dice. Student A says that a 12 will be rolled first. Student B says that two consecutive 7s will be rolled first. The students keep rolling until one of them wins. What is the probability that A will win?
Q5. 4 vertices of a regular tetrahedron are such that every three of them form an equilateral triangle. How many points in 3-space can you find so that every three of them form an isosceles triangle?
Q6. In a partnership, two men invest $2000 and $3000. If the net profit of $13500 at the end of the year is divided in accordance with the amount each partner invested, then the man who invested more gets
Q7. In Ms. Brownâ€™s class the ratio of boys to girls is 3:5 . On a recent exam, the pass to fail ratio for the boys was 5:6 and the pass to fail ratio for the girls was 3:1 . What fraction of the entire class passed the exam?
1.
Q8. The angle of elevation of the top of a tower 30 m high, from two points on the level ground on its opposite sides are 45 degrees and 60 degrees. What is the distance between the two points?
A. 30 B. 51.96 C. 47.32 D. 81.96
Q9. Consider the following series.
For which values of n is S(n) rational?
Q10. In the equation (10x + x)x = 100 y^{2} + 10 y^{3} + y^{2}, if y = 2 then x is
Q11. A club has 8 male and 8 female members. The club is to choose a committee of 6 members. The committee must have 3 male and 3 female members. How many different committees can be chosen?
a) 112896 b) 3136 c) 720 d) 112
Q12. What is the unitâ€™s digit of the number 31771?
a) 1 b) 7 c) 3 d) none of these
Q13. What is the sum of the series till 3n Series...
1 + 3 - 5 + 7 + 9 -11+ 13 + 15 - 17
Q14. A square is inscribed in a circle of radius r. The area of the region in the circle but not in the square, expressed as fraction of the area of the circle, is:
1) 1 - 2/ï° 2) 4/ï° 3) ï°/4 - 1 4) dependent upon the radius of the circle
Q15. A club has 8 male and 8 female members. The club is choosing a committee of 6 members. The committee must have 3 male and 3 female members. How many different committees can be chosen?
1) 2896 2) 3136 3) 720 4) 112
END

## QBM027

Q1. In a plane a set of n points (n ≥ 3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length d.
The number of diameters of the given set is at most?
Q2. An integer n has property P if there are integers p and q such that
0 < p < q < n and the sum p+(p+1)+. . . +q is divisible by n.
then n has property P if and only if
a. n is not a power of 2. b. n is a power of 2. c. n is not a power of 3.
d. n is a power of 3.
Q3. A triomino is an arrangement of three squares with a "corner" as sketched below -- note that the figure may be rotated through any multiple of 90 degrees
_____
| | |
|__|__|
| |
|__|
What is the largest number of squares on an 8 X 8 checkerboard which can be colored green so that any triomino on the board has at least one of its squares not colored green?
Q4. We will call a number that consists of the same sequence of digits repeated twice, such as 11 or 12391239 a doublestring number. Find the smallest doublestring number which is a cube.
Q.5 A technician mixes a solution having water and oil in the ratio 3 : 4 with another solution having oil and turpentine in the ratio 5 : 6. How much solution should he mix so that the amount of oil in the new mixture is 9 litres?
Q6. If n is a positive integer, then n^5 + 5 and (n+1)^5 + 5 are relatively prime.
a. Always true b. True for specific cases c. Always false d.Cant be said
Q7. In an examination, the average marks obtained by students who passed was x%, while the average of those who failed was y%. The average marks of all students taking the exam was z%. Find in terms of x, y and z, the percentage of students taking the exam who failed.?
Q8. Two-fifths of the voters promise to vote for P and the rest promise to vote for Q. Of these, on the last day, 15% of the voters went back on their promise to vote for P and 25% of voters went back of their promise to vote for Q, and P lost by 2 votes. Then the total number of voters is:
(1)100 (2)110 (3) 90 (4) 95
Q9. In the game of picking the parcel, 4 players stand at the corners of a square and a parcel is kept at the center of the square. As soon as the signal goes up, a player has to run and pick up the parcel and proceed towards the diagonally opposite corner. The parcel changes hand, and the third player now runs with the parcel and taking a quarter-circular pat h lands at the spot vacated by the 1st player. He then places the parcel at the center and second player takes a quarter-circular path and passes the parcel to the player on his right returns of his spot. If the distance between opposite corners is 14 m, what is the total distance traveled by the parcel?
(a)36m (b)44m (c)45m (d)47m
Q10. When (629)^{24 }is divided by 21, find the remainder.
(1)1 (2)2 (3)5 (4)11
Q11. Which of the following is the highest?
(1)12^{2} +9^{2} (2) 13^{2} +8^{2} (3) 14^{2} +7^{2} (4) 15^{2} +6^{2}
Q12. What is the largest integer x so that x^{3} divides 40,000?
(a) 10 (b) 20 (c) 25 (d) 26 (e) None of the above
Q13. In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many students solved only problem B?
Q14. In the game G(m; n), for given integers m; n with 2 ≤ m≤ n, players A and B alternately subtract any positive integer less than m from a running score which starts at n. Player A starts, and the winner is the player who brings the score to zero.
For given m; n there is always one player who can force a win. Find who, and explain how.
Q15. This past weekend I installed a new bi-fold door on a closet in the attic. The door consists of two panels, each one foot wide, hinged together in the middle. One side of one panel is fixed to the wall, while the other side of the other panel runs along track along the top of the door. As the door opens and closes, it rubs against a carpet on the attic floor, tracing out a path on the carpet pile.
What is the equation of the path traced out by the door on the carpet?
END

## QBM028

Q1. We will call a number that consists of the same sequence of digits repeated three times, such as 555 or 705570557055 a triplestring number. Find the smallest triplestring number which is a square.
Q2. Find an arrangement of eight points in the plane so that the perpendicular bisector of the line segment joining any two of the points passes through exactly two of the points.
Q3. Traveling at 90 km/hr, I reach my destination in 5 hours. If my speed becomes 72 km/hr, what is the time taken? Ans 6.25
Q4. Using positive chips, P, and negative chips, N, model the process and solution for finding the difference between - 4 and +2
Q5. The surface area of the three coterminous faces of a cuboid are 6, 15, 10 sq.cm respectively . Find the volume of the cuboid.
a.30
b.20
c.40
d. 35
Q6. A and B together can do a piece of work in 6 days. A alone can do it in 10 days. What time will B require to do it alone?
a)20 days b)15 days c)25 days d)30 days
Q7. The sum of the interior angles of a polygon is 16200. The number of sides of the polygon must be:
a. 9
b. 11
c. 15
d. 12
Q8. Find a ten digit integer, N, with the property that the one's digit of N is the number of 9's in N, the ten's digit is the number of 8's, and so on, with the leading digit being the number of 0's.
Q9. Amar transport corporation has five trucks each of which can carry 10 tonnes. The schedule of the trucks is such that the first truck leaves every day, thesecond leaves every alternate day; the third every third day & so on. Find out how many days in the year 2000 at least four trucks left on the same day (Assume that all the trucks left for the first time on the 1st of Jan 2000)
a.49
b.60
c.63
d.70
and here is one on alligation.
Q10.It took 1461 days to construct a building. The construction started on the first of January of 19AB and was completed by the 31st of December of 19CD. How many February days were there in the above period?
a.112
b.113
c.114
d.115
Q11. The sides of a rhombus ABCD measure 2 cm each and the difference between two angles is 90° then the area of the rhombus is:
a. √2 sq cm
b. 2 √ 2 sq cm
c. 3√ 2 sq cm
d. 4 √2 sq cm
Q12. Two positive integers differ by 4 and the sum of their reciprocals is 10/21. One of the numbers is:
a. 3
b. 1
c. 5
d. 21
Q13.Rajdhani express leaves Mumbai towards Delhi at 3.10 p. m. and travels
uniformally at 120 kmph. August Kranti Express leaves Delhi towards Mumbai at
12.20 p. m. and travels uniformally at 80 kmph. Both trains cross at Baroda at
4.30 p. m. On a particular day, Rajdhani leaves at 3.20 p. m. When will the two
trains cross?
a. 4.32 p. m.
b. 4.36 p. m.
c. 4.28 p. m.
d. 4.40 p. m
Q14. a, b and c are the sides of a triangle. Equations ax^{2} + bx + c = 0 and 3x^{2} + 4x + 5 = 0 have a common root. Then angle C is equal to
a. 60
b. 90
c. 120
d. None of these
Q15. How many different necklaces can be made by stringing 4 red beads and 2 blue beads together?
a. 3
b. 4
c. 5
d. 6
Q16. You are given that ALL primes that are one more than a multiple of 4 can be written as the sum of two squares. For example, 13 = 22+32.
Assuming that a prime is expressible as the sum of two squares , then in how many ways this can be done ?
a. One way for all the numbers
b. More than one way for some numbers
c. Both a and b
d. Cant be determined.
END

