122important questions quant

2

14. All possible words are formed using all the letters in the word EQUATION exactly once and are arranged such that no two vowels and no two consonants are in alphabetical order. How may words are there?

(A) 56 (B) 140 (C) 720 (D) 4914

15. How many 5 - digit numbers, comprising only the digits 4 and 5 are divisible by 3?

(A) 10 (B) 11 (C) 12 (D) More than 12

16.

There is a rectangular grid consisting of unit cells as shown in the figure given below. One needs to travel from point A to point C. One can travel only to the right and upwards. In how many ways can one go from point A to point C without traveling via B? (A) 3915 (B) 3150 (C) 2520 (D) 6434

17. How many three-digit odd number can be formed such that if 2 is one of the digits, the following digit is 4?

(A) 5 (B) 365 (C) 300 (D) 372

18. A coaching institute was to send 9 parcels for its postal students, four of whom were from Bihar and the remaining five were from Delhi. The dispatch clerk made a mistake in addressing the parcels. He addressed them in such a way that no student received the correct parcel although all the parcels for Bihar were sent to Bihar and all the parcels for Delhi were sent to Delhi. Find the total number of ways in which he could have addressed those 9 parcels. (A) 396 (B) 2737 (C) 475 (D) 89496

19. Three, four and five points are selected on the sides AB, BC and CD respectively of ABC. None of the 12 points coincide with A, B and C. How many triangles can be formed by using these 12 points as vertices? (A) 220 (B) 217 (C) 210 (D) 205

Geometry & Mensuration

Directions for questions 1 to 40: Select the correct alternative from the given choices.

1. PQR is a triangle. PQ = 19.5 cm and QR = 11 cm. PS is the altitude of PQR of length 5 cm. C is a circle circumscribing the triangle PQR. Find the radius (in cm) of C.

(1) 18.05 (2) 28.85 (3) 27.25 (4) 21.45 (5) 31.25

2. ABC is an obtuse angled triangle whose sides are 7 cm, 16 cm and y cm where y is an integer. Find the number of possible values of y.

(1) 9 (2) 10 (3) 6 (4) 16 (5) 17

3. PQRS is a square. M and N are the midpoints of PS and QR respectively. X and Y are points lying on the line joining M and N inside PQRS, such that PXS = QYR = 60. Find the ratio of the area of the hexagon PQXRSY and the remaining area inside PQRS.

(1) 4 3 (2) 2 3 + 3 (3) 2 3 1 (4) 1 + 3 (5) 3 + 1

4. Each of two circles passes through the centre of the other. The radius of either circle is 2 cm. Find the area of the intersecting region (in sq.cm).

(1) 34pi

3 (2) 3

8pi+ 2 3

(3) 3

8pi 2 3 (4)

38pi

+ 3

(5) 23

32

pi

5. The radius of the base of a right circular cone is 6 cm and the height is 21 cm. The cylinder having the maximum possible total surface area is placed inside the cone such that one of its flat surfaces rests on the base of the cone. Find the total surface area of the cylinder, in sq.cm.

(1) 8

1353pi (2)

81321pi

(3) 8

1383pi

(4) 8

1473pi (5)

5441pi

6. C1 and C2 are two equal circles whose centers are M and N respectively. They intersect at X and Y. Neither of the centers lies inside the other circle. If XMN = , what is the range of possible value of ? (1) 0 30 (2) 0 45 (3) 0 60 (4) 0 75 (5) 0 90

7. Two identical circles intersect each other. The radii of the circles and the distance joining the centres of both the circles are in the ratio of 1 : 3 . Find the ratio of the area of the region common to both the circles, to that of a circle.

(A) pi

pi

43

(B) pi

pi

12323

(C) pi

pi

63

(D) pi

pi

6332

8. A large number (greater than 1200) of equilateral triangles of side 2 units are available to form equilateral triangles. Equilateral triangles of sides 16 units, 18 units, 30 units are formed using the available triangles. Find the total number of triangles used. (A) 1050 (B) 1100 (C) 1150 (D) 1200

B

C

A

3

9. Find the side of the smallest equilateral triangle that can contain a semicircle of radius 8 3 cm. (in cm) (A) 16 3 (B) 32 (C) 32 3 (D) 40

10. In the figure below, ABC is equilateral. Three identical circles are shown in it. Each circle touches the other two circles and has two of the sides of the triangle being tangents to it. Find the ratio of the side of the triangle and the radius of each circle. (A) ( )322 + (B) ( )3212 + (C) ( )312 + (D) ( )3312 +

11. N points are marked on the circumference of a circle. The number of triangles which can be drawn by using these points as vertices is 210 more than the number of triangles which can be drawn by using all but one of these points as vertices. Find N. (A) 21 (B) 22 (C) 23 (D) 24

12. Let X denote the product of the sides of a triangle. Let Y denote the product of its semi-perimeter,

circumradius and its inradius. XY

=

(A) 41

(B) 61

(C) 31

(D) Cannot be determined

13. In the figure above, PQRS is a square. A sector of a circle with centre S is shown. TQVU is a rectangle such that TU = 9 and UV = 32. Find PS.

(A) 60 (B) 65 (C) 70 (D) Cannot be determined

14. Two intersecting circles are said to be orthogonal, if a pair of tangents drawn to them at any point of intersection are perpendicular to each other. C1 and C2 are orthogonal circles. Each of them has a radius of 8 cm. Find the area of the region common to both the circles. (in sq cm) (A) 16(2pi 1) (B) 16(2pi 3) (C) 32(pi 1) (D) 32(pi 2)

15. On the periphery of a square grass field ABCD, two poles are fixed. One of them is fixed at the midpoint of side AB while the other one is fixed at the midpoint of side BC. Side of ABCD is 16 m. A cow is tied to one of the poles and another is tied to the other pole. If each cow is tied with a 8 m long rope, then find the ungrazable area in the field. (in sq. m) (A) 256 32pi (B) 216 32pi (C) 192 32pi (D) 240 32pi

16. PQR is an isosceles triangle with PQ = QR. QA is the median to PR. B is a point on QR (or QR extended) such that AQ = AB. PQA = 30. Find ABQ. (A) 20 (B) 30 (C) 40 (D) Cannot be determined

17. A triangle has an area of 960 sq m. Two of its sides are 68 m and 32 m. Find its third side if it is given that the triangle is right angled (in m). (A) 52 (B) 58 (C) 62 (D) None of these

18. A right circular cone has a radius of 18 cm and a height of 54 cm. From it, a right circular cylinder of radius 15 cm was cut. Find the volume of the conical part of the remaining solid (in cubic cm). (A) 4540pi (B) 4375pi (C) 4780pi (D) 3375pi

19. Two plastic cubes each of which have an integral side (in cm), have the sum of their lateral surface areas equal to 468 sq cm. The sum of their volumes is 945 cubic cm. Find the volume of the hemisphere whose radius equals the sum of the edges of the two cubes (in cubic cm) (A) 2150pi (B) 2250pi (C) 2325pi (D) 2400pi

20. In ABC, AB = 32, BC = 24 and AC = 40. Using B as a centre, a circle is drawn. This circle cuts AB and BC at D and E respectively. The radius of this circle equals the circumradius of ABC. Find AD : EC. (A) 3 : 1 (B) 2 : 1 (C) 3 : 2 (D) 5 : 2

21. There are 13 spherical balls, each of radius 3 2 . Nine of the balls are arranged in layer in a 3 x 3 square formation. Four balls are placed in a second layer over the four depressions which are formed. Find the height of the smallest cuboid which can enclose these balls. (A) 6 + 6 2 (B) 6+6 3 (C) 2 3 +2 6 (D) 3 2 +6

22. Four points are chosen at random in a region comprising an equilateral triangle of side 4 3 and its interior. Which of the following is true? (A) There are at least two points P, Q such that

PQ 4. (B) There are at least two points P, Q, such that

PQ 4. (C) There are at least two points PQ, such that

PQ > 4. (D) There are at least two points PQ, such that

PQ < 4. 23. In the figure below, PQRS is a rectangle with

PQ = 9 and PS = 6. MNOP is a smaller rectangle with MN parallel to PQ. Also the lengths of MN and MP are integers and A, B, C, D are the midpoints of the sides of the rectangle MNOP shown in the figure. The area of the shaded region is 162/3% of the area of the rectangle PQRS. Find the length of AB.

