MBA - Masters In Business Administration FAQ's: XAT - O B J E C T I V E -T Y P E Q U E S T I O N S QUANTITATIVE APTITUDE

Directions: In each question below, choose the correct

alternative from the four options provided.

1. Last year Mr Basu bought two scooters. This year

he sold both of them for Rs 30,000 each. On one, he earned

20% profit, and on the other he made a 20% loss. What was

his net profit or loss?

(A) He gained less than Rs 2000

(B) He gained more than Rs 2000

(D) He lost more than Rs 2000

2. 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.

(A)

( – )

z x

y x

(B)

( – )

x z

y z

(C)

( – )

y x

z y

(D)

( – )

y z

x z

3. Three circles A, B and C have a common centre O.

A is the inner circle, B middle circle and C is outer circle. The

radius of the outer circle C, OP cuts the inner circle at X and

middle circle at Y such that OX = XY = YP. The ratio of the

area of the region between the inner and middle circles to

the area of the region between the middle and outer circle

is:

(A)

3 (B)

(C)

5 (D)

4. 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

5. If Sn denotes the sum of the first n terms in an

Arithmetic Progression and S1 : S4 = 1 : 10 then the ratio of

first term to fourth term is:

(A) 1 : 3 (B) 2 : 3

6. The curve y = 4x2 and y2 = 2x meet at the origin O

Questions asked in XLRI Examination held on January 9, 2005

and at the point P, forming a loop. The straight line OP

divides the loop into two parts. What is the ratio of the areas

of the two parts of the loop?

(A) 3 : 1 (B) 3 : 2

7. How many numbers between 1 to 1000 (both

excluded) are both squares and cubes?

(A) none (B) 1

8. An operation ‘$’ is defined as follows:

For any two positive integers x and y,

x$y = +

FH G

IK J

x then which of the following is an

integer?

(A) 4$9 (B) 4$16

9. If f(x) = cos(x) then 50th derivative of f(x) is:

(A) sin x (B) – sin x

10. If a, b and c are three real numbers, then which of

the following is NOT true?

(A) a+b ≤ a + b

(B) a–b≤a+b

(D) a–c≤a–b+b−c

11. If R = {(1, 1), (2, 2), (1, 2), (2, 1), (3, 3)} and

S = {(1, 1), (2, 2), (2, 3), (3, 2), (3, 3)} are two relations in the

set X = {1, 2, 3}, the incorrect statement is:

(A) R and S are both equivalence relations

(B) R∩S is an equivalence relations

(C)R−1∩S−1is an equivalence relations

(D) R∪S is an equivalence relations

12. If x > 8 and y > – 4, then which one of the following

is always true?

(A) xy <>

(B) x2 < – y

(D) x > y

13. For n = 1, 2, .... let Tn = 13 + 23 + ... + n3, which one

of the following statements is correct?

621 ◆ FEBRUARY 2006 ◆ THE COMPETITION MASTER

O B J E C T I V E -T Y P E Q U E S T I O N S

(A) There is no value of n for which Tn is a positive

power of 2.

(B) There is exactly one value of n for which Tn is a

positive power of 2.

a positive power of 2.

(D) There are more than two values of n for which Tn

is a positive power of 2.

14. 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:

(A) 256 3 sq cm

(B)

256

3 sq cm

(C)

3 sq cm

(D) 64 3 sq cm

15. The length of the sides of a triangle are x + 1,

9 – x and 5x – 3. The number of values of x for which the

triangle is isosceles is:

(A) 0 (B) 1

16. The expression

x x a b

2 2 2

– + +

+ + +

lies between:

(A)

a b

and

a b

2 2

+ +

+ −

– + +

(B) a and b

(C)

a b

and

2 2

+ +

+ –

(D)

a b

2 2

+ −

+ +

17. What is the sum of first 100 terms which are

common to both the progressions

17, 21, 25, ... and 16, 21, 26, .... :

(A) 100000 (B) 101100

18. Two people agree to meet on January 9, 2005

between 6.00 P.M. to 7.00 P.M., with the understanding that

each will wait no longer than 20 minutes for the other. What

is the probability that they will meet?

(A)

9 (B)

(C)

9 (D)

19. If the roots of the equation x a

x a c

x b

x b c

+ +

= 1

are equal in magnitude but opposite in sign, then:

(A) c≥a (B) a≥c

20. Steel Express runs between Tatanagar and Howrah

and has five stoppages in between. Find the number of

different kinds of one-way second class ticket that Indian

Railways will have to print to service all types of passengers

who might travel by Steel Express?