## QBM030

Q1. What is the value of m which satisfies 3m^{2} - 21m + 30 < 0?
(1) m < 2, or m > 5 (2)m > 2 (3) 2 < m < 5 (4) m < 5
Q2. When 3/4th of a unitâ€™s digit is added to the tenâ€™s digit of a 2-digit number, the sum of the digits becomes 10. If 1/4th of the tenâ€™s digit added to the unit digit, then the sum of the digits is 1 less than previous. Find the number
(a)94 (b)84 (c)48 (d)88
Q3. How many triangles are possible which satisfies the following conditions?
A. The side length of the triangles are consecutive integers
B. One of the angle of the triangle is twice as large as another.
a. 1 b. 2 c. More than 2 d. Such triangle is not possible
Q4. What is the smallest positive integer with the property that if the digit on the exteme right (i.e. the one's digit) is moved to the extreme left, the resulting number is one-and-a-half times the original number.
Q5. Determine the positive numbers are such that (3+âˆša)^{1/3} + (3-âˆša)^{1/3} is an integer.
Q6. Find the smallest integer N such that N and N3 together contain all 10 digits from 0 through 9 the same number of times.
Q7. Find the value of the infinite product
Q8. The sequence {1, 3, 2} has the property that the average of the first two entries is an integer, as is the average of the first three entries.
Is there a permutation of the positive integers, a1, a2, a3, . . . , such that the average of each initial segment is an integer?
Q9. If a, b and c are three real numbers, then which of the following is NOT true?
(A)|a+b| â‰¤ |a| + |b|, (B)|a-b| â‰¤ |a| + |b|, (C)|a-b| â‰¤ |a| - |b|, (D)|aâ€“c| â‰¤ |aâ€“b| + |bâ€“c|
Q10. The sides of a triangle are 5, 12 and 13 units respectively. A rectangle is constructed which is equal in area to the triangle and has a width of 10 units. Then the perimeter of the rectangle is
(1)30 (2)26 (3)13 (4)None of these
Q11. If x > 8 and y > â€“ 4, then which one of the following is always true?
(A) xy < 0 (B) x^{2} < â€“y (C)â€“x<2y (D) x>y
Q11. Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together. What length of time will elapse before they chime together again?
(1)2 hours 24 minutes (2) 4 hours 48 minutes (3) 1 hour 36 minutes (4) 5 hours
Q12. Sum of all prime numbers less than 50 is
1) more than 500 (2) less than 200 3) a prime number (4) an even number
Q13. N = 10x +y, where x and y are single-digit natural numbers. To find N, which of the following informationâ€™s is/are necessary/sufficient?
Y is a multiple of 3 and x is a multiple of 2 N is a prime Number.
N is a perfect square.
(1) only A and B together (2) only B and C together (3) only A and C together (4) All even together are not sufficient
Q14. If f(x âˆ’ 2) = x^{2} + 1, which of the following is equal to f(x + 1)?
(a) f(x) = x^{2} + 7x + 6 (b) f(x) = x^{2}+ 7x + 8 (c) f(x) = x^{2} + 5x + 9 (d) f(x) = x^{2} + 6x + 10
(e) None of the above
Q15. There are 32 computers in a network numbered with 5 bit-integers 0000, 0001, ..... 11111. There is a one-way connection from computer A to computer B, if and only if A and B share four of their bits with the remaining bit being 0 at A and 1 at B. Which of the following computers CANNOT be sent a message from the computer 10101
(a)11101 b)10111 (c)10001 (d)All of these can be sent
END

## QBM031

Q1. Let S_{n} = 1 + 1/2 + 1/3 + _ _ _+ 1/n (n = 1; 2; _ _ _). then S_{n} âˆ’ S_{m} is

(i)Always a prime number for m(ii)Always an odd number for m (iii) Always an even number for m(iv)never an integer for m2 - 10x - 22 ?
Q4. You may recall that a real number is called rational if it can be written as a fraction with integers in both the numerator and denominator, otherwise it's called irrational . There are many irrational real numbers; for example, if n is any positive integer which is not a perfect square, then is irrational. Find two irrational numbers, a and b, so that ab is an integer.
Q5. Find the smallest integer N such that N and N2 together contain each digit its own number of times, that is, 0 zeros, 1 one, 2 twos, ..., 9 nines.
Q6. Two players take turns choosing one number at a time (without replacement) from the set {-4, -3, -2, -1, 0, 1, 2, 3, 4}. The first player to obtain three numbers (out of three, four, or five) which sum to 0 wins. Does either player have a forced win?
Q7. Determine all the square integers whose decimal representations end in 2001. What is the smallest of these numbers?
Q8. In an apartment building there are seven elevators, each stopping at no more than six floors. If it is possible to go from any one floor to any other floor without changing elevators, what is the maximum number of floors in the building?
Q9. How many numbers between 1 to 1000 (both excluded) are both squares and cubes?

(A)none

(B)1

(C)2

(D)3
Q10. 5^{6} - 1 is divisible by

(1)13

(2)31

(3)5

(4) None of these
Q11. Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M3 to get N3.
Q12. Is there a set S of positive integers such that a number is in S if and only if it is a sum of two distinct members of S or a sum of two distinct positive integers not in S?
Q13. At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?
Q14. 1/m + 1/n = 1/94, where m and n are positive integers. Find m + n, given that m is half of n.

(1)49

(2)282

(3)423

(4)927
Q15. The units digit of 2006^{ 29} + 97^{2006} is:

a.2

b.4

c.6

d.8

e.0
END

## QBM032

Q1. The digits of 53, 125, can be rearranged to form 83, 512. Find the smallest cube whose digits can be rearranged to form 2 other cubes.
Q2. The sides of a rectangular field are in the ratio 3 : 4 with its area as 7500 sq. m. The cost of fencing the field @ 25 paise per meter is:

a)Rs. 87.50

b)Rs. 86.50

c)Rs. 67.50

d) Rs. 55.50
Q3.There is a work to be done by three friends A,B,C. Three of them together take 5 hours less than A alone would have taken,one-third that B alone would have taken and two-ninths the time C alone would have taken. How long does the three of them take to finish the work?