Q T P

S R

U V

4

M A N

B

O C P

D

Q P

S R

P B

C F

G J

R I H E D A Q

A

C B

D F

E

G

(A) 5 (B) 5)2/3( (C) 52 (D) Cannot be uniquely determined

24. There are two concentric circles with their centre at O. Chord AB of the outer circle is tangent to the inner circle. If the measure of the line AB is 40 cm and the radii of both the inner and the outer circle are integral number of centimeters, then which of the following cannot be the length of the diameter of the outer circle?

(A) 50 (B) 58 (C) 96 (D) 104

25. In the triangular field PQR, PQ = 54 m, QR = 240 m and Q =900. [PQ, AB, CD, EF, GH, and IJ are all equispaced and all the lines are perpendicular to QR. Also HI = IR]. If one moves along PQABCDEFGHIJR, then what is the total distance covered (in meters)?

(A) 270m (B) 594m (C) 351m (D) None of these

26. The maximum number of acute angles in a convex octagon is

(A) 1 (B) 2 (C) 3 (D) None of these

27. There are 6 boys B1, B2, B3, .B6 standing at the vertices A to F respectively of a regular hexagon. They start walking simultaneously along the perimeter in the clockwise direction with speeds in the ratio 1 : 2 : 3 : 4 : 5 : 6. When B6 completes 3 full rounds, which are the vertices at which there are no boys?

(A) A, C (B) B, E (C) A, C, E (D) B, D

28. On square ABCD, points P, Q, R and S are on sides DA, AB, BC, CD such that AP = AQ = CR = CS X and Y are points on AB and CD respectively, which are equidistant from P and R.

Find CRAPYDXB

+

+

(A) 0.5 (B) 0.75 (C) 1 (D) Cannot be determined

29. In the figure below, circles with centres A, B and C have equal radii and similarly circles with centres D, E and F have equal radii. Find the ratio of the radii of circles with centres at D and G.

(A) 11

31013 + (B)

1131017 +

(C) 11

31019 (D)

11)22(10 +

30. Which of the following statements is/are true? () A triangle exists with altitudes measuring 6,

9 and 15 () A triangle exists with altitudes measuring 4,

9 and 12 which of the following is true. (A) Only (B) Only II (C) Both and (D) Neither I or

31. A, B, C and D are four friends who are standing at the four corners of a rectangular field in the anticlockwise order. They decide to meet at a point P inside the field. If the shortest distance of the point P from the initial position of A, B, C, and D are 20 m, 40 m, 60 m and x m respectively, then find the value of x. (A) 30 2 (B) 20 6 (C) 40 3 (D) 50

32. When a rectangle ABCD (with AB < BC) is folded such that the vertex C touches vertex A, the length of the crease XY formed is equal to the length of the rectangle. How many times the length of the rectangle is its breadth?

(A) 2

1

212

+ (B)

21

212

(C) 2

1

215

+ (D)

21

215

33. Two perpendicular chords PQ and RS intersect at T PT= 4 and QT=18. If TS is 2 times TR, then find the radius of the circle (A) 110 (B) 2 30 (C) 130 (D) 5 6

34. On side AB of rectangle ABCD, P is a point such that APD : DPC: CPB = 2 : 5 : 5. Find AB : BC.

(A) 3 (B) 232 (C) 2 (D) Cannot be determined

35. A cuboid has a square base. The length of its longest diagonal is 2502 . All dimensions of the

5

cuboid are integral. How many such cuboids are possible? (A) 1 (B) 2 (C) 3 (D) 4

36. In a cyclic quadrilateral, the diagonals AC and BD intersect at P at right angles. If PA = 8, PC = 9 and the radius of the circumcircle of ABCD is

25.81 , then find PD given that where PD < PB. (A) 12 (B) 9 (C) 8 (D) 6

37. In trapezium ABCD, the diagonals AC and BD intersect at P. Three times the area of the pentagon APBCD is equal to 7 times the area of PCD. Find AP : PC. (A)

32

(B) 43

(C) 34

(D) 23

38. The semiperimeter of a right angled triangle is 60 and the inradius is 8. Find the length of the smaller leg. (A) 30 (B) 20 (C) 24 (D) 18

39. There are two vertical poles AB and CD where B and D are on the ground. AB = 4, CD = 7.5, BD = 11.5. On line BD, P is a point where the points A and C subtend a right angle and Q is a point that is equidistant from A and C. Find PQ. (A) 3 (B) 3.5 (C) 4 (D) Cannot be determined

40. In triangle PQR, Q =900 and PQ = QR. D is a point on the same side of PR as Q such that PR = 2 QD. Find the measure of angle PDR.

(A) 221/2 (B) 30

(C) 45 (D) Cannot be uniquely determined

Special Equations


1. In a certain country, the currency used was called femto. Raju had his lunch in one of the hotels in it. His bill came to 47 femtos. He had notes in only 3 denominations 1-femto, 2-femto and 20-femto. In how many ways can he settle the bill? (1) 24 (2) 18 (3) 48 (4) 16 (5) 42

2. A bank teller was making the payment for a cheque presented by Mohan. As he was in a confused state of mind, he transposed the rupees and paise and hence gave more than what he should have. Mohan left the bank and bought a biscuit from a nearby store for `1.50. The amount remaining with him was 4 times the amount on the cheque. The amount remaining with him must have been between (1) `70 and `71 (2) `79 and `80 (3) `85 and `94 (4) ``93 and `94 (5) `98 and `99

3. Nagu bought a new bike and went for a drive, liding at a uniform speed on a highway. At 9 am he passed a milestone. He turned his head back and read the number on the milestone. He continued driving and at 10 am, he passed another milestone. Again he turned his head back and saw that the number was the reverse of the number on the first milestone. He continued driving and at 11 am he passed a third milestone. Again, he turned his head back to note the number. The sum of the digits of the number on this milestone was equal to the sum of the digits of the number on the first milestone. Which of the following is the speed at which he travelled? (Assume that the numbers he saw were increasing). (A) 9 km/hr (B) 45 km/hr (C) 18 km/hr (D) Cannot be determined.

Equations, Ratio, Proportion, Variation


1. Amar, Bhuvan, Chetan and Dinesh are four friends. Amar has m marbles with him. He gives Bhuvan 1 less than half the number of marbles he has. Then he gives Chetan 1 less than half the remaining number of marbles he has. Finally, Amar gives Dinesh 1 less than half the remaining number of marbles he has and is left with 4 marbles. Which of the following best describes the value of m?