(A) 49 (B) 42

21. The horizontal distance of a kite from the boy flying

it is 30 m and 50 m of cord is out from the roll. If the wind

moves the kite horizontally at the rate of 5 km per hour

directly away from the boy, how fast is the cord being

released?

(A) 3 km per hour

(B) 4 km per hour

(D) 6 km per hour

22. Suppose S and T are sets of vectors, where

S = {(1,0,0), (0, 0, –5), (0, 3, 4)} and T = {(5, 2, 3), (5, –3, 4)}

then:

(A) S and T both sets are linearly independent vectors

(B) S is a set of linearly independent vector, but T is not

not

(D) Neither S nor T is a set of linearly indepedent

vectors

23. Suppose the function ‘f’ satisfies the equation

f (x + y) = f(x) f(y) x and y. f(x) = 1 + xg(x) where

x→0

lim g(x) = T, where T is a positive integer. If fn (x) = kf(x)

then k is equal to:

(A) T (B) Tn

24. Set of real numbers ‘x, y’, satisfying, inequations

x – 3y ≥ 0, x + y ≥ –2 and 3x – y≤ – 2 is:

(A) Empty (B) Finite

25. ABCD is a trapezium, such that AB, DC are parallel

and BC is perpendicular to them. If ∠DAB=45°, BC = 2 cm

and CD = 3 cm then AB = ?

(A) 5 cm (B) 4 cm

26. If F is a differentiable function such that F(3) = 6

and F(9) = 2, then there must exist at least one number ‘a’

between 3 and 9, such that:

(A) F’(a) = 3

2 (B) F(a) = – 3

2 (D) F’(a) = – 2

27. A conical tent of given capacity has to be

622 ◆ FEBRUARY 2006 ◆ THE COMPETITION MASTER

O B J E C T I V E -T Y P E Q U E S T I O N S

constructed. The ratio of the height to the radius of the base

for the minimum amount of canvas required for the tent is:

(A) 1 : 2 (B) 2 : 1 (C) 1: 2 (D) 2:1

28. If n is a positive integer, let S(n) denote the sum of

the positive divisors of n, including n and G(n) is the greatest

divisor of n. If H(n) =

G n

S n

( )

then which of the following is

the largest?

(A) H(2009) (B) H(2010)

29. If the ratio of the roots of the equation

x2 – 2ax + b = 0

is equal to that of the roots

x2 – 2cx + d = 0, then:

(A) a2b = c2d (B) a2c = b2d

30. X and Y are two variable quantities. The

corresponding values of X and Y are given below:

X : 3 6 9 12 24

Y : 24 12 8 6 3

Then the relationship between X and Y is given by:

(A) X+Y∝X–Y

(B) X Y

X Y

+ ∝ 1

–

(D) x

∝ 1

Read the following and answer questions 31 to 34

based on the same.

Eight sets A, B, C, D, E, F, G and H are such that

A is a superset of B, but subset of C.

B is a subset of D, but superset of E.

F is a subset of A, but superset of B.

G is a superset of D, but subset of F.

H is a subset of B.

N(A), N(B), N(C), N(D), N(E), N(F), N(G) and N(H) are

the number of elements in the sets A, B, C, D, E, F, G and H

respectively.

31. Which one of the following could be FALSE, but

not necessarily FALSE?

(A) E is a subset of D

(B) E is a subset of C

(D) E is a subset of H

32. If P is a new set and P is a superset of A and N(P)

is the number of elements in P, then which of the following

must be true?

(A) N(G) is smaller than only four numbers

(B) N(C) is the greatest

(D) N(P) is the greatest

33. If Q and Z are two new sets superset of H and

N(Q) and N(Z) is the number of elements of the sets Q and

Z respectively, then:

(A) N(H) is the smallest of all

(B) N(E) is the smallest of all

(D) Either N(H) or N(E) is the smallest

34. Which of the following could be TRUE, but not

necessarily TRUE?

(A) N(A) is the greatest of all.

(B) N(G) is greater than N(D).

(D) N(F) is less than or equal to N(H).