(a)3 hrs

(b) 4 hrs

(c) 5 hrs

(d) None of these
Q4. A car is at rest at point A. The speed of the car increases by
5 m/min at the end of every minute. How long does it take to reach the point B which is at a distance of 50 m from A?

1) 4 min

2) 5 min

3) 6 min

4) 4Â½ min
Q5.An equilateral triangle is formed by joining the middle points of the sides of a given equilateral triangle. A third equilateral triangle is formed inside the second equilateral triangle in the same way. If the process continues indefinitely, then the sum of areas of all such triangles when the side of the first triangle is 16 cm is:
Q6. Which one of the following cannot be the ratio of angles in a right-angled triangle?

(1) 1: 2 : 3

(2) 1 : 1 : 3

(3) 1 : 3 : 6

(4) None of these
Q7.Two teams participating in a competition had to take a test in a given time. Team B chose the easier test with 300 questions, and team A the difficult test with 10% less questions. Team A completed the test 3 hours before schedule while team B completed it 6 hours before schedule. If team B answered 7 questions more than team A per hour, how many questions did team A answer per hour?

(a)15

(b)18

(c)21

(d) 24
Q8. The length of the sides of a triangle are x + 1, 9 â€“ x and 5x â€“ 3. The number of values of x for which the triangle is isosceles is:

(A) 0

(B)1

(C)2

(D)3
Q9. A, B, C and D are four towns, any three of which are non -collinear. The number of ways to construct three roads each joining a pair of towns so that the roads do not form a triangle is:

(1)7

(2) 8

(3) 9

(4) more than 9
Q10. Sum of all prime numbers less than 50 is

1) more than 500

(2) less than 200

3) a prime number

(4) an even number
Q11. A worm crawls up a 53 foot pole 3 feet each day, but slips back 2 feet each night. After how many days will he reach the top?

(a) 51

(b) 52

(c) 53

(d) 54

(e) None of the above
Q12. Which of the following statements is true?

(1) 8^{7} â€“ 8 is divisible by 7

(2) 9^{10} â€“9 is divisible by 10

(3) (10)^{11} â€“10 is divisible by 10

(4) None of these
Q13. N = aebfcg is a six- digit number where a, b, c, e, f, g are six digit;
If a = e, b = f and c = = g then which of the given informations is not correct?
(1)If g = 4 then N is divisible by 44.

(2) If a + b + c = 6 then N is divisible by 33.

(3) If g= 8, then for different values of a, b, c and e, N may be a perfect square

(4) If b = c = 0, then N is not a perfect square.
Q14. For the product n(n + 1) (2n + 1), n E N, which one of the following is necessarily false?

(1) It is always even

(2) Divisible by 3.

(3) Always divisible by the sum of the square of first n natural numbers

(4) Never divisible by 237.
Q15. If x + y + z = 1 and x, y, z are positive real numbers, then the least value of (1/x â€“ 1)(1/y â€“ 1)(1/z â€“1 )is:

(A)4

(B)8

(C)16

(D) None of the above
END

## QBM033

Q1. I have a d-digit positive integer where each digit from 1 to d appears exactly once, and in which each digit (except for the leftmost) differs from one further to the left by +1 or -1.
For example, if d = 3, the only numbers satisfying the conditions are, in increasing order, 123, 213, 231 and 321.
Find a formula in terms of d for the number of integers satisfying these conditions.
Q2. Determine the smallest integer that is a square and whose decimal representation starts with 2005.
Q3. A 3x4 rectangle has sides 3,4,3,4 and diagonals 5,5. Its "weight" is said to be 3+4+3+4+5+5 = 24. Find a quadrilateral with all sides and diagonals having integer length and whose weight is smaller than 24.
Q4. Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that DBC = 60°. Point E is on side AB such that ECB = 50°. Find, with proof, the measure of EDB.
Q5. It's given that the number Z = n^{4} + a is not prime for any natural number n. Then which of the following is true about a
i. There is no such value for a

ii. There is only value for a

iii. Infinity many natural numbers are possible.

iv.None of the above.
Directions Q. 6 to 7: are based on the following information:
There are three different cable channels namely Ahead, Luck and Bang. In a survey it was found that 85% of viewers respond to Bang, 20% to Luck, and 30% to Ahead. 20% of viewers respond to exactly two channels and 5% to none.
Q6. What percentage of the viewers responded to all three?
(1) 10

(2) 12

(3) 14

(4) None of these
Q7. Assuming 20% respond to Ahead and Bang, and 16% respond to Bang and Luck, what is the percentage of viewers who watch only Luck?
(1) 20

(2) 10

(3) 16

(4) None of these
Q8. Pipe A can fill a tank in "a" hours. On account of a leak at the bottom of the tank it takes thrice as long to fill the tank. How long will the leak at the bottom of the tank take to empty a full tank, when pipe A is kept closed?
A. (3/2) a hours

B. (2/3) a hours

C. (4/3) a hours

D.(3/4) a hours
Q9. The sum of the reciprocals of two real numbers is âˆ’1, and the sum of their cubes is 4. What are the numbers ?
Q10. In a zoo, there are rabbits and pigeons. If their heads are counted, these are 90 while their legs are 224. Find the number of pigeons in the zoo.
a) 70

b) 68

c) 72

d) 22
Q11. LCM of two distinct natural numbers is 211. What is their HCF?
a) 37

b) 211

c) 1

d) data insufficient
Q12. a six digit number is formed by writing 3 consecutive two digit number
side by side in ascending order. If the number so formed is divisible by
2,3,4,5,6,8, then what is the hundreds digit of the number?
Q13. If a^{2} = b^{2}, then a = b is
1) always true

2) sometimes true

3) always false

4) none of these
Q14. A man buys shares at a discount of Rs. X. Later he sells all but 10 of the shares he purchased at a premium of Rs. X. If his investment was Rs. 4500 and proceeds from the sale were Rs. 6250, how many shares did he buy originally? [Assume face value of shares as Rs. 100]
(a)50

(b)40

(c)60

(d)90
Q15. Dhruv claims that he in 1994, his age is equal to the sum of the digits of his year of birth. What is the sum of the digits of Dhruva's age in 1994?
1) 3

2) 5

3) 7

4) 17
Q16. ABCD is a square whose side is 2 cm each; taking AB and AD as axes, the equation of the circle circumscribing the square is:
(A) x2 + y2 = (x + y)

(B) x2 + y2 = 2(x + y)

(C) x2 + y2 = 4

(D) x2 + y2 = 16

END

## QBM034

Q1. The remainder obtained when a prime number greater than 6 is divided by 6 is: (1) 1 or 3

(2) 1 or 5

(3) 3 or 5

(4) 4 or 5
Q2. N = (A7A)17 is a perfect square. Which of the following statement is FALSE? (1) A is an even digit.

(2) A is divisible by 3.

(3) When N is divided by 13 we get remainder 3.