(1) 1 m 4 (2) 5 m 9 (3) 9 m 13 (4) 10 m 14 (5) m 14

2. If a b1

and b c2, then find the value of

9a + 5c, when b = 27. Given that when b = 3 then a = 4 and c = 6. (A) 94 (B) 49 (C) 76 (D) 38

3. If 3a + 5b + 7c = 1.25 k and 2a + b + 3c = 0.75 k, then 7b + 5c is what percentage of k?

(A) 25% (B) 50% (C) 35% (D) 75%

4. There are some two rupee coins and five rupee coins in a bag. If the number of five rupee coins is tripled, then the amount in the bag is increased by 75%. Which of the following can be the number of five rupees coins in the bag? (A) 13 (B) 20 (C) 18 (D) 32

5. There are 100 questions in a test paper. Four marks are awarded for each right answer and two marks are deducted for each wrong answer. If Abhilash attempts more than 85 questions and get 70 marks, What is the minimum number of questions that he could have answered correctly? (A) 40 (B) 38 (C) 39 (D) 41

6

6. Arun, Varun and Kiran have a total of `8,000 with them. They spend `50, `100 and `200 respectively, Now they have a money is the ratio of 14 : 22 : 15. What is the amount with the Varun initially? (A) 2150 (B) 3400 (C) 3300 (D) 2100

Time and Distance

Directions for question 1: Select the correct alternative from the given choices.

1. P and Q are 2 stations. Raju plans to drive from city R, located 840 km directly to the north of Q, to Q, at a speed of

3210

kmph so that he can

catch a train arriving there from P. The train leaves P at 9 a.m. and travels at a speed of 84 3 kmph. P is between east and south-east of R with RQ at 60 to RP. Also P is between the north and the north-east of Q with PQ at 30 to RQ. If Raju has to reach P at least 25 minutes before the train, then find the approximate latest time at which he should start from R.

(1) 6:50 a.m. (2) 6:40 a.m. (3) 6:20 a.m. (4) 6:30 a.m. (5) 7:00 a.m.

Directions for questions 2 and 3: Answer these questions based on the information below.

Cities P and Q are 4800 km apart. P is to the west of Q. Both cities are in different time zones Best Airlines was an airline which operates non-stop flights between the cities. All its planes cruise at the same speed in both directions. However, the effective speed of any plane is influenced by a steady wind blowing from west to east at 100 kmph. The table below shows the departure time of the planes from each city and their arrival time at the other city.

(Given below are local times of the respective cities)

Departure Arrival City Time City Time

P 7:00 a.m. Q 5:00 p.m. Q 6:00 p.m. P 4.00 a.m.

2. Find the time in P when the plane landed in Q. (1) 4:00 p.m. (2) 3:00 p.m.

(3) 3:30 p.m. (4) 2:30 p.m. (5) Cannot be determined

3. Find the cruising speed of the plane (in kmph). (1) 600 (2) 550 (3) 500 (4) 450 (5) Cannot be determined


4. A starts from point P at 9:00 a.m. and travels east at 45 km/hr. B starts a bit later from P and travels south at 30 km/hr for 48 minutes. At that instant, he is 74 km from A. When does B start from P?

(A) 9:42 a.m. (B) 9:45:20 a.m. (C) 9:48 a.m. (D) 9:51:40 a.m.

5. In an x-metre race, A beats B by 180 m and C beats by 351 m. In the same race, B beats C by 198 m. Find x?

(A) 1200 (B) 1000 (C) 1122 (D) 1320

6. Towns A and D are 36 km apart. Three friends, Tarun, Varun and Arun start together from A towards D. While Arun sets off on foot, Varun takes Tarun along on his bike and travels at 45 km/hr. He drops Tarun at a point C and turns back for Arun. He meets Arun at a point B, and then turns back towards D. All the three friends reached D together. If both Arun and Tarun walked at 5 km/hr, how long do the 3 take to cover AD?

(A) 3 (B) 332

(C) 391

(D) 15131

7. Two friends P and Q start simultaneously from the opposite ends of a race track AB of length 150 m at speeds 60 m/s and 40 m/s respectively. P starts from A and Q starts from B. Once each reaches an end, he immediately turns back and moves towards the other end. They keep moving to and from between the two ends. Find the difference between the distance covered by P and Q by the 8th meeting. (A) 300 m (B) 450 m (C) 600 m (D) 1200 m

8. Ajay and Sanjay start simultaneously from the same point on a circular track. If they travel in opposite directions, they meet at 7 distinct points on the track whereas if they travel in the same direction, then they meet at n distinct points on it where n is a prime number. If Ajay is faster than Sanjay and Sanjay's speed is P% less than that of Ajay, then which of the following can be a value of P? (A) 25 (B) 50 (C) 75 (D) 831/3

Quadratic Equations

Directions for questions 1 and 2: Select the correct alternative from the given choices.

1. Three consecutive positive integers are taken in descending order. The first, second and the third are raised to the first, second and third powers respectively. The powers are added and the result is multiplied by 3. The square root of the result is found to be the sum of the original integers. The least of the integers is denoted by L. Which of the following holds true?

(1) 1 L 4 (2) 5 L 8 (3) 9 L 12 (4) 13 L 15 (5) L > 15

2. The roots of the equation x3 px2 + qx r = 0 are a, b and c, which are consecutive integers. Find the least possible value of q.

(1) 0 (2) 1 (3) 2 (4) 1 (5) 2

7

Directions for questions 3 and 4: Answer these questions using the information below.

Let g(x) = px2 + qx + r, where p, q and r are constants and r 0. One root of g(x) = 0 is 4 and 8g(3) = 3g(6).

3. Find the other root of g(x) = 0. (1) 1 (2) 3 (3) 6 (4) 2 (5) Cannot be determined

4. Find the sum of p, q and r. (1) 21 (2) 20 (3) 19 (4) 18 (5) Cannot be determined

Directions for questions 5 and 6: Answer these questions based on the given information.

Raju makes and sells an item in a market every day. He sells each unit of that item at `106. The cost of producing x units per day is 200 + px + qx2, where p and q are constant. If Raju increased his daily production from 10 units to 15 units, his daily production cost would increase by 888/9%. If he increased his daily production from 15 units to 20 units, his daily production cost would increase by 6412/17%. Assume that there is a high demand for the item and that Raju can sell whatever he produces. He wishes to maximize his profit.

5. Rajus daily production must be (in units). (1) 10 (2) 15 (3) 8 (4) 5 (5) 12

6. Find the maximum daily profit that Raju can obtain (in Rs). (1) 400 (2) 425 (3) 350 (4) 375 (5) 184


7. E(y) is a quadratic expression. It has the minimum value of 1 when y = 3 and E(2) = 2. Find E(4). (1) 40 (2) 45 (3) 50 (4) 55 (5) 60

8. If the equations ax2 + bx + c = 0; bx2+ cx + a = 0 have one root in common, then which of the following is definitely true?

(A) a3 + b3 = 3abc (B) a3 + b3 + c3 = 3abc

(C) a3 b3 c3 = 3abc (D) a2 + b2 + c2 = 2ab + 2bc + 2ca

9. If , and are the roots of the equation x3 + 2x2 5x 6 = 0 and , , are the roots of the equation x3 + px2 + qx + r = 0; then the value of p is (A) 6 (B) 5 (C) 5 (D) 6

10. Let and be the roots of a quadratic equation and = 9 and || || = 5 then find the product of roots.

(A) 14 (B) 21 (C) 16 (D) 7

11. What is the minimum value of the square of the difference of the roots of the quadratic equation x2 (k + 7) x (3k 15) = 0?

(A) 160 (B) 180 (C) 160 (D) 180

12. A certain number of cups of tea are available for `90. If the price of each cup increases by `1.50, the number that can be bought for the same amount decreases by 10. Find the actual cost of each cup of tea (in rupees).

(A) 3 (B) 3.50 (C) 4 (D) 4.50

13. The expression x4 + y4 2x2 y2 32x2 32y2 + 256 is completely factorized into real factors. Which of the following statements about the factors is true? (A) There are 2 irreducible quadratic factors.

(B) There is a cubic factor and a linear factor. (C) There are 4 linear factors.