35. If x + y + z = 1 and x, y, z are positive real numbers,

then the least value of ( – )( – )( – )

x y z

is:

(A) 4 (B) 8

36. 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)

(D) x2 + y2 = 16

37. Two players A and B play the following game. A

selects an integer from 1 to 10, inclusive of both. B then adds

any positive integer from 1 to 10, both inclusive, to the

number selected by A. The player who reaches 46 first wins

the game. If the game is played properly, A may win the

game if:

(A) A selects 8 to begin with

(B) A selects 2 to begin with

(D) None of the above

Read the following and answer questions 38 and 39

based on the same:

The demand for a product (Q) is related to the price (P)

of the product as follows:

Q = 100 – 2P

The cost (C) of manufacturing the product is related to

the quantity produced in the following manner:

C = Q2 – 16Q + 2000

As of now the corporate profit tax rate is zero. But the

Government of India is thinking of imposing 25% tax on the

profit of the company.

38. As of now, what is the profit-maximizing output?

(A) 22 (B) 21.5

39. If the government imposes the 25% corporate profit

tax, then what will be the profit maximizing output?

(A) 16.5 (B) 16.125

40. If X =

+ +

a a a

n ( ( )

...

1 r) 1r2 (1r)

, then what is

the value of a + a (1 + r) + ... + a(1 + r)n–1?

623 ◆ FEBRUARY 2006 ◆ THE COMPETITION MASTER

O B J E C T I V E -T Y P E Q U E S T I O N S

(A) X [(1 + r) + (1 + r)2 + ... + (1 + r)n]

(B) – X (1 + r)n

(1+ r) –1

(D) X (1 + r)n–1

41. The first negative term in the expansion (1+2x)7

is the:

(A) 4th term (B) 5th term

42. The sum of the numbers from 1 to 100, which are

not divisible by 3 and 5.

(A) 2946 (B) 2732

Read the following and answer questions 43 to 47

based on the same.

Five numbers A, B, C, D and E are to be arranged in an

array in such a manner that they have a common prime

factor between two consecutive numbers. These integers are

such that:

A has a prime factor P

B has two prime factors Q and R

C has two prime factors Q and S

D has two prime factors P and S

E has two prime factors P and R

43. Which of the following is an acceptable order, from

left to right, in which the numbers can be arranged?

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

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

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

44. If the number E is arranged in the middle with

two numbers on either side of it, all of the following must

be true, EXCEPT:

(A) A and D are arranged consecutively

(B) B and C are arranged consecutively

(D) A is arranged at one end in the array

45. If number E is not in the list and the other four

numbers are arranged properly, which of the following must

be true?

(A) A and D can not be the consecutive numbers.

(B) A and B are to be placed at the two ends in the

array.

(D) C and D can not be the consecutive numbers.

46. If number B is not in the list and other four numbers

are arranged properly, which of the following must be

true?

(A) A is arranged at one end in the array.

(B) C is arranged at one end in the array.

(D) E is arranged at one end in the array.

47. If B must be arranged at one end in the array, in

how many ways the other four numbers can be arranged?

(A) 1 (B) 2

Questions 48 to 50 are followed by two statements

labelled as (1) and (2). You have to decide if these statements

are sufficient to conclusively answer the question. Give

answer:

(A) If statement (1) alone or statement (2) alone is

sufficient to answer the question

(B) If you can get the answer from (1) and (2) together

but neither alone is sufficient

alone is also sufficient

(D) If neither statement (1) nor statement (2) is sufficient

to answer the question

48. Around a circular table six persons A, B, C, D, E

and F are sitting. Who is on the immediate left to A?

Statement 1: B is opposite to C and D is opposite to E

Statement 2: F is on the immediate left to B and D is to the

left of B

49. A, B, C, D, E are five positive numbers.

A + B <>

Is ‘A’ the greatest?

Statement 1: D + E <>

Statement 2: E <>

50. A sequence of numbers a1, a2 ..... is given by the

rule an

2 = an+1. Does 3 appear in the sequence?

Statement 1: a1 = 2

Statement 2: a3 = 16

ANSWERS AND EXPLANATIONS

1. (D) The actual calculations for such a problem are

too lengthy

By the direct approach, % Loss =

100

= 4

Actual loss = Rs 60,000 × 4%

= Rs 2400 and (2400 > 2000)

2. (A) Again, for this problem, direct approach (allegation

diagram) can be used

Let, x > y

z > y

Total = z – y + x – z = x – y

∴ % failed =

failed

total

x z

x y

×100=

–

z x

y x

–

3. (C) Area of circle = ðr2

∴ Required ratio

ð ð

( ) – ( )

3 2

2 2

x x

4 1

9 4

( – )