(4) None of these
Q3. ABC is a three-digit number in which A, B and C are three different prime digits. The number formed by the first two digits, i.e. AB and the numbers formed by the last two, i.e. BC, are also prime numbers. Find the sum of the digits of the number.
(1) 12

(2) 15

(3) 21 or 15

(4) None of these
Q4. A group of people buy larks together (on a lark.) If each person paid 9 Franc, there would be 11 Franc left over after the purchase. If, however, each person contributed only 6 Franc, there would be a shortfall of 16 Franc. How many people are in the group?
(a) 10

(b) 11

(c) 12

(d) 13 (e) None of the above
Q5. A 7 foot ladder is leaning against a wall and just resting on a box which is 1 foot on a side. See figure. Precisely how high up the wall is the ladder sitting, assuming it is touching higher up the wall than it is out from the wall? The answer must be exact, no approximations!
Q6.How may number of squares with total area less than or equal to 1/2 can be packed into a square S of area 1 (in such a way that any point belonging to two of the packed squares is on the boundary of both).
a. 1

b.2

c.Any

d. None
Q7. Suppose a type of glass is such that:
70% of light shining from one side is transmitted through to the other.

20% of the light is reflected.

The remaining 10% is absorbed in the glass.

How much of an original light source will be transmitted through 3 panes of glass?
Q8. rhombus, ABCD, has sides of length 10. A circle with center A passes through C (the opposite vertex.) Likewise, a circle with center B passes through D. If the two circles are tangent to each other, what is the area of the rhombus?
Q9. In a race of 200 meters, A beats S by 20 meters and N by 40 metres. If S and N are running a race of 100 metres with exactly the same speed as before, then by how many metres will S beat N?
(1)11.11 metres

(2) 10 metres

(3) 12 metres

(4) 25 metres
Q10. A mother purchase three shirts of the same color but of different size for her three sons. All the three shirts were kept in a box in a dark room. The three boys took one shirt each at random from the box. What is the probability that none of the boys this own shirt?
(1)1/2

(2)1/3

(3)2/3

(4)1/4
Q11. A square flat piece of metal was used to construct a 2-inch high open box. This was accomplished by cutting 2-inch by 2-inch squares from each corner and folding up the sides. The resulting box holds 18 cubic inches. How wide was the original square?
(a)6 inches

(b) 7 inches

(c) 8 inches

(d) 9 inches

(e) None of the above
Q12. Predict the next number in the sequence 1, 2, 4, 7, 11, 16, 22, 29, 37, 46,
a.55

b.56

c.57

d.58

e.59
Q13. A man can do a piece of work in 60 hours. If he takes his son with him and both work together then the work is finished in 40 hours. How long will the son take to do the same job, if he worked alone on the job?
A.20 hours

B.120 hours

C.24 hours

D.None of these
Q14.What is the ratio of the sides of the inner and outer square, where the boundaries of the inner square are lines drawn from the vertices of the square to the midpoint of the opposite side.
1) 1:2

2) 1:3

3) 1: √(5)

4) None of these
Q15. Given that x is a positive integer prove that f(x) = x^{2} + x + 1 will never divide by 5.
Q16. If Ram and Shyam can do a piece of work in 15 days, Shyam and Sohan in 20, and Sohan and Ram in 30 days. Ram, Shyam and Sohan can individually do the work respectively in how many days?
a) 12, 10, 13

b) 60, 20, 45

c) 40, 24, 120

d) 10, 12, 15
END

## QBM035

Q1. A person starts from the origin of the coordinate axis. He travels in
this pattern. 1 unit to right , (1/2) units up , (1/4) units right, (1/8)
units down , and continues the above pattern . At what point will he
ultimately come to stop?
(a)(4/3,2/5)

(b)(3/2,4/3)

(c)(2/5,4/3)

(d)(4/5,4/3)
Q2. Each meal costs Rs. 9 in a hotel. If one takes a monthly coupon (1 meal a day and 30 days a month), the total cost is Rs. 225. The discount per meal is
1)16 2/3%

2)Re 1

3) Rs 45

4) 15%
Q3. Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together. What length of time will elapse before they chime together again?
(1) 2 hours 24 minutes

(2) 4 hours 48 minutes

(3) 1 hour 36 minutes

(4) 5 hours
Q4. A mathematics professor and four mathematics students are in a square room, 10 feet on a side. The four students are stationed at the room's four corners, each student armed with a water pistol having a range of 10 feet. What is the area of that portion of the room in which the professor is simultaneously in range of all four water pistols?
Q5. Several years ago, three suitors from three countries were vying for the hand of a lovely maiden. They agree to fight a pistol duel under the following conditions:
Q6. Let Hn = 1/1 + 1/2 + ... + 1/n.
Show that, for n > 1, H_{n}is not an integer.
Q7. When Darby remarked that the apples seemed very small this week, the seller offered to throw an extra apple into the dollar basket. Darby noted that this reduced the price per dozen by 5 cents and she then purchased the basket. How many apples did she get for her dollar?
(a)15

(b)16

(c)17

(d)18

(e) None of the above
Q8. What units digits exist in the product of all prime numbers between 10 and 30?
(1)5

(2)4

(3)3

(4)2
Q9. A tree was planted when it was 4 feet tall and thereafter grew an equal number of feet each year. At the end of 6 years it was twice as tall as it had been at the end of 2 years. How tall was the tree at the end of 4 years?
(a)12 feet

(b)13 feet

(c)14 feet

(d)15 feet

(e) None of the above
Q10. The captain of a luxury cruise ship wants to install windshield wipers on the portholes of the ship. The straight wipers are to be attached at a point on the circumference of the circular portholes, furthermore, the entire length of the blade must remain in contact with the flat glass at all times. How long should the wiper blade be so as to clean half of the porthole?
Q11. One may perform the following two operations on a natural number:

1. Multiply it by any natural number;

2. Delete zeros in its decimal representation.

For any natural number n, can one perform a sequence of these operations that will transform n to a one-digit number.
Q12. In an attempt to copy down from the board a sequence of six positive integers in Arithmetic Progression, a student wrote down the five numbers,
113, 137, 149, 155, 173
accidentally omitting one. He later discovered that he also miscopied one of them. Which number was miscopied?
1) 137

2) 149

3) 155

4) 173
Q13. How many whole numbers between 200 and 700 begin and/or end with 3? a. 150

b.160

c.140

d.170

e.None of these
Q14. Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?
A. 18 days

B. 36 days

C. 54 days

D. None of these
Q15.It is between 2 and 3 o' clock, and in 10 minutes the minute hand will be as much after the hour hand as it is behind it now, what is the time (in minutes, post 2 o'clock)?
END

## QBM036

Q1. The sum of a two-digit number and its reverse is equal to 99. How many such two-digit numbers are there?
a)6

b) 9

c) 8

d) none of these
Q2. The rate of inflation was 1000%. What will be the cost of an article, which costs 6 units of currency now, two years from now?
(1) 666

(2)660

(3)720

(4) 726
Q3. Is x/y prime?
I. x is divisible by 3 but not by 9
II. y is a multiple of 6
**Direction for Q4 and Q5**
A cube is divided into 4 equal cubes. Each of these cubes is further sub-divided into 4 equal cubes.
Q4. What is the ratio of the surface area of the smallest cube as a percentage of the original cube?
(a)0.625
(b)0.0625
(c)0.0156
(d) 0.25
Q5. The original cube’s sides are painted blue, then what is the probability that exactly 2 sides of a small cube is painted blue?
Q6. 30 playing cards of length 12 cm and 6 cm are used to make a pyramid with 4
cards in the base. Find the area covered by the front side of the pyramid.
(a) 288^{3} cm^{2}