(D) None of these

14. The expression ax2 + bx + c takes a maximum value of 5 at x = 1 and takes the value of -1 at x = 0. Find the value of the expression at x = 5. (A) 89 (B) 91 (C) 210 (D) 211

15. The roots of the equation x3 + ax2 + bx + c = 0, where c > 0, are k, kr, kr2 where k and r are integers. If the sum of the squares of the roots is 364, then find the value of c (A) 180 (B) 72 (C) 108 (D) 216

16. If the sum of the roots (not necessarily real) of a quadratic equation is 6 and the sum of the squares of the roots is 16, then find the product of the roots. (A) 8 (B) 10 (C) 9 (D) Such an equation doesn't exist

17. The roots of x3 21x2 + px 280 = 0 are in arithmetic progression. Find the value of p. (A) 138 (B) 128 (C) 118 (D) Cannot be determined

18. If cubic equation ax3 + bx2 + cx + d = 0 has two positive roots where a, b, c, d are real and d0, then which of the following is true? (A) c and b are of opposite signs.

(B) a and d are of the same signs. (C) b and c are of the same sign. (D) a and d are of opposite signs.

Progressions / Series


1. Find the number of common terms of the sequences 24, 29, 34, .. 474 and 25, 29, 33, , 485

(1) 20 (2) 21 (3) 25 (4) 24 (5) 23

2. Nilgiris coffee cost `(110 + 0.2N) per kg on the Nth day of 2006 where 1 N 200. Its price

8

remained constant that year from the 200th day. Coorg coffee cost `(97 + 0.25N) per kg on the Nth day of 2006 where 1 N 365. Find the date in 2006 on which the prices of the two varieties of coffee were equal. (1) June 25 (2) August 10

(3) July 31 (4) August 20 (5) July 21

Directions for questions 3 and 4: Answer these questions based on the given information

M and N are positive quantities. Let g1 = M and h1 = N. When x is even, gx = Mhx

1 and hx = Mgx

1

When x is odd, gx = Nhx 1 and hx = Ngx

1

3. Which of the following equals gx + hx when x is even?

(1) M(MN) x21

(M + N) (2) MN 1x2

1 (M + N)

(3) M(MN) 1x21

(M + N) (4) MM 2

1

(M + N) (5) M(MN) 2x2

1 (M + N)

4. If M =41

and N =43

, find the least odd x for which

gx + hx < 0.02? (1) 3 (2) 5 (3) 7 (4) 9 (5) 11


5. If k = 13.12.11.10

1+

14.13.12.111

+

+ 102.101.100.99

1, then the value

of k is

(A) 5616600

2149 (B)

56661001249

(C) 5666100

1429 (D)

56166004219

6. Find the sum of all the five digit numbers which leave a remainder of either 4 or 6 when divided by 8.

(A) 659100000 (B) 561900000 (C) 1318200000 (D) 1237522500

7. If the sum of the first n terms of an arithmetic progression in 2400 and the sum of next n terms as 7200, then find the ratio of first term and common difference. (A) 3 : 2 (B) 2 : 1 (C) 1 : 2 (D) 2 : 3

8. If the nth term of a series is given by tn =2tt 1n1n + +

,

then 27

31

tt

is


9. Find the sum of the first 40 terms of the series 2 + 3 + 4 + 6 + 8 + 9 + 16 + 12 + . . .

(A) 621 (B) 221 + 628 (C) 220 + 628 (D) 620 + 629

10. If S =

124126

6365

2628

79

ad infinitum, S =

(A) 23

(B) 21

(C) 35

(D) 3

11. If S1 = 21027

...

309

207

125

63

21

++++++ and

S2 = 141

...

41

31

21

++++ , then find the value of

2S2 S1 (A)

54

(B) 43

(C) 21

(D) 23

12. If a, b, c, are 3 positive numbers in geometric progression and a12 = b15 = cx, then x = (A) 18 (B) 20 (C) 21 (D) 24

13. 3.2.1

1+

4.3.21

+5.4.3

1+ . . . +

15.14.131

is equal

to

(A) 10526

(B) 10531

(C) 71

(D) 21059

14. If S= 11

21

+ 31

41

+ 51

61

+. . . + 991

-

1001

, the value of S is also equal to

(A) 11

+ 21

+ 31

+ 41

+... + 251

(B) 511

+ 521

+ 531

+ ... + 100

1

(C) 501

+ 511

+ 521

+ ...100

1

(D) None of these

15. The nth term tn of a sequence is defined by tn = tn-1 tn-2. If t1 = 3, t2 = 4, and tn tm is a multiple of 10, which of the following is not a possible value of (m, n)? (A) (10, 16) (B) (16, 10) (C) (7, 1) (D) (8, 2)

16. The sum of n terms of an arithmetic progression, starting with the 11th term is directly proportional to n2. The sum of 2 of the terms of the arithmetic progression is 0. Which of the following cant be either of these two terms? (A) 5th (B) 11th (C) 21st (D) 16th

17. )10(12...)3(5)2(4)1(3)12(10....)5(3)4(2)3(1

2222

2222

++++

++++=

(A) 37954784

(B) 1115

(C) 2329

(D) 56

9

Functions


1. G(x) is a function satisfying

=

baG)b(G

)a(G for all

real a, b If G(3) =91

, find G(9).

(1) 811

(2) 91

(3) 81 (4) 9 (5) Cannot be determined

2. G(y) is a function satisfying the following conditions. G(1) = 5400 and

=

x

1yG(y) = X2G(X). where X is a

natural number greater than 1. Find G(6). (1)

72100

(2) 7

2400 (3)

71800

(4) 7

1500 (5) 240

3. If f(xy) = f(x) + f(y) + f(x) f(y) 9

10and f(6) =

32

,

then the value of f(1/6) is (A)

32

(B) 89

(C) 9

17 (D)

817

4. If f(x) = x1

x

+, then the value of f(f(f(f(f(f( 3))))) =

(A) 265

(B) 254

(C) 193

(D) 376

5. If f(x +2) = f(x) + 7 and f(1) = 2; f(2) = 5 then the ratio of f(150) to f(75) is (A) 523 : 261 (B) 253 : 621 (C) 427 : 673 (D) None of these

6. If f (x + y) = f(x) + f(y) , where x 1, y 1 and f(7) = 24.5, then find the value of f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7).


7. If f(x+1) f(x) = ax + b, then which of the following is true about the graph of y = f(x)? (A) A line with a slope of a (B) A line with a slope of b (C) A parabola (D) None of these

8. Consider the function f(x) = (x + 3)p (x + 2)q (x - 1)r (x 3)s, where p. Q. R. s are nonnegative integers. f(3.5) = a, f(2.5) = b, f(0.5) = c

f(1.5) = d, f(3.5) = e Also, ab > 0,

c

b 0 and eded =+ .

Which of the following can be the value of (p, q, r, s)?