( – ) = 3

x y

z - y : x - z

Pass Fail

Ratio =

x x x

624 ◆ FEBRUARY 2006 ◆ THE COMPETITION MASTER

O B J E C T I V E -T Y P E Q U E S T I O N S

4. (B) From the adjoining diagram,

x + y = 90°

x – y = 45°

x = 67.5° and y = 22.5°

Consider Ä ABC

sinA = b

sinB

= c

sinC

Thus, area of rhombus = 2 2 cm2

5. (C) Use Sn = n

[2a + (n –1)d] and Tn = a + (n – 1) d

= 1

a d]

[2 3

6a = 6d or a = d

∴

a 3d

= a

= 1

6. (D) The curves y = 4x2 and y2 = 2x meet at x = 0 and

x = 1

(Solve simultaneously)

At x = 1

, y = 1

Equation of OP = y = 2x – 2

Ratio of areas = A

= =

area between y x andy x

area betweeny x z andy x

2 2 4

2 2

– 2

–

Now, for A1

Put 2x – 2 = 4x2 ∴ x =

2 , x = 1

and for A2

Put 2x – 2 = 2x ∴ x =

2 , x = 2

∴ Ratio =

z z

( – ) – ( )

2 2 4

2 2 2

1 2

x dx x dx

x dx xdx

= 1 : 1

7. (B) * Try with whole cubes as they are fewer in number

43 = 64 and 82 = 64

8. (D) By direct substitution.

9. (D)

d y

2 ,

d y

3 and

d y

4 are respectively:

– sin x, – cos x, sin x and cos x

After this, there is repetition of values.

For 50th derivative,

= 12 2

Remainder = 2

i.e. 50th derivative = same as

d y

2 = – cos x

10. (C)

11. (A) An equivalence relation is reflexive, symmetric and

transitive.

12. (C) Here x = 9, 10, 11 .... ∞

y = – 3, – 2, – 1, 0, 1, 2, 3, .... ∞

13. (A)

n(n+1)

2 = odd x even no. ≠ 2x

14. (B) Required area =

[162 + (16)

2+ (16)

2 + ... ∞]

and sum of GP = a

1– r (when n = ∞)

∴ Area = 3

[

–

] =

256 3

15. (D) Equate any 2 values and solve.

16. (C) Let,

x x a b

2 2 2

– ( )

( )

+ +

+ + +

= m

This becomes a quadratic equation when

discriminant, D ≥ 0

17. (B) Sn =

2 [2a + (n – 1)d]

Common terms are 21, 41, 61, etc., d = 20

∴ Sn = 100

[2 × 21 + (100 – 1)20]

= 101100

18. (D) They can meet when A comes between 6 : 00 = 6 : 40.

and so B can join him between 6 : 20 = 7 : 00

Similarly, the process can be reversed

∴ Required p = (

min

)

utes 2

utes

19. (C) a = – b, or a + b = 0

Use discriminant, D = b2 –4ac

20. (B) We have 5 stations + (T + H) = 7 stations

Out of the 7 stations, we have to print tickets

connecting any 2; i.e. arrangements of 7 things, any

2 at a time, i.e. No. of tickets = 7P2 = 42

21. (D)

= 5

> 1, i.e. > 5, i.e. 6

y = 4x2

y2 = 2x

2y C

2 2

B a

67.5° C

b=2

22.5°

625 ◆ FEBRUARY 2006 ◆ THE COMPETITION MASTER

O B J E C T I V E -T Y P E Q U E S T I O N S

22. (D)

23. (C)

24. (D) x – 3y ≥ 0 x + y ≥ – 2 3x – y ≤ – 2

4x ≥ – 4 (from equations 2 and 3)

x ≥ – 2

x ≤ 2

25. (A) Draw DX. As can be seen easily,

AX = DX (IsoscelesÄ ).

∴ AX = 2

AB = 2 + 3 = 5 cm

26. (D) From Lagrange’s mean value theorem,

there is c in (a, b), such that:

f b f a

b a

( )– ( )

– = f’(c)

Here, f(a) = f(3) = 6

f(b) = f(9) = 2

f’(c) or f’(a) = −

−

2 6=

9 3

–

27. (D) v =

ðr2h for a cone

or h =

ðr

....... equation 1

Amount of canvas = curved area = S

= ðr l = ðr(r2 h2)

+ 2

S2 = ð2r2 (r2 + h2) = ð2r2 (r )

2 4

9 +

Let S2 = z

dr = ð2 (4 – )

3 18

2 3 r

ð r

and

d z

2 = ð2(12 2 54 )

r 2v4

Put

= 0

4r3 –

18 2

2 3

ð r = 0

2r6 =

9 2

9v2 = 2ð2 r6

∴ d z

= ð2 (12r2 + 12 2 6

2 4

) = ð2 (24r2)

= positive quantity

z (i.e. S2) has minimum value if 9v = 2ð2r6

D(0,2) C(2,2)

A(0,0) B(2,0)

i.e. 9( )

ðr2h

= 2ð2 r6 i.e. h2 = 2r2 i.e.

