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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
(C) He lost less 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)
1
3 (B)
2
5
(C)
3
5 (D)
1
5
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
(C) 3 2 sq cm (D) 4 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
(C) 1 : 4 (D) 1 : 5
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
(C) 2 : 1 (D) 1 : 1
7. How many numbers between 1 to 1000 (both
excluded) are both squares and cubes?
(A) none (B) 1
(C) 2 (D) 3
8. An operation '$' is defined as follows:
For any two positive integers x and y,
x$y = +
F
HG
I
KJ x
y
y
x then which of the following is an
integer?
(A) 4$9 (B) 4$16
(C) 4$4 (D) None of the above
9. If f(x) = cos(x) then 50th derivative of f(x) is:
(A) sin x (B) � sin x
(C) cos x (D) � cos 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 � ≤ +
(C) 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 S − − ∩1 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 < 0
(B) x2 < � y
(C) � x < 2y
(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.
(C) There are exactly two values of n for which Tn is
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
3 sq cm
(C)
64
3
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
(C) 2 (D) 3
16. The expression
x x a b
x x a b
2 2 2
2 2 2
2
2
� + +
+ + +
lies between:
(A)
a b
a b
and
a b
a b
2 2
2 2
2 2
2 2
1
1
1
1
+ +
+
+ −
+ + �
(B) a and b
(C)
a b
a b
and
2 2
2 2
1
1
1
+ +
+ �
(D)
a b
a b
2 2
2 2
1
1
+ −
+ +
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
(C) 111000 (D) 100110
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)
5
9 (B)
7
9
(C)
2
9 (D)
4
9
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 ≥
(C) a + b = 0 (D) a = b
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
(C) 21 (D) 7
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
(C) 5 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
(C) T is a set of linearly independent vectors, but S is
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
(C) log T (D) (log T)n
24. Set of real numbers 'x, y', satisfying, inequations
x � 3y ≥ 0, x + y ≥ �2 and 3x � y≤ � 2 is:
(A) Empty (B) Finite
(C) Infinite (D) Cannot be determined
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
(C) 3 cm (D) 2 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
(C) F'(a) = � 3
2 (D) F'(a) = � 2
3
27. A conical tent of given capacity has to be
A
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)
(C) H(2011) (D) H(2012)
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
(C) a2d = c2b (D) d2b = c2a
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
�
(C) X Y ∝
(D) x
Y
∝ 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
(C) E is a subset of A
(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
(C) N(B) is the smallest
(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
(C) N(C) is the greatest 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).
(C) N(H) is the least of all.
(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 ( � )( � )( � )
1
1
1
1
1
1
x y z
is:
(A) 4 (B) 8
(C) 16 (D) None of the above
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)
(C) x2 + y2 = 4
(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
(C) A selects any number greater than 5
(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
(C) 20 (D) 19
39. If the government imposes the 25% corporate profit
tax, then what will be the profit maximizing output?
(A) 16.5 (B) 16.125
(C) 15 (D) None of the above
40. If X =
+
+
+
+ +
+
a a a
n ( ( )
...
( ) 1 1 1 2 r) r r
, 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
(C) X
( ) � 1 1 + r
r
n
(D) X (1 + r)n�1
41. The first negative term in the expansion ( ) 1 2 7 + x
is the:
(A) 4th term (B) 5th term
(C) 6th term (D) 7th term
42. The sum of the numbers from 1 to 100, which are
not divisible by 3 and 5.
(A) 2946 (B) 2732
(C) 2632 (D) 2317
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
(C) B, C, D, E, A
(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
(C) B and E 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.
(C) A and C 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.
(C) D 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
(C) 3 (D) 4
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
(C) If statement 1 alone is sufficient and statement (2)
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 < C + D, B + C < D + E, C + D < E + A.
Is 'A' the greatest?
Statement 1: D + E < A + B
Statement 2: E < C
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 =
x2
100
=
20
100
2
= 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
�
�
or
z x
y x
�
�
3. (C) Area of circle = πr2
∴ Required ratio
=
π π
π π
( ) � ( )
( ) � ( )
2
3 2
2 2
2 2
x x
x x
=
π
π
x
x
2
2
4 1
9 4
( � )
( � ) = 3
5
x y
z
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 ∆ ABC
a
A sin = b
B sin
= c
C sin
Thus, area of rhombus = 2 2 cm2
5. (C) Use Sn = n
2
[2a + (n �1)d] and Tn = a + (n � 1) d
S
S
1
4
= 1
10
=
+
a
a d]
4
2
2 3 [
6a = 6d or a = d
∴
T
T
1
4
=
+
a
a d 3
= a
a 4
= 1
4
6. (D) The curves y = 4x2 and y2 = 2x meet at x = 0 and
x = 1
2
(Solve simultaneously)
At x = 1
2
, y = 1
Equation of OP = y = 2x � 2
Ratio of areas = A
A
1
2
=
= =
= =
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 =
1
2 , x = 1
and for A2
Put 2x � 2 = 2x ∴ x =
1
2 , x = 2
∴ Ratio =
z z
z z
( � ) � ( )
( � ) � ( )
2 2 4
2 2 2
1
2
1 2
1
2
1
1
2
2
1
2
2
x dx x dx
x dx xdx
=
17
12
17
12
= 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)
dy
dx
,
d y
dx
2
2 ,
d y
dx
3
3 and
d y
dx
4
4 are respectively:
� sin x, � cos x, sin x and cos x
After this, there is repetition of values.