(A) (2, 3, 5, 1) (B) (4, 3, 2, 0) (C) (3, 2, 5, 1) (D) (0, 1, 2, 3)

9. The continuous function f(x) satisfies the following conditions x

f(x + 2) f(x) = 4 for x 2 f(x + 2) + f(x) = 6 for 0 < x < 2 and f(x + 2) f(x) = 4 for x 0 also f(0) = 5 Which of the following is f(x)? (A) 14x2 + (B) 5x2 + (C) 32x4 + (D) 14x2 +

10. If f(x + 1) f(x) = x and f(x) is an nth degree expression in x, then n = (A) 2 (B) 3

(C) 4 (D) Cannot be determined

11. If f1(x) = 2x71x

+

, fn(x) = f1[fn-1 (x)], find f8 (x)

(A) 1x71x2

+

(B) 1x71x2

+

(C) 1x71x2

+ (D)

1x71x2

Averages


1. Twelve years ago, the average of the ages of the members of a joint family having ten members was 25 years. Four years later a member aged 50 years died and a child was born in the family that year. Four years after that, another member aged 50 years died and another child was born. Find the present average age of the members of the family (in years). (1) 26 (2) 27 (3) 28 (4) 29 (5) 30

2. There are two groups (A & B) of children in a joint family. There are 3 more children in B as compared to A. While the average age of the children in A is 6 years more than that of B, the combined age of the children in A is less than that of B. After 4 years, the difference in the combined ages of group A and group B will double. If there are less than 10 children in the family, which of the following can be the average age (in years) of the children in B? (A) 9 (B) 8 (C) 7 (D) 5

3. The number and the average weights of different groups of children are given below: Number Average Weight (Kg.) M 40 M + 2 45 M + 2 50 7 60 If the average weight of the entire class is 50 kg. Find the total number of students. (A) 20 (B) 21 (C) 22 (D) 23

4. There are two vessels P and Q, P containing 120 L of milk and Q containing 120 L of water. In the first operation 30 L is removed from P and poured

10

into Q and then 30 L from Q is poured back into P. Like this, one more operation takes places i.e. transferring 30 L from P to Q and then transferring 30 L from Q to P. What is the ratio of milk and water in P, finally? (A) 2 : 1 (B) 13 : 6 (C) 15 : 7 (D) 17 :8

5. Dinku Beora was a chronic alcoholic and because of persistent, health problems he decided to quit drinking. He devised an ingenuous way of doing so. He bought a 750 ml goodbye bottle of Old Monk. On the first day, he drank 5% of the contents in it and replaced that quantity with water. Next day he drank 10% of the contents in the bottle and replaced it with water. Like this he continued on the 19th day he drank 95% of the contents in the bottle and replaced it with water and on the 20th and last day he drank the entire contents in the bottle. Find the ratio of the total quantity of alcohol and water that he drank in the entire process. (A) 2 : 21 (B) 2 : 19 (C) 3 : 28 (D) None of these

Quant Based Reasoning

Directions for questions 1 and 2: Answer these questions based on the following information. P, Q, R, S and T are 5 horses. They participated in a race. The following are the rules of the race.

(1) A person who bets on the winning horse gets 4 times the bet amount.

(2) A person who bets on the horse coming second gets 3 times of the bet amount.

(3) A person who bets on the horse coming in third gets back his amount.

(4) Other persons lose their amount.

Mohan had placed his bets on Q, R and T. The amounts he bet on Q, R and T were `2000, `4000 and `6000 respectively. He ended up with no gain and no loss.

1. Which of the following is not possible? (1) At least two horses finished before P. (2) There were three horses between Q and P. (3) T finished last. (4) S came in second. (5) There were three horses between T and R.

2. Suppose S finished in the fourth position. Then which of the following is not possible?

(1) P finished first. (2) One horse finished between Q and R. (3) Q came in second (4) T finished last. (5) R came second.

Data Sufficiency

Directions for questions 1 and 2: Answer these questions based on the instruction below.

In the questions below, each question has two statements A and B following it. Mark your answer as

(1) if the question can be answered from A alone but not from B alone.

(2) If the question can be answered from B alone but not from A alone.

(3) if the question can be answered from A alone as well as from B alone.

(4) if the question can be answered from A and B together but not from any of them alone.

(5) if the question cannot be answered even from A and B together.

A certain number of players participated in a tournament, played according to the following rules. The number of players at any stage is denoted as N.

(i) if N is even, the players are grouped into 2N

pairs.

The players in every pair play against each other. The resulting winners move on to the next round.

(ii) If N is odd one player is allowed to move on to the next round. He is said to be given a bye. The remaining N 1 players are grouped in to

21N

pairs who play against each other. The resulting winners move on to the next round. The players who lose are eliminated from the tournament.

From the rules above, it follows that if there are

N players in a round, then 2N

players move on to

the next round if N is even and 2

1N + players

move on to the next round if N is odd. This process continues until the final round, which is played between two players. The winner in this round is the champion.

1. Find the number of matches played by the champion. A. In the first round, there were 169 players. B. The champion was given a bye only once.

2. The number of players in the first round was M where 129 < M < 256. Find M. A. One player received a bye while moving from

the third to the fourth round. B. Only one player received a bye in the entire

tournament.

Directions for questions 3 to 6: Each question is followed by two statements A and B. Indicate your responses based on the following directives:

Mark (1) if the question can be answered using A alone but not using B alone.

Mark (2) if the question can be answered using B alone but not using A alone.

Mark (3) if the question can be answered using A and B together, but not using either A or B alone.

Mark (4) if the question cannot be answered even by using A and B together.

3. Class X has 80 students. The average height of the students in it is 140 cm. It has two sections, A and B, with equal number of students in each section. The average height of A exceeds that of B. Mohan is the tallest in A and Sohan is the

11

shortest in B. If each of these students is transferred to the other section, the average heights of the sections would get interchanged. Find the height of Mohan. (A) The average heights of A and B differ by 2 cm. (B) If Sohan shifted from B to A, the average

heights of the sections would become equal.

4. A company has to store at least 270pi kilolitres of water at all times to meet safety and regulatory requirements. It is considering having a spherical tank whose wall thickness is uniform and whose outer radius is 6 meters for this purpose. Will the tank meet the company requirements? (A) When empty, the tank weighs 36000pi kg. It is

made of a material whose density is 4 gm/cc. (B) The tanks inner radius is at least 4.5 metres.

5. P, Q and R are three integers. Find the maximum value of PQ + QR + PR. (A) P = Q R (B) P + Q + R = 84

6. P is a point on AB. Rohit wanted to draw a square ABCD but failed to do so. Why did he fail? (A) PC = 3 cm (B) PC =

3PD

Percentage, Profit & Loss

Directions for questions 1 and 2: Answer these questions based on the information below.

Mohan was considering three alternatives for investing a certain amount. He wanted to get the maximum possible assured return on his investment. The three alternative are given below. He could make use of each completely or partially along with the others. Alternative 1: Invest in the mutual funds of PQR Ltd. A rise in the stock market will result in a return of 8% and a fall will result in a return of 10%.

Alternative 2: Invest in the mutual funds of RQP Ltd. A rise in the stock market will result in a return of 5% and a fall will result in return of 4%. Alternative 3: Invest in a public sector bank which promises a 0.4% return.

1. Find the greatest assured return for Mohan. (1) 0.4% (2) 0.5% (3)

32 %

(4) 0.8% (5) 65 %

2. Find the strategy which will maximize the guaranteed return to Mohan. (1) 100% in alternative 3.

(2) Equal investment in each alternative. (3) Investments in alternatives 1 and 2 in the

ratio 1 : 2 (4) Investments in alternatives 1 and 2 in the

ratio 2 : 1 (5) Investments in alternatives 1, 2 and 3 in the

ratio 2 : 3 : 4


3. The price of coffee was increased by 40% but Raj was willing to increase his expenditure by 12%. Find by what percentage should he decrease his consumption. (A) 10% (B) 15% (C) 20% (D) 25%

4. The ratio of the populations of cities X, Y and Z in 2008 was 3 : 5 : 6. The percentage increases in the populations of X, Y and Z from 2008 to 2009 were 10%, 12.5% and 15% respectively. Find the percentage increase in their total population from 2008 to 2009. (A) 12% (B) 14% (C) 13% (D) None of these

5. If the cost of a ball pen reduces by 20%, Raj can buy 90 more ball pens for `3600. Find the cost (in `) of a ball pen. (A) 12.5 (B) 10 (C) 15 (D) 20