(b)180^{3} cm ^{2}

(c) 360^{3} cm ^{2}

(d) 576^{3}cm ^{2}
Q7. How much does it cost @ Rs. 2/ft^{2} to paint the roof and the walls of a 10 ft x 8 ft x 10 ft (length x width x height) room?
1) Rs. 440

2) Rs. 880

3) Rs. 720

4) Rs. 540
Q8. Given that 'a' and 'b' are positive integers and 'a' is not equal to 'b'
and (4^{a}+1)(4^{b}+1)=3^{c}+1
find the value of a^{b}+b^{a}
(a)-1

(b) 0

(c) 1

(d) indeterminate
Q9. Shim can do a job in 20 days, Ram in 30 days and Singh in 60 days. If Ram and Singh help Shim every 3rd day, how long will it take for them to complete the job?
A. 12 days

B.16 days

C.15 days

D.10 days
Q10. In a class of 40 students, 15% are absent today. How many students are in class today ?
a.30

b.31

c.32

d.33

e.34
Q11. The number of right-angled triangles that can be formed by using any three of the six points (0,0), (0,2), (0,4), (2,0), (4,0), (2,2) as vertices is:
a.9

b.10

c.11

d.12

e. None of these
Q12. Find the area of the triangle whose vertices are (-6, -2), (-4, -6), (-2, 5).
A.36

B.18

C.15

D.30
Q13. The pressure of wind on a plane surface varies as the square of the velocity of the (in kmph). The pressure on a square cm is 2 2/9 gm when the wind is moving at a rate of 24 kmph. Find the velocity of the wind when the pressure on a square metre is 18 kg.
a) 32.4

b) 30.1

c) 21.6

d) 26.5
Q14. How many zeroes will be there at the end of 36!^{36!}?
a) 8

b) 64

c) 8 x 36!

d) none of these
Q15. A set S consists of 12 distinct elements a_{1},a_{2},...,a_{12}.
How many subsets can be made of S with the restriction that the subscripts of elements in any subset starting from the second element is an integral multiple of the subscript of the first element.
eg: if the 1st element is a1, we could have the set as {a1, a2 , a3..} if the 1st element is a2, we could have the set as {a4 , a6..}
(a) 1221

(b) 2112

(c) 2101

(d) 2102
END

## QBM037

Q1.If A and B together can complete a task in 6 2/3 days, B and C together can complete it in 12 days, and C and A together can complete it in 7 1/2 days, then how many days will it take for A alone to complete half the task?
1) 2( 8/11) days

2) 5 days

3) 7 (8/11) days

4) 5 (8/11) days
Q2.In a race of 200 meters, A beats S by 20 meters and N by 40 metres. If S and N are running a race of 100 metres with exactly the same speed as before, then by how many metres will S beat N?
(1)11.11 metres

(2) 10 metres

(3)12 metres

(4) 25 metres
Q3.A rod of length 1 is broken into four pieces. What is the probability that the four pieces are the sides of a trapezoid?
Q4.Take a deck of 52 playing cards consecutively numbered from 1 to 52. A perfect shuffle occurs when you divide the deck in two, one half in each hand, and riffle them together in an alternating fashion; after the shuffle the cards in the deck will be ordered as follows:
1, 27, 2, 28, 3, 29, ... , 24, 50, 25, 51, 26, 52
Notice that the top card always stays on top and the bottom card always stay on the bottom. Imagine now that you repeatedly shuffle the cards, performing a perfect shuffle every time. How many times will you have to do this before all the cards in the deck are restored to their original position?
Q5. Is the number 2438100000001 prime or composite? No calculators or computers allowed!
Q6. A man has to invite six of his friends on his birthday for dinner. He can send invitations by post, by phone or by his servant. In how many ways can he send invitations to his friends so that each of his friends gets only one invitation?
(1)120

(2)720

(3)729

(4)216
Q7. On 1st January 1969, a person purchases Rs. 10,000, 4% debentures. On 1st Jan. 1970, he sells 3/4 of it at a discount of 6% and invests the proceeds in steel shares at Rs. 470 per share. He sells the remaining debentures at Rs. 105 and purchases bonds at a price of Rs. 75 per bond. Each bond pays Rs. 5. Each share pays Rs. 16. Find the alternation in his annual income in 1970?
(1)Rs. 15 decrease

(2)Rs. 25 increase

(3)Rs. 15 increase

(4) None
Q8. A 3 x 2 grid is placed so that its corners are at (0,0) and (3,2). A legal move is defined as a move either in the positive x or positive y direction. The darkened point has been removed from the grid. Now, how many possible paths are there from (0,0) to (3,1)?
1) 3

2) 6

3) 11

4) 7
Q9. How many four digit numbers exist which can be formed by using the digits 2, 3, 5 and 7 once only such that they are divisible by 25?
A.4! - 3!

B.4

C.8

D.6
Q10. The sum of A and B is 12 and their reciprocals add up to 3/8. Then A and B are:
a) 5, 7

b) 3, 9

c) 4, 8

d) 2, 10
Q11. An elastic ball is propped from a height of 20 m. If the height of fall is twice the height of bounce, the total distance travelled by the ball before it comes to rest is
1) 40 m

2)60 m

3)80 m

4) infinite
Q12. The largest value of min (2 + x^{2}, 6 - 3x) when x > 0 is
(1)1

(2)2

(3)3

(4) 4
Q13. When is the product of two consecutive natural numbers a non-trivial integer power; that is, for which natural numbers n, t, k is n(n+1) = tk, for natural numbers t and k > 1?
Q14. An absentminded professor buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the professor opens a matchbox and finds that it is empty. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained n matches, what is the probability that the other box currently contains k matches? (Where 0 k n.)
Q15. 2/3rd of the balls in a bag are black, the rest are red. If 5/9th of the black balls and 7/8th of the red balls are defective, find the total number of balls in the bag, if the number of non-defective balls is 146.
(a)216

(b)432

(c)648

(d)578
END

## QBM038

Q1. There are 20 boys and 10 girls in class. The average score of the class in an examination is 40. If the average score of the girls is 46, the average of the boys is
1)36

2)37

3)38.5

4)39
Q2. Vijay writes first hundred whole numbers. How many times does he write zero?
a)12

b)11

c)9

d)10
Q3. There is a five-volume dictionary among 50 books arranged on a shelf in random order. If the volumes are not necessarily kept side-by-side, the probability that they occur in increasing order from left to right is:
(A)1/5

(B)1/550

(C)1/50

(D) None of these
Q4. A spherical ball, when immersed in a cylinder of base radius 7cms. Raises the level of water in the cylinder by 2cms. Find the radius of the ball?
(a)3 (210/6)

(b)3 (147/2)

(c)94/7 (d) Cannot be determined
Q5. A monster prowls around the perimeter of a perfectly circular pond. You are sitting on a raft in the middle of the pond. The monster can move four times the speed at which you can swim, but if you can reach the shore in front of the monster, you can out run him (because, of course, monsters cannot see people who are standing on dry ground). Can you escape from your watery prison, and, if so, how?
Q6. Find all solutions to c^{2} + 1 = (a^{2}-1)(b^{2} - 1), in integers a, b, and c.
Q7.A man can buy 10 kg more rice when the price of reduces by 10%. If the price increases by 12.5%, how much less can he buy for Rs. 1800. What was the original price?
a. 15kg, Rs. 12

b. 10kg, Rs. 20

c. 9kg,Rs.20

d.12 kg, rs.15

e. insufficent data
Q8. There are some original documents and some photocopies in a bunch of five sheets. No original has less than one copy and no two originals have an equal number of copies. There is more than one original document in the bunch. The number of originals in the set is
1)4

