Download quintessence

INFOMATHS
PUNE-2015
27
then find 13. The digit at unit place of the number (13)
0
(a) 1
(b) 7
(c) 9
(d) 3
na
-|x|
14.
The
function
f(x)
=
e
is
0 f  x   ?
(a) continuous everywhere but not differentiable at x
(a) nk
(b) (n+1)k (c) (n-1)k (d) 0
=0
2. Three square are chosen at random on the chess
(b) continuous and differentiable everywhere
board. The chance that they are in diagonal line is
(c) only continuous at x = 0
7
3
7
3
(d) None of these
(a)
(b)
(c)
(d)
1
84
64
744
744
15. lim 1  ax  x  ?
x 0
dy
3. Consider the differential equation
 2y  0
(a) 0
(b) e-a
(c) ea
(d) None of these
dx
2
16. If (3n +690) = (492k04) where n is a number and k
(a) Every solution of equation is identically zero
is a digit. Then what is the possible value of n?
(b) All solutions of equation are unbounded
(a) greater than – 1 and less than 3
(c) All solutions of equation approaches to zero
(b) greater than 0 and less than 10
when x  
(c) greater than 9 and less than 18
(d) No solution of equation approaches to zero when
(d) N.O.T
x
17. Find the angle between the tangents to the curve y =
4. For what value of k the line y = 9x be the tangent to
x2 – 5x + 6 at the points (3, 0) and (2, 0) is
k x 1
ke




the curve y 
at some point on the xy-plane
(a)
(b)
(c)
(d)
x 1
3
6
2
4
with constraint that x > - 1
18. S is a set and P(S) denotes the power set of S then
(a) k < 0
(b) k > 0 and k < 1
(a) P[P(S)] = P(S)
(b) P[P(S)]  P(S) = P(S)
(c) k > 1 and k < 3
(d) k > 3
(c) P[P(S)]  P(S) = {} (d) S  P(S)
5. If m things are distributed among ‘a’ men and ‘b’
dy
y
x-y
at (1, 1)?
women, then the chance that men get odd number of 19. If x = e then find
dx
things?
(a) 1
(b) 2
(c) 0
(d) 4
m
m
m
m
2
1 b  a   b  a 
1 b  a   b  a 
20.
Find
the
area
bounded
by
x
=
8y
and the line x – 2y
(a) .
(b) .
m
m
+
8
=
0
2
2
b  a 
b  a 
(a) 25
(b) 48
(c) 36
(d) 32
m
m
1 b  a   b  a 
21.
There
are
four
machines
and
it
is
known that exactly
(c) .
(d) None of these
m
two of them are faulty. They are tested one by one,
2
b  a 
in a random order till both the faulty machines are
6. Find the total number of distinct relations on the set
identified. Then, the probability that only two tests
containing 3 elements?
are needed is
(a) 6
(b) 128 (c) 18
(d) 512
(a) 1/3
(b) 1/6
(c) 1/2
(d) 1/4
x