6. Ashok made a loss of 15% by selling 96 apples for `2040. How many apples must he sell for `2600 to make a 30% gain? (A) 80 (B) 100 (C) 65 (D) 104

7. Three persons A, B and C have their monthly incomes in the ratio 6 : 7 : 8. Their monthly expenditures are in the ratio 5 : 6 : 10. The monthly savings of C is 37.5% of his monthly income. Find what percent of Bs savings was As savings? (A) 50% (B) 662/3% (C) 831/5% (D) 87.5%

8. Pradeep bought a puppy for a certain price. He sold it to his neighbour at a profit percent whose magnitude was equal to the profit realised by him in the transaction. But after a couple of days the neighbour sold it back to Pradeep at 20% loss. Effectively, 27.5% of the cost price was refunded to Pradeep. At what profit percentage did he sell the puppy to his neighbour? (A) 30% (B) 37.5% (C) 27.5% (D) 32.50%

Simple Interest Compound Interest


1. The difference between the simple interest and the compound interest for two years on a sum invested at 16% p.a. is `384. Find the sum (in `)

(A) 13500 (B) 15000 (C) 14250 (D) 12750

2. Ramu took a certain loan at Simple Interest in 2000 for a period of 4 years. The rate of interest was constant throughout the loan period whereas had he cleared the loan after 9 years, he would have paid `90,000. Where as had he cleared the loan after 12 years, he would have paid `105,000. Find the amount (in `) he paid to clear the loan.

(A) 45000 (B) 65000 (C) 70000 (D) 75000

12

Time and Work


1. Pipe X can fill a tank in a certain time. It was opened at 12 pm. Due to a leak at the bottom of the tank, the tank was filled only at 12:50 pm. If the leak can empty the tank in 200 minutes, then find the time (in minutes) in which X can fill the tank.

(A) 30 (B) 45 (C) 40 (D) 50

2. 8 men can build a wall, 8 m long in 8 days, working 8 hours a day. Find the number of hours per day for which 16 men are required to work, to build a wall 16 m long in 16 days.

(A) 2 (B) 8 (C) 16 (D) 4

3. 50 men can complete a job in 20 days working 9 hours a day. They started the job. They worked 9 hours each day for the first x days. At the end of x days, 5 men left. The remaining job was completed by the remaining men working 8 hours a day in 12.5 days. Find x.

(A) 12.5 (B) 8 (C) 9 (D) 10

4. If A, B, C, D work independently, the amount that A is paid for 3 days is equal to that paid to B for 4 days. The amount paid to C for 3 days is equal to that paid to D for 2 days. B and D are paid equal amounts for equal duration. A and B together complete a piece of work for which they are paid `1680. If all 4 had completed the same work together, what would A's share have been (in rupees)? Assume that equal amounts are paid for equal work.

(A) 960 (B) 800 (C) 560 (D) 630

5. For workers working on a construction site, the rate of doing work for men and women increases in winter with respect to that in summer by

3331 % and 50% respectively. 4 men and 12

women can complete a certain piece of work in 120 days in summer. The time taken triples if the women do not turn up for the work. If a men and b women can complete two times the given work in 180 days in winter then find the possible number of ordered pairs of (a, b). (A) 2 (B) 20 (C) 10 (D) 5

6. A and B are two daily labourers who work on a maintenance site. The daily wage of A is 40% less than the daily wage of B. A and B together worked on a certain project and completed it in 72 days. As a result, at the end of the project they together received a certain amount. If that entire amount were to be earned by A alone, then for how many days would he need to work ? (A) 144 (B) 216 (C) 288 (D) 192

7. A and B can together do a piece of work in 12 days. If A completed half the work and the other half is completed by B with only one of

them working each day, the total time taken would be 241/2 days. Find the number of days A alone would take to complete the work if B is faster than A. (A) 28 (B) 21 (C) 24 (D) 25

Venn diagrams


1. In a class, 30% of the students like tea and 40% of the students like coffee. 20% of the students who like tea also like coffee. Find the percentage of the studies who like neither tea nor coffee.

(A) 32% (B) 34% (C) 38% (D) 36%

2. In a locality, 180 residents watch only Sony TV, 210 residents watch only Star Plus and 150 residents watch only Zee TV. 540 residents watch atleast one of Sony TV and Star Plus. At most 340 residents watch Star Plus. 90 residents watch all the three channels. Find the minimum possible number of students who watch Sony TV and Zee TV but not Star Plus.

(A) 40 (B) 50 (C) 20 (D) 30

3. In a group, 60% of the boys and 50% of the girls like cricket. 45% of the boys and 55% of the girls like volleyball. Number of students who like cricket is 7 more than that of students who like volleyball. The difference of the number of boys and girls who like cricket and that of the number of boys and girls who like volleyball are in the ratio 16 : 5. Find the strength of the group.

(A) 90 (B) 100 (C) 110 (D) 120

Trigonometry


1. T1 and T2 are two towers and Raju was on the top of the tower T1. He realized that there were two points on the ground such that the angle of elevation of T1s top from each of those points was . The distances from, T2s bottom to the top of T1 as well as to each of the points was 30 feet. The area of the triangle formed by the top of T1 and each of the points can be (in feet). (T2 's bottom and the two points on the ground are collinear) (A) 225 (B) 960 (C) Both (A) and (B) (D) Neither (A) nor (B)

2. A ladder has a length of 10 m. It makes an angle of 45 with a wall. It touches the wall at a point P. There are two points on the ground. The angle of elevation of P from each of these points is 60. Find the distance between these points (in m). (The two points on the ground and the bottom of the wall are collinear)

(A) 3

67 (B)

3610

(C) 3

68 (D)

364

13

3. Given that ,rqprpqrpq

cos 222 ++

++= where p, q, r are

real numbers such that the sum of any two exceeds the third, which of the following is not a possible value of ?

(A) 20 (B) 40 (C) 50 (D) 80

Indices, Logarithms, Surds


1. 33x + 2 92x 1 = 118098, find the value of 6x + x6. (A) 945 (B) 559 (C) 2403 (D) None of these

2. Find the value of

1211201201211

...

181717181

171616171

+++

++

+

(A) 1520

(B) 447

(C) 4437

(D) 1513

3. If ,3x

1x =+ find the value of x6 + .

x

16

(A) 729 (B) 326 (C) 322 (D) 324

4. If a = 5 2 6 and b = 5 + 2 6 , evaluate

33

44

baba

+

+.

(A) 49

4901 (B)

4854799

(C) 10 (D) 4854801

5. If log2log4x = log4log2x, find x. (A) 16 (B) 8 (C) 8 2

(D) 24

6. If 1, log7(4x + 5), log7(4x + 1 1) are in arithmetic progression, which of the following is a possible value of x?

(A) 1 (B) 1.5 (C) 2 (D) 2.5 7. Let t = 3(log3a)2 9log3a + 9. Which of the

following is true about the equation at=27? (A) It has exactly one distinct solution for t.

(B) It has exactly two distinct solutions for t. (C) It has exactly three distinct solutions for t. (D) It has no real solutions for t.

Modulus / Inequalities


1. How many integer values of x exist that do not satisfy the inequality (x2 + 4x 32) (x2 + 2x 8) > 0?

(A) 7 (B) 8 (C) 6 (D) 4

2. Under which of the following conditions the

inequation baa

bba 22

is true?