2)3

3)1/4

4) none of these
Q9.A cistern can be filled separately by two pipes A and B in 40 minutes and 30 minutes resp. A tap C at the bottom can empty the full cistern in 60 mins. Taps A and B are opened at 3.10pm and tap Cis opened 10 minutes later. find when the cistern will b full.
a. 4.00 pm

b. 3.40pm

c. 3.20pm

d. 3.30pm

e.none of these
Q10. Three labeled boxes containing black and white cricket balls are all mislabeled. It is known that one of the boxes contains only white balls and other only black balls. The third contains a mixture of black and white balls. You are required to correctly label the boxes with the labels black, white and white by picking a sample of 1 ball from only 1 box. What is the label on the box you should sample?
1]White

2] Black

3] Black and white

4] Not possible determine from a sample of 1 ball
Q11. Two gamblers take turns rolling a fair N-sided die, with N at least 5. The faces of the die are labeled with the numbers 1 to N. The first player starts. If he rolls an N or an N-1, he wins and the game is over. Otherwise, the other player rolls the die; if she rolls a 1, 2, or 3, she wins and the game is over. Play continues, with the players alternating rolls until one of them wins.
What is the probability that the first player will win? Are there any values of N for which the first player has at least an even chance of winning?
Q12. How many ways can 90316 be written as
a + 2 b + 4 c + 8 d + 16 e + 32 f where the coefficients can be any of 0, 1, or 2?
Q13. Consecutive fifth powers (or, indeed, any powers) of positive integers are always relatively prime. That is, for all n > 0, n5 and (n + 1)5 are relatively prime. Are n5 + 5 and (n + 1)5 + 5 always relatively prime? If not, for what values of n do they have a common factor, and what is that factor?
ANS: n ^{5} + 5 and (n + 1)^{5} + 5 are relatively prime only if n ≠ 533360 (mod 1968751).
If n ≠ 533360 (mod 1968751), their greatest common divisor is 1968751.
Q14. For the product n(n + 1) (2n + 1), n is an integer, which one of the following is necessarily false?
(1)It is always even

(2)Divisible by 3.

(3)Always divisible by the sum of the square of first n natural numbers

(4)Never divisible by 237.
Q15.
END

## QBM039

Q1. There are two vessels, one containing 1 litre of pure milk and the other containing 1 litre of water. If half a litre of the latter is transferred to the former and upon stirring the contents well, half a litre of the former is transferred to the latter, what is the ratio of milk to water in the latter?
1)4 : 5

2) 3 : 4

3) 2 : 3

4) 1 : 2
Q2. What is the remainder when (2222)^{5555}+ (5555)^{2222} is divided by 7 ?
a)3

b)5

c)6

d)0
Q3. P is a prime number greater than 5. What is the remainder when P is divided by 6?
a)5

b)1

c)1 or 5

d) none of these
Q4. Divide 60 into two parts, so that three times the greater may exceed 100 by as much as 8 times the less falls short of 200. What is the greater part?
a)40

b)36

c)32

d) 31
Q5. The average age of a family of 5 members is 20 years. If the age of the youngest member were 10 years then what was the average age of the family at the time of the birth of the youngest member?
A.13.5

B.14

C.15

D.12.5
Q6. The playoffs in Major League Baseball begin in just about a week, culminating in the World Series which selects the champion team. The World Series is a tournament of seven games with the champion being the first team to win four games. How many different ways can the World Series be played?
Q7.Boxes numbered 1, 2, 3, 4, and 5 are kept in a row, and they are to be filled with either a red or a blue ball, such that no two adjacent boxes can be filled with blue balls. How Many different arrangements are possible, given that all balls of a given color are exactly identical in all respects?
(1)8

(2)10

(3)15

(4)22
Q8. A priest and a pirate are shipwrecked on an island with a boat which is large enough to carry only one of them to the mainland. They decide to let Lady Luck decide which one of them will be rescued. On the island they find two coins. Assuming that at least one of the coins is fair -- that is, gives heads or tails 50% of the time -- but not knowing which one that may be, how can they use the coins to decide fairly who gets off the island?
Q9.If you'd like me to notify you by email when the Problem of the Week resumes in the fall, please drop me a line, including your email address. I'll send you a note in late August or early September.
Q10. What is interesting about the following sequence?
1/89, 1/9899, 1/998999, 1/99989999,
Q11. Roll a standard pair of six-sided dice, and note the sum. There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12. Find all other pairs of six-sided dice such that:
a. The set of dots on each die is not the standard {1,2,3,4,5,6}.
b. Each face has at least one dot.
c. The number of ways of obtaining each sum is the same as for the standard dice.
Q12. If 0 < x > 9, y>10, 94, 3B. abc < xyz

C. (x + y + z) > (a + b + c)
(1) All of A, B, and C

(2) only B

(3) only C

(4) only Band C
Q13. A bag contains 40 jellybeans, of which 10 are red, 10 are black, 10 are green, and 10 are yellow. The least number that a blindfolded person must eat to be certain of having eaten at least one of each color is:
a.31

b.22

c.4

d.5

e. None of these
Q14. If the average of three numbers a, b, and c is A. What is the average of the four numbers a, b, c and A?
A. A/4

B. A/2

C. 2A

D. None of these
END

## QBM040

Q1. A field in the shape of a right triangle is to be subdivided into two fields of equal size by building a straight fence between two sides of the field. If the lengths of the sides of the field are 300 and 400 feet long, respectively, with the hypotenuse being 500 feet long, what is the shortest fence that will do the job and where should the fence be built?
Q2. A straight bar of metal, initially 800 feet long, expands 8 inches in length. The ends are fixed so that the bar becomes distorted into the shape of an arc of a circle, for which the original bar would now be a chord. What is the approximate height above the center of the original bar of this new distorted bar?
Q3. Which of the following is NOT true?
(1)|a + b| = | b + a|

(2) |a - b| = |b - a|

(3) |a + b | < |a| +|b|

(4) | a - b | > |a| - |b|
Q4. The ratio of two numbers is 7:3. The sum of these two numbers is 30. What is the difference of these two numbers?
a.9

b.10

c.11

d.12

e.13
Q5. The average monthly salary of 12 workers and 3 managers in a factory was Rs. 600. When one of the manager whose salary was Rs. 720, was replaced with a new manager, then the average salary of the team went down to 580. What is the salary of the new manager?
A.570

B.420

C.690

D.640
Q6. The sides of a triangle are 6 cm, 11 cm and 15 cm. The radius of its incircle is:
a) 5√2/4 cm

b) 3√2 cm

c) 6√2 cm

d) 4√2/5 cm
Q7. Three consecutive positive even numbers are such that thrice the first number exceeds double the third by 2; then the third number is:
(1)10

(2)14

(3)16

(4)12
Q8. If in the previous question, Akshay's speed from home to bank was 30 m/min and from bank to market was 20 m/ 10 min, find his average speed.
(a)2.8 m/10 min