4
7. Find the equation of tangent to the curve y  4e at 22. If  v " u  uv ' u ' v  k then find the value of k?
the point where it crosses Y-axis is
(a)  v ' u (b)  u " v (c)  u ' v ' (d)  uv "
(a) 4x + y = 4
(b) 4x – y = 16
23. If three numbers are selected from (2n + 1)
(c) x – y = 4
(d) x + y = 4
3
2
consecutive integers then find the probability that the
8. The curve y = x – 6x + 9x + 1 has symmetric
numbers from and A.P.?
rotation about the point
3n
3
(a) (0, 0) (b) (3, 2) (c) (2, 3) (d) None of these
(a)
(b)
2
bx
4n  1
4n  1
9. Find the differential equation of the curve y  ae
3n
after eliminating arbitrary constants.
(c)
(d) None of these
2
2
4n 2  1
d 2 y  dy 
d 2 y  dy 
(a) y 2     0
(b) x 2     0
24. The error is measuring radius of a circle is 0.5%.
dx  dx 
dx
 dx 
Then what will be the percentage change in the area
2
 d 2 y  dy
d2y
 dy 
of the circle is?
 y   0
(c) y  2  
(d)
0
dx 2
(a) 0.1%
(b) 0.0025%
 dx 
 dx  dx
(c) 0.01%
(d) 0.5%
10. If f(x + y) = f(x) . f(y) and f(6) = 3, f'(0) = then f'(6)
25.
Five
person
entered
in
a
lift cabin on the ground
=?
floor
of
8
floor
house.
Suppose
each of them with
(a) 15
(b) 22
(c) 28
(d) 30
1/3
equal
probability
and
independently
can leave the
11. Find the maximum value of the function (x + 1) –
1/3
cabin
at
any
floor
beginning
with
the
first.
Then find
(x – 1) is
the
probability
that
all
five
person
leaving
at
(a) 0
(b) 1
(c) 2
(d) 3
different
floors.
12. If f(9) = 9 and f'(9) = 3 then find the value of
7
7
7
P
P
P
f  x  3
(a) 55 (b) 75 (c) 55 (b) N.O.T
lim
7
7
7
x 9
x 3
(a) 3
(b) 0
(c) 1
(d) 9
1.
If f(x) = f(x + ) and
 f  x  k,
a
1
INFOMATHS/MCA/MATHS/
INFOMATHS
26. The probability that a student passes in Mathematics,
B: I. I never stolen the horse
Physics and Chemistry are m, p and c respectively of
II. G stolen the horse
these subjects, the student has 75% chance of
(a) B stolen the horse
(b) G stolen the horse
passing in at least once, a 50% chance of passing in
(c) R stolen the horse
(d) Data insufficient
atleast, two and a 40% chance of passing in exactly 34. A cricket match is played and you want to know the
two, which of the following relations are true.
result of the match immediately you can know the
result only by asking your friend. Your friend tries to
19
27
(a) p  m  c 
(b) p  m  c 
have some fun and tally you that you can ask only
20
20
questions which he answer as ‘yes’ or ‘no’. Your
1
1
friend tells you that he will lie but at most once. A
(c) pmc 
(d) pmc 
20
4
cricket match took place in which one team lost and
27. A is a set containing n elements. A subjects P of A is
so another won. You want to know the result. What
chosen at random. Then, returning back the elements
is the minimum number of questions you need to ask
of P the set A is completed. A subset Q is again
your friend so you can know the result?
chosen at random. Find the probability that have no
(a) 2
(b) 3
(c) 4
(d) None
element common.
35. P is a moving point and tangents from P meet a
n
n
n
circle. Whose centre is at the origin and radius is
1
2
3
(a)   (b)   (c)   (d) N.O.T
unity, at A and B [where, A and B are point of
2
3
4
contact] and angle AOB = 60. Then, locus of P is
28. If A has (n + 1) coins and B has n coins. Coins are
2
flipped simultaneously, then what is the probability
(a) circle of radius
(b) circle of radius 3
3
that a gets more heads than B?
(c) circle of radius 2
(d) None
1
1
1
(a)
(b) 
(c) 
(d) None
36.
A
and
B
are
two
candidates
seeking admission for an
2
2
2
interview. The probability that A is getting selected
29. The maximum number of cricket ball, which are
is 0.5 and the probability that both A and B are
placed, such that touches the remaining is 4. Then
selected is atmost 0.3. The probability of B getting
the maximum number of coins that are placed such
selected is
that touches the remaining is
(a) P(B)  0.8
(b) P(B)  0.8
(a) less than 5
(c)
P(B)
=
0.9
(d) Data Insufficient
(b) greater than 4 but less than 8
37.
50
defective
bulbs
and
50
non-defective bulbs are
(c) Greater than 7 but less than 11
selected
at
random,
one
by
one
with replacement. If
(d) None
X, Y and Z are the events defined as
30. f(x) is a differentiable function satisfying f(x3) = 4x4,
X : Chance of getting 1st bulb defective
then find the value of f'(8)?
Y : Chance of getting 2nd bulb non – defective
64
32
16
16 2
Z : Chance of getting both defective or both non(a)
(b)
(c)
(d)
3
3
3
3
defective
31. If f and g are two positive functions having relation
Then, which of the following is true.
(a) X, Y, Z are independent
f  x
R such that f – g, it is given that lim
1,
(b) X, Y, Z are pairwise independent
x  g  x 
(c) X, Y, Z are mutually exclusive
which one of the following is not consequence?
(d) None of these
(a) f2 ~ g2
(b) f ~ g
38. Numbers is selected from the numbers 00, 01, ……,
99 and the product obtained of the 2 digits of the
(c) ef ~ eg
(d) (f + g) ~ 2f
numbers is 18. If 4 numbers are selected, then what
32. A chess match starts between 6 pm – 7 pm and ends
is the chance that atleast 3 numbers are such that
between 9 pm – 10 pm. Observing in 12 hours wall
their digits have products 18?
clock, when the match starts the minute hand points
at position m and hour hand points at position h.
97
7
3
97
(a)
(b)
(c)
(d)
4
4
4
When the match ends, the hour hand points at m and
100
 25
 25
 25
minute hand at h, then what is the minimum time the
39. Let a  z+ is element of A and a relation a R b
game is played?
defined on A such that a = bk where k is a positive
(a) less than 130 minutes
integer then, which of the following belongs to this
(b) Greater than 131 minutes and less than 181
relation?
minutes
(a) (8, 128)
(b) (16, 256)
(c) Greater than 180 minutes and less than 210
(c)
(169,
13)
(d) (11, 3)
minutes.
40.
A
ring
of
diameters
20
cm
is suspended horizontally
(d) none
by
6
strings
of
equal
length,
such that the height
33. One day a horse is stolen and G, R, B are rounded up
between
the
centre
of
the
ring
and
top of the string is
for questioning. Each of them made two statement
24
cm,
and
the
strings
are
attached
in such a way that
and none of them made more than one false
they
are
equidistant
from
each
other
then, what is the
statement cosine of the