(A) a b and ab < 0 (B) a b and ab > 0 (C) Either (a) or (b) (D) None of these

3. What is the set of values of x for which the

inequation 6x5x8x9x4

2

2

+ > 3 is satisfied

(A) (6, 1) (1,) (B) (, 6) (1, ) (C) (, ) (D) (, 1) (6, )

4. If x1, x2, x3, x4, x5 and x6 are positive and x1 x2 x3 x4 x5 x6 = 1; then the minimum value of (x1 + 4) (x2 + 4) (x3 +4) (x4 +4) (x5 +4) (x6 +4) is

(A) 15625 (B) 1425 (C) 13225 (D) 11025

5. If S = 512

1...

41

31

21

11

+++++ , then S lies in

which of the following ranges? (A) (4, 5.5) (B) (5.5, 9) (C) (9, 10.5) (D) (10.5, 12)

6. P is any positive number such that it is possible to find 4 positive numbers whose product is P and whose sum S P. Among the numbers 4, 5, 6, 7, how many are possible values of P?

(A) 0 (B) 1 (C) 2 (D) 3

7. If x3 + y3 + z3 = 125, a2 + b2 + c2 = 16, ax + by + cz = 20 and x, y, z, a, b, c are integers, then which of the following cannot be the value of

( )222 zyx4 cba ++ ++ ? (A)

251

(B) 271

(C) 331

(D) 491

8. How many integral values of x satisfy the following inequality

7x3x5x14 ++++< < 25? (A) 9 (B) 11 (C) 13 (D) 16

9. Find the area described by the inequality 2yxyx ++

(A) 3 (B) 2 (C) 2.5 (D) 4

10. In triangle PQR, p, q and r are the lengths of the sides opposite P, Q, and R respectively. If (p + q + r)2 = 3(pq + qr + rp), what can be said about the triangle PQR?

(A) It is equilateral. (B) It is isosceles. (C) It is scalene. (D) Cannot be uniquely determined

Operator Based Questions


1. Given that a b = 4a + 3b + 7ab and a a < b b, then which of the following is true?

(A) (a b) (a + b + 1) < 0 (B) (a + b) (a b + 1) > 0 (C) (a b) (a + b+ 1) > 0 (D) (a+ b) (a b + 1) > 0

14

2. If x @ y =

++

+

y3y4x

x

3x4y , then the value

of 6 @ 61

is

(A) 36

3891 (B) 1

(C) 36

4179 (D)

361297

3. The operations * and are defined as a * b = a + b + ab and a b = ab (a + b). Find the value of (3 * 4) (4 5). (A) 19 (29) (113) (B) (85) (30) (C) 19 (180) (199) (D) 84 (29) (113)

Numbers


1. Raju wrote the first 50 natural numbers one after another on a black board. He then carried out the following procedure 49 times. In each instance, he erased two numbers, say p and q and replaced them by a single number p + q 1. Find the final number left on the board.

(1) 1224 (2) 1275 (3) 1276 (4) 1274 (5) 1226

2. Find the last two digits of 73024. (1) 41 (2) 81 (3) 21 (4) 01 (5) 61

3. N is a natural number. Div(N), a function of N, is defined as follows:

Div(N) = N if N 9 = Div(S(N)), otherwise, where S(N) is the sum of the digits of N. For instance, Div(8) = 8, Div(625) = Div(6 + 2 + 5) = Div(13) = Div(1 + 3) = Div (4) = 4 etc.

Find the number of positive integer values of N less than 600, for which Div (N) = 9.

(1) 44 (2) 55 (3) 66 (4) 77 (5) 67

4. Find the value of 2222 31

211

21

111 +++++ +

upto 2005 terms

(1) 2005 2006

1 (2) 2006

20061

(3) 2006 2005

1 (4) 2005

20051

(5) 2006 2007

1

5. Set A = {3, 4, .. 2N + 1, 2N + 2}, where N is a natural number. Each odd element in it was increased by 3 and each even element in it was increased by 1. P denotes the average of the resulting odd elements and Q denotes the average

of the resulting even elements, then P Q =

(1) 2 (2) 1 (3) 1 (4) N (5) N1

6. X and Y are natural numbers. X is odd and less than 100. Find the number of solutions of

Y3

181

X1

= .

(1) 4 (2) 3 (3) 2 (4) 1 (5) 5

7. A tournament had 2N + 1 teams t1, t2, .. t2N+1 where N > 6. Each team had x players where x > 4. The following pairs of teams have a common player : t1 and t2N+1, t2 and t2N, .. tN and tN+1. These are the only pairs of teams who have a common player. Find the total number of players in the 2N + 1 teams.

(1) N(x 1) + x (2) N(2x 1) + x (3) N(2x 2) + x (4) N(x + 1) + x (5) N(x + 2) + x

8. A four digit number has the form AABB. It is also a perfect square. How many (A, B) are possible?

(1) 0 (2) 3 (3) 4 (4) 1 (5) 2

9. Let M = 3(3!) + 4(4!) + .. + 15(15!) What is the remainder when M 15 is divided by 14! 2?

(A) 14! 443 (B) 14! 459 (C) 459 (D) 443

10. A number when divided by 5, 6, 7 and 8 leaves remainders of 3, 4, 5 and 6 respectively. How many such 4-digit numbers are there?

(A) 11 (B) 10 (C) 9 (D) 12

11. If x = ....41

31

21

222 +++ , what is the value of

....

71

51

31

222 +++ in terms of x?

(A) 4

1x3 (B)

4x3

1 (C) 2

1x (D)

2x

1

12. If k = 79798080

42494249

+

, then

(A) 0 < k 1 (B) 1 < k 4 (C) 4 < k 7 (D) k > 7

13. If A = 71421 . 98 105 112 ...189 196, what is the remainder when A is divided by 9?

(A) 1 (B) 3 (C) 7 (D) 5

14. If 2a + 5b = 7(a2b5)1/7 and x = 2

ba +, what is

the value of 2(x + a)2 + 5(x b)5? (A) 7 (B) 168 (C) 3 (D) None of these

15. Which of the following is a rational number? (A) (7 + 4 3 )50 + (7 4 3 )50 (B) (7 4 3 )50 + (7 + 4 3 )50

(C) 734log34 + 347log7 (D) None of these

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16. Which of the following is prime? (A) 270 + 1 (B) 296 + 1 (C) 2160 + 1 (D) None of these

17. A 200 page book is formed by using 50 sheets, folded in the middle and stapled along the fold, with each sheet providing 4 pages. The pages are then numbered from 1 to 200. The nth sheet is removed from the book. What is the sum of the pages on this sheet?

(A) 400 2n (B) 400 n (C) 200 + 4n (D) None of these

18. If u + v + w + x + y = 15, what is the maximum value of uvx + uvy + uwx + uwy?

(A) 125 (B) 144 (C) 3125 (D) 243

19. The remainder when 130110 1301 is divided by 21 is

(A) 1 (B) 2 (C) 19 (D) 20

20. If a, b c, d are natural numbers such that 23 < a < b < 40 < c < d < 50, how many values are possible for the quadruplet (a, b, c, d)?

(A) 4284 (B) 4320 (C) 4200 (D) 4165

21. A number N leaves a remainder of 7, 10 and 17 respectively when divided by 15, 21, and 35. What is the remainder when N is divided by 105?

(A) 73 (B) 59 (C) 52 (D) 55

22. How many five digit numbers with digits that are not necessarily distinct are divisible by 6 but not by 12?

(A) 7500 (B) 7499 (C) 14999 (D) 15000

23. Which of the following numbers does not divide (412 1)?

(A) 5 (B) 63 (C) 255 (D) 127

24. Find the number of factors of 243243 which are multiples of 21.

(A) 20 (B) 21 (C) 22 (D) 24

25. What is the remainder when 4911 + 5011 + 5111 + 5211 is divided by 202?

(A) 0 (B) 101 (C) 201 (D) 1

26. Q is the number formed by knocking off all the terminal zeros from 256!. What is the index of the highest power of 12 that divides Q?