(b)3 m/min

(c)3.6 m/10 min

(d) 4.8 m/min
Q9. A rectangular block of wood has for its three dimensions a different odd prime number of inches. Its volume and (total) surface area are, respectively, a three digit number and a four digit number. What are the dimensions of the block?
Q10. Fix a positive integer n. How many sequences (a1, a2, ... , an) of positive integers are there with the property that at most i of the terms are greater than n-i, for all i = 0,1,...,n?
Q11. Show that n^{4} + 4^{n} is composite for all integers n > 1.
Q12. The side of equilateral triangle is 96 cm. the mid points of its side are joined to form another triangle whose mid points are turn joined to form still another triangle. This Process is repeated indefinitely. Find the sum of the perimeter of all the triangles.
(1)1176 cm

(2)1275 cm

(3)576 cm

(4) None of these
Q13. The largest value of min (2 + x^{2}, 6 - 3x) when x > 0 is
(1)1

(2)2

(3)3

(4)4
Q14. Let x, y and z be natural numbers satisfying x < y < z and x + y + z = k. which of the following is the smallest values of k which does not determine x, y, z uniquely?
(1)9

(2)6

(3)7

(4) 8
Q15. Two ferry boats sail back and forth across a river, each traveling at a constant speed, and turning back without any loss of time. They leave opposite shores at the same instant, pass for the first time 700 feet from one shore, continue on their way to the banks, return and pass for the second time 400 feet from the opposite shore. What is the width of the river?
END

## QBM041

Q1. If x and y are positive real numbers, show that xy + yx > 1.
Q2. Given any sequence of n integers, show that there exists a consecutive subsequence the sum of whose elements is a multiple of n.
For example, in sequence {1,5,1,2} a consecutive subsequence with this property is the last three elements; in {1,âˆ’3,âˆ’7} it is simply the second element.
Q3. How many positive integers not greater than 600 are multiples of 2 but not 3?
a.300

b.250

c.200

d.150

e.None of these
Q4. 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
Directions 5-9: Of 1500 people surveyed, 800 read the Times of India, 700 read Indian Express and 550 read Hindustan Times. 300 read the Times of India and Indian Express, 400 read Indian Express and Hindustan Times and 350 read Hindustan Times and the Times of India. There were 250 people who read all the tree newspapers.
Q5. How many people read only the Times of India?
a)800

b)300

c)400

d)200
Q6. How many people read Indian Express but not the Times of India?
a)300

b)200

c)250

d)400
Q7. The number of people subscribing to Hindustan Times only is:
a)200

b)50

c)100

d)150
Q8.The number of people who do not subscribe to any of these three newspapers is:
a)250

b)100

c)150

d)200
Q9.If x = 777...777 ( 101 7 are there ) then what is x mod 440 ?
Q10. A and B starting from x and y simultaneously, heading for y, x respectively. The ratio of their speeds is 2:1. If each one does 6 return trips (if A return trip is x to y; y to x) how many times would they met each other?
(a) 3

b)4

(c)5

(d)6
Q11. The profit margin on readymade shirts is 20%. If there is a defect, they are disposed off at Rs. 100 per shirt. It costs Rs. 120 to make a shirt, and 5 shirts out of 100 are defective. The profit realised on 100 shirts coming out of the tailoring unit is
1) Rs. 2180

2) Rs. 2400

3) Rs. 2280

4) Rs. 4180
Q12. In a certain year, April has exactly four Tuesdays, and exactly 4 Saturdays. On what day could April 1 possibly fall that year?
a.Monday

b.Tuesday

c.Wednesday

d.Friday

e.Saturday
Q13. 5^{333} + 25^{222} - 625^{111} is same as
a)1

b)25

c)125

d)0
Q14. At a given time clock A shows 1.20.20 and clock B shows 1.20.40. With respect to clock C, which is running correctly, A is gaining 1 s/min and B is losing 1 s/min. After how much time will A and B show the same time?
1)10s

2)20s

3)10min

4)20 min
Q15. A person travels through 5 cities - A, B, C, D, E. Cities E is 2 km west of D. D is 3 km north-east of A. C is 5 km north of B and 4 km west of A. If this person visits these cities in the sequence B - C - A - E - D, what is the effective distance between cities B and D?
(1)13km

(2)9km

(3)10km

(4)11 km
Q16. A man has 10 pairs of shows in his cupboard. One morning he picks 4 shoes (one by one) at random. Find the probability that there is at least one pair complete so that he can go to office without making any further withdrawal from the cupboard.
(1)224/323

(2)99/323

(3)112/323

(4)None of these
END

## QBM042

Q1. Find the smallest natural number greater than 1 billion (109) that has exactly 1000 positive divisors. (The term divisor includes 1 and the number itself. So, for example, 9 has three positive divisors.)
Q2. Describe as explicitly as you can all cubic polynomials with integer coefficients having
(a)three distinct real roots,

(b) local maximum and minimum values at integers, and

(c) point of inflection at an integer.
Q3. In a 16-team basketball league, each team plays every other team exactly 4 times. Find the total number of games played in the league.
a.480

b.960

c.900

d.450

e. None of these
Q4. If (x + 2) 2 = 9 and (y + 3) 2 = 25, then the maximum value of x/y is.
A.1/2

B. 5/2

C.5/8

D.1/8
Q5. The rate of inflation was 1000%. What will be the cost of an article, which costs 6 units of currency now, two years from now?
(1)666

(2)660

(3)720

(4)726
Q6. You are being evaluated in three different categories. In category I you can receive either a 0, 1, or 2. In categories II and III you can receive either a 0, 1, 2, 3, or 4. Your ultimate rating is the sum of the points you receive in each category. In how many different ways could you end up with a rating of 5?
(a)10

(b)11

(c)12

(d)13

(e) None of the above
Q7. Of 100 students, 42 took mathematics, 38 took chemistry, and 20 took both
mathematics and chemistry. How many took neither mathematics nor chemistry?
a.20

b.25

c.30

d.35

e. None of these
Q8. What is the angle between the minute hand and the hour hand when the time is 1540 hours?
A.150

B.160

C.140

D.130
Q9. In a game, a basket is kept in the centre, with 16 potatoes placed on either side of the basket in a single line at equal intervals of 6 feet. How long will a competitor take to bring the potatoes one by one into the basket, if he starts from the basket and runs at an average speed of 12 feet a second?
a)144s

b)72s

c)84s

d) 272s
Q10. The income of A is twice that of B. A spends 90% of his income while B spends 80% of his income. What is the ratio of their savings?
1)2 : 1

2) 1 : 2

3) 1 : 1

4) 8 : 9
Q11. Question deleted
Q12. Take any two positive integers N and a. Show that Na is the sum of N consecutive odd integers.
As an easy example, note that
713 = 96889010407 = 13841287195 + 13841287197 + 13841287199 + 13841287201 + 13841287203 + 13841287205 + 13841287207
Most solvers noted that the result is not true if a = 1 and N is even, but is otherwise correct.
Since Na is the sum of N times the integer Na-1 it is also the sum of N integers whose average is Na-1. Take these integers to be the N odd integers centered at the odd integer Na-1 - N + 1.
Q13. Suppose n fair 6-sided dice are rolled simultaneously. What is the expected value of the score on the highest valued die?
Q14. Let a, m, and n be positive integers, with a > 1, and m odd.
What is the greatest common divisor of am âˆ’ 1 and an + 1?
Q15. A man invests Rs. 3000 at a rate of 5% per annum. How much more should he invest at a rate of 8%, so that he can earn a total of 6% per annum?
(1)Rs. 1200

(2)Rs.1300

(3)Rs.1500

(4)Rs. 2000
END

## QBM043

Q1. When 36 divide a number, it leaves a remainder of 19. What will be the remainder when 12 divide the number?
A.10

B.7

C.11

D.None of these
Q2.An isosceles triangle with equal angles = 75o is cut from a cardboard. How many such triangles need to be joined together to form a regular polygon?
1)10

2) 12

3)6

4) A regular polygon cannot be formed
Q3.Find a number such that if 5, 15 and 35 are added to it, the product of the first and third results may be equal to the square of the second.
a)10

b)7

c)6

d) 5
Q4.What is the ten's digit of 43^{6}?
a)4

b) 1

c) 9

d) none of these
Q5. A can do a piece of work in 36 days, B in 54 days and C in 72 days. All of them began together but A left 8 days and B left 12 days before the completion of the work. How many days in all did C put in till the entire work was finished?
(1)48 days

(2)24 days

(3)12 days

(4) None
Q6. The remainder obtained when a prime number greater than 6 is divided by 6 is: (1)1 or 3

(2) 1 or 5

(3) 3 or 5

(4) 4 or 5
Q7. The length of a ladder is exactly equal to the height of the wall it is resting against. If lower end of the ladder is kept on a stool of height 3 m and the stool is kept 9 m away from the wall, the upper end of the ladder coincides with the top of the wall. Then, the height of the wall is:
(1)12m

(2)15m

(3)18m

(4)11m
Q8-Q9
Pinky enters a shop to buy almonds, biscuits and chocolates. She has to buy at least 7 units of each. She buys more biscuits than she does almonds and more chocolates than she does biscuits. She picks up a total of 26 items.
Q8. How many almonds does she buy?
(1)7

(2)8

(3)9

(4) Cannot be determined
Q9.Which of the following is not a valid value for number of chocolates bought?
(1)9 (2)10 (3)11 (4) All are valid
Q10. Four bells begin to toll together and then at intervals of 6, 7, 8 and 9 seconds. The number of times they will toll together in 2 hours and the intervals at which they will are
(a) 14 times, 504 sec

(b) 15 times, 600 sec

(c) 13 times, 650 sec

(d) 11 times, 720 sec
Q11. Ram purchased a flat at Rs. 1 lakh and Prem purchased a plot of land worth Rs. 1.1 lakh. The respective annual rates at which the prices of the flat and the plot increased were 10% and 5%. After two years they exchanged their belongings and one paid the other the difference. Then:
(1) Ram paid Rs. 275 to Prem

(2) Ram paid Rs. 475 to Prem

(3) Ram paid Rs. 2750 to Prem

(4) Prem paid Rs. 475 to Ram
Q12. Mr. Akshay stays in a triangular colony. He stays at one vertex, with a bank and a market at the other two vertices. The roads meeting at the market are at 90. Akshay goes to the bank from his home and then to the market totally covering 160 m. He then walks straight back home from the market. If distance from home to bank is 40 m more then what he covered from bank to market, find the distance he walked from market to home.
(a) 80 m

(b) 120 m

(c) 100 m

(d) 155 m
ANS:
Q13. If you walk for 30 minutes at a rate of 4 miles per hour and then run for 15 minutes at a rate of 10 miles per hour, how many miles have you gone at the end of 45 minutes?
a. 5 miles

b.6 miles

c. 5.5 miles

d. 4.5 miles

e. None of these
Q14. How many zeros contained in 100!?
A. 100

B.24

C.97

D.Cannot be determined
Q15. Divide 183 into 3 parts in GP such that the sum of first and third terms is 2 1/20 times the second. What is the middle term?
a)84

b)75

c)60

d) 48

## QBM044

Q1. How many positive integers divide at least two of the following integers: 410, 620, 320?
(1)39

(2)40

(3)41

(4)None of these
Q2. A bucket with a capacity of 5 liters is used to draw water from a cylindrical tank of radius 7 m in which and water is filled upto a height of 2 m. The density of the bucket is 0.5 and the weight 2 kg. What is the level of water in the tank when the bucket is dipped the third time to draw water? (Density =Volume/Weight)
(a)1.8m

(b)1.9m

(c)1.96m

(d)2m
Q3. The remainder obtained when a prime number greater than 6 is divided by 6 is:
(1)1or3

(2)1or5

(3)3or5

(4)4or5
Direction for questions 4 and 5: Read the data below
The climb from the foot to the top of a hill is 800 m. Amar climbs at 16 m/min and tests for 2 minutes or 20 m in 2 min and rest for 1 minute. Bonny can climb at 10 m/min and rest for 1 minute or 16 m/min and rest for 2 minutes.
Q4. If Amar has to reach the top in exactly 2 hours, what is the maximum number of tests that he can take?
(a)40

(b)41

(c)39

(d)38
Q5. If both Amar and Bonny climb as fast as they can, then how far would Bonny be from the top when Amar reaches the top?
(a)630m

(b)170m

(c)640m

(d)160m
Q6. A merchant purchases 25% more goods spending Rs 240 after the price of the
goods fall by 20%. What are the old and new prices of the goods?
(a) 22, 20

(b) 24, 19.2

(c) 10, 8

(d) 30, 24
Q7. A trader buys goods from Delhi to sell in Bombay where he gets 20% higher price realisation. Per journey, he spends Rs. 2000 on travelling, Rs. 2500 as bribe to the parcel authority and makes a net profit of Rs. 20,000. Find the total value of goods purchased by the dealer in 7 journeys to Delhi.
(1)Rs.1,22,500

(2)Rs.2,45,000

(3)Rs.8,57,500

(4) None
Q8. What is the shortest distance between M1 and M2?
(1)11km

(2)7ï€ 2km

(3)7km

(4)14 km
Q9.A man invests Rs. 3000 at a rate of 5% per annum. How much more should he invest at a rate of 8%, so that he can earn a total of 6% per annum?
(1)Rs.1200

(2)Rs.1300

(3)Rs.1500

(4)Rs.2000
Q10. This is the last problem of the fall semester. The Problem of the Week will return for the spring semester. Hope to hear from everyone then.
What is the probability that the sum of three integers, each between 1 and 999, yields no "carry"?
Q11.What is the greatest integer so that each of its interior digits is less than the average of its neighbors?
Q12.The length of a ladder is exactly equal to the height of the wall it is resting against. If lower end of the ladder is kept on a stool of height 3 m and the stool is kept 9 m away from the wall, the upper end of the ladder coincides with the top of the wall. Then, the height of the wall is:
(1)12m

(2)15m

(3)18m

(4)11 m
Q13. If A and B are nonzero digits, the number of digits (not necessarily different) in the sum of 9876 + A32 + B1 is:
a.4

b.5

c.6

d.9

e. None of these
Q14. The Sum of the internal angle of a n-sided convex polygon is An + B, where A and B are constants. What is the value of A/B?
(a)â€“2

(b)â€“1/2

(c)2

(d)1/2
Q15. A starts from patna to bilaspur and walks at the rate of 5 kmph; B starts from bilaspur 3 hrs. later and walks towards- patna at the rate of 4.5 kmph; if they meet in 11 hrs after B started. After how much time of A's meeting with B, will A reach bilaspur?
(1)8 hrs

(2)9 hrs

(3)10 hrs

(4)11 hrs
END