= 2

* Such answers must be tabulated and learnt for ready

reference.

28. (C) Gn = n, Hn = n

29. (C) x = a�} a2 –b = a+ a2 –b and a− a2 –b

and y = c�} c2 –d = c+ c2 –d and c− c2 –d

a a b

+ 2−

– 2–

c c d

+ 2−

– 2–

By reversing componendo and dividendo,

a2 −b

c2 −d

Squaring,

a b

2 –

c d

2 –

, i.e. a2d = bc2

30. (D)

31. (D) Given B<A≤C, E<B≤D <G≤F

B<F≤A

H <>

From this information,

H<B<F≤A≤C ..... (1)

(<Q,Z) (A<P)

and E<B≤D<G≤F ..... (2)

A is false from (2), B from (1), C from (1 and 2)

D “may be” true

32. (D) From 1 and 2

33. (D) From 1 and 2

34. (C) B is true, A and D are false, C “may be” true

35. (D) In any case, since x, y, z >1,

1 1 1

x y z

, , < (i.e. negative)

(–) × (–) × (–) = – (negative quantity)

36. (B) The ends of diameter AC are:

A(0, 0) and C(2, 2)

Equation of the circle with

ends of diameter as (x1, y1)

and (x2, y2) is:

(x – x1) (x – x2) + (y – y1) (y – y2) = 0

(x – 0) (x – 2) + (y – 0) (y – 2) = 0

x2 – 2x + y2 – 2y = 0

x2 + y2 = 2(x + y)

A B

D C

2 X 3

45°

2 2

45°

]given

626 ◆ FEBRUARY 2006 ◆ THE COMPETITION MASTER

O B J E C T I V E -T Y P E Q U E S T I O N S

37. (D) Since repetition of numbers is allowed, both are

equally free to win the game

38. (C) C = Q2 – 16Q + 200.

Put Q = 100 –2P

C = 4P2 – 368P + 8600

and Profit = Price – Cost

= (100–2P) – (4P2 + 368P + 8600)

Differentiating, – 8P = 366

P =

366

8 and Q = 100 – 2P, etc.

39. (C)

40. (B) X

r r r n =

1 + −

1 1 [ ....

( )

]

= a

[

( ) –

( )

]

1 1

and Sn = a + a(1 + r) + ... = a

r [1 – (1 + r)n], using GP

∴

n = –(1 + r)n

Sn = – X (1 + r)n

41. (C) Let x = (1+2x)7 =(1+2)

x 2

Using Binomial expansion, we have:

x = 1+

2 7

.x+ ( –1) (2x)2+.......

till

( –4)(2x)5

Negative term will come when we have

2 <>

i.e. n = 4. This happens with the 6th term

42. (D) Sum of all numbers,

S = 100

[1+100] = 5050 .... using AP

Similarly, sum of multiples of 3,

S3 = 33

2 [3 + 99] = 1683 Required sum

Similarly, sum of multiples of 5, = 5050–1683–1050

S5 = 20

2 [5 + 100] = 1050 = 2317

43. (C) We have 3 options:

No. 1 A D E B C

OR D A E B C

P P

B A

No. 2 A E B C D

No. 3 A E D C B

44. (D) See option (1) [a ✓ b ✓ c ✓ d ×]

45. (B) A—D—C—B

46. (C) E—A—D—C

P P

OR A—E—D—C

47. (B) B—C—D—E— A

P P

OR B—E—A—D—C

P P

48. (B)

49. (B) A + B <>

B + C <>

C + D <>

D + E <>

E <>

Adding, A + 2B <>i.e. B <>

50. (C) Put n = 1 in an

2 = an+1

2 = a2, a2

2 = a3, a3

2 = a4, etc

From statement 1: a1

2 = a2

i.e. 22 = a2 or a2 = 4

Now, a2

2 = a3

i.e. 42 = a3 or a3 = 16, etc

Thus, a1 = 2, a2 = 4, a3 = 16, etc

— —

MBA - Masters In Business Administration FAQ's

XAT - O B J E C T I V E -T Y P E Q U E S T I O N S QUANTITATIVE APTITUDE

0 comments:

Blog Archive

Infolinks In Text Ads

Labels