27. Let N = (31)(32)(33)(98)(99)(100) If N is divisible by 12n where n is a natural number, find the maximum value that 'n' can take.

(A) 30 (B) 34 (C) 35 (D) 40

28. 'a' and 'b' are prime numbers and n is an integer such that 1 n ab. What is the sum of all the possible values of n such that n and ab are coprime to each other?

(A) (a 1) (b 1) (B) 2

1baab +

(C) ab (D)2

)1b)(1a(ab

29. Set A = {1, 4, 7, 10, 13, , 100}. B is a non empty proper subset of A such that all the elements of B are divisible by 7. Find the number of such subsets.

(A) 5 (B) 31 (C) 32 (D) 63

30. There are 4 distinct integers p, q, r and s, such that the equation (n - p) (n - q) (n r) (n s) = 9 is satisfied for some integral values of n. How many integral values of n satisfy the given equation for a particular set of values of p, q, r, and s?

(A) 0 (B) 1 (C) 2 (D) More than 2

31. Find the remainder when (3333344444 + 4444433333) is divided by 7 (A) 0 (B) 2 (C) 3 (D) 6

32. Find the remainder when 4 + 44 + 444+ 4444+ ... + 444 . . . (50 digits) is divided by 9. (A) 0 (B) 3 (C) 6 (D) 8

33. Consider the equation 61

b1

a

1=+ , where a and b

are integers. How many ordered pairs (a, b) exist which satisfy the given relation? (A) 9 (B) 10 (C) 17 (D) 18

34. The difference between the squares of two integers is 420. How many such pairs of integers exist? (A) 4 (B) 8 (C) 12 (D) 16

35. How many 9 digit multiples of 6 can be formed using only the digits 8 or 9? (A) 56 (B) 84 (C) 86 (D) None of these.

36. (a + b + c + d)5 (a5 + b5 + c5 + d5) is always divisible by

(A) 24 (B) a + b + c + d (C) 9 (D) 5

37. If A = 1! + 2! + 3! + 4! + .......+ 49! and B = 1! + 2! + 3! + 4! +......+168!, then which of the following is true? (A) A is a perfect square. (B) (B) is a perfect square. (C) Both A and B are perfect square. (D) Neither A nor B is a perfect square.

38. What is the sum of the even factors of 3600? (A) 4030 (B) 12896 (C) 6046 (D) 12090

39. Consider all 5-digit numbers for which the sum of the digits is 41. How many of these are divisible by 11? (A) 10 (B) 12 (C) 16 (D) More than 16

40. In how many ways is it possible to express 36 as a product of 3 positive integers? (A) 8 (B) 6 (C) 12 (D) None of these

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41. W is a whole number. The function root(W) is defined as follows root (W) = W if W 2 = root (W 3) if 3 W 9 = root (sum (W)) otherwise, where sum (W) denotes the sum of the digits of W For example Root (7) = root (7 3) = root (4) = root (4 3) = root (1) = 1 Root (245) = root (2 + 4 + 5) = root (11) = root (1 + 1)= root (2) = 2 How many values of W less than 500 satisfy the condition root (w) = 0? (A) 165 (B) 166 (C) 167 (D) 168

42. The 18-digit number X has a 7 in the units place. If this 7 is transposed to the front of the number, the resulting 18-digit number is 5 X. Find X. (A) 124587124587124587 (B) 124857124857124857 (C) 142857142857142857 (D) 142587142587142587

43. How many numbers less than or equal to 1108 are coprime to 252? (A) 311 (B) 312 (C) 316 (D) 318

Directions for questions 44 to 46: These questions are based on the following data.

Professor Calculus devised a game to pass his time. The game was played in the following manner. He kept 500 coins numbered from 1 to 500, on a table in sequence all showing heads.

In round 1, he flipped all the coins such that each showed a tail. In round 2, he flipped only those coins whose position was a number divisible by 2. In round 3, he flipped only those coins whose position was a number divisible by 3 and so on. He continued flipping the coins for 500 rounds.

44. What was the greatest number of consecutive coins all showing heads at the end of the game? (A) 16 (B) 41 (C) 42 (D) 43

45. Among the following coins, which coin was flipped the greatest number of times? (A) 210 (B) 324 (C) 288 (D) 240

46. How many coins did Professor Calculus flip in round 13, which were flipped only once in the previous rounds? (A) 7 (B) 8 (C) 11 (D) 38


47. If the sum of all integers between 3n and 3n+3(both exclusive) is divisible by 70 where n is a positive integer, then which of the following is necessarily true about n?

(A) It is divisible by 4 (B) It is a multiple of 6 (C) It is divisible by 7 (D) It is divisible by 5.

48. Find the units digit in 13 + 23 + 33+ +903. (A) 0 (B) 5 (C) 6 (D) 1

49. If S(N) denotes the sum of the digits of N, then what is the remainder when S(1) + S(2) + S(3) + ..+ S(99) is divided by 99?

(A) 0 (B) 9 (C) 18 (D) 36

Probability


1. There are 50 tokens numbered 1 to 50 kept on a table. You are asked to pick two tokens at random from the table one after the other. What is the probability that the difference between the numbers on the tokens picked by you lies between 1 and 6 (both excluded)?

(A) 24536

(B) 1225186

(C) 1225372

(D) 2450186

2. A number is selected at random from 1 to 105. Consider the following 3 events.

P: X is coprime to 3` Q: X is coprime to 5 R: X is coprime to 7

Which of the following is true about the events P, Q, R? (A) They are pairwise independent and mutually

exclusive. (B) The are pairwise independent but not

mutually exclusive. (C) They are not pairwise independent but are

mutually exclusive. (D) They are neither pairwise independent nor

mutually exclusive.

3. Alsi Arora worked in a private company where the working hours were from 9:00 am to 6 pm. In a year on 25% of the days he arrived late to office whereas on 35% of the days he left early from office. If p is the probability of the number of days that he worked for the entire working day, then which of the following best describes the possible values of p? (A) P = 0.4875 (B) 0.25 P 0.65 (C) 0.25 P 0.65 . (D) 0.40 P 0.65

4. Ramesh doesn't always have time to read the daily newspaper and whenever he reads it he does it either in the morning or in the evening. The probability that he reads it in the morning is 10% and the probability that he reads it in the evening is 30%. Find the expected number of days in a month of 30 days on which he doesn't read the paper at all. (A) 16 (B) 18.9 (C) 21 (D) 20.5

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Miscellaneous


1. There is a sequence of 7 sets A1, A2, A4, each consisting of 6 elements and another sequence of n sets B1, B2, ,,,, Bn each consisting of 3 elements. The union of all the A's is equal to the union of all the B's. If each element of this union occurs in exactly 3 of the A sets and 6 of the B sets, then find n. (A) 28 (B) 21 (C) 14 (D) Cannot be determined

Coordinate Geometry


1. If the equation 9x2 y2 + 6y = A where A is a constant, represents a pair of straight lines, then which of the following gives the point of intersection of those two lines? (A) (1, 2) (B) (0, 3) (C) (0, 3) (D) cannot be determined

Inequations


1. A triangle with sides x, y and z is such that x3 + y3 +z3 = 3xyz. Which of the following is true regarding the triangle? (A) It is an isosceles triangle (B) It is an equilateral triangle (C) It is an obtuse angled triangle (D) It is a right angle triangle

Statistics


1. The median of 11 numbers is x. If the greatest number is removed the median of the remaining numbers is 10.5. If the smallest is removed the median of the remaining numbers is 13. If x (or one of the numbers which is equal to x) is removed the median would be

(A) 11 (B) 11.5 (C) 12 (D) Cannot be determined

122important questions quant

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