For 50th derivative,
50
4
= 12 2
4
Remainder = 2
i.e. 50th derivative = same as
d y
dy
2
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 =
3
4
[162 + ( ) 16
2
2+ ( ) 16
4
2 + ... ∞]
and sum of GP = a
r 1� (when n = ∞)
∴ Area = 3
4
16
1
1
4
2
[
�
] =
256 3
3
15. (D) Equate any 2 values and solve.
16. (C) Let,
x x a b
x x a b
2 2 2
2 2 2
2
2
� ( )
( )
+ +
+ + +
= m
This becomes a quadratic equation when
discriminant, D ≥ 0
17. (B) Sn =
n
2 [2a + (n � 1)d]
Common terms are 21, 41, 61, etc., d = 20
∴ Sn = 100
2
[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
min
)
40
60
2 utes
utes
=
4
9
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)
y
x
= 5
3
dy
dx
> 1, i.e. > 5, i.e. 6
A1
y = 4x2
A2
y = 2x 2
O
P
2
B
A
C 2y
2x
2
2 2
B a
A
C 67.5�
b=2
c
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 (Isosceles∆ ).
∴ 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
2
3
�
27. (D) v =
1
3
2 πr h for a cone
or h =
3
2
v
r π
....... equation 1
Amount of canvas = curved area = S
= πr l = πr( ) r h 2 2
1
2 +
S2 = π2 2 r (r2 + h2) = π2 2 r ( ) r
v
r
2
2
2 4
9 +
π
Let S2 = z
dz
dr = π2 ( � ) 4
18 3
2
2 3 r
v
r π
and
d z
dr
2
2 = π2( ) 12 54 2
2
2 4 r v
r
+
π
Put
dz
dr
= 0
4r3 �
18 2
2 3
v
r π = 0
2r6 =
9 2
2
v
π
9v2 = 2 2 π r6
∴ d z
dr
2
2
= π2 (12r2 + 12 2 6
2 4
π
π
r
r
) = π2 (24r2)
= positive quantity
z (i.e. S2) has minimum value if 9v = 2 2 6 π r
D(0,2) C(2,2)
A(0,0) B(2,0)
i.e. 9( )
πr h 2
2
3
= 2 2 π r6 i.e. h2 = 2r2 i.e.
h
r
= 2
* Such answers must be tabulated and learnt for ready
reference.
28. (C) Gn = n, Hn = n
Sn
29. (C) x = a a b � 2 � = a a b + 2 � and a a b − 2 �
and y = c c d � 2 � = c c d + 2 � and c c d − 2 �
a a b
a a b
+ −2
2 � �
=
c c d
c c d
+ −2
2 � �
By reversing componendo and dividendo,
a
a b 2 −
=
c
c d 2 −
Squaring,
a
a b
2
2 �
=
c
c d
2
2 �
, i.e. a2d = bc2
30. (D)
31. (D) Given B A C < ≤ , E B D < ≤ < ≤ G F
B
B F A < ≤
B
H < B
From this information,
H B F A C < < ≤ ≤ ..... (1)
( , ) 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
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
C D
X 3 2
45�
3
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
a
r r r n =
+
+
+
+
+ − 1
1
1
1
1
1 1 [ ....
( )
]
= a
r
[
( ) �
( )
]
1 1
1
+
+
r
r
n
n
and Sn = a + a(1 + r) + ... = a
r [1 � (1 + r)n], using GP
∴
S
X
n = �(1 + r)n
Sn = � X (1 + r)n
41. (C) Let x = + ( ) 1 2 7 x = + ( ) 1 2
7
2 x
Using Binomial expansion, we have:
x = 1+
7
2
2 7
2
7
2
1 2 2 . ( � ) ( ) ....... x x + +
till
7
2
7
2
4 2 5 ( � )( ) x
Negative term will come when we have
7
2 < n,
i.e. n = 4. This happens with the 6th term
42. (D) Sum of all numbers,
S = 100
2
[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
P
S
P
P
R
R
Q
Q
S
OR D A E B C
S
P P
P
R
R
Q
Q
S
A B
C
D
E
F
No. 2 A E B C D
P
P
R
R
Q
Q
S
S
P
No. 3 A E D C B
P
R
P
P
S
S
Q
Q
R
44. (D) See option (1) [a ✓ b ✓ c ✓ d �]
45. (B) A�D�C�B
P
P
S
S
Q
Q
R
46. (C) E�A�D�C
R
P P
P
S
S
Q
OR A�E�D�C
P
R
P
P
S
Q
S
47. (B) B�C�D�E� A
Q
R
S
Q
P
S
R
P P
OR B�E�A�D�C
Q
R
R
P P
P
S
S
Q
48. (B)
49. (B) A + B < C + D
B + C < D + E
C + D < E + A
D + E < A + B
E < C
Adding, A + 2B < 2A + B i.e. B < A
50. (C) Put n = 1 in an
2 = an+1
a1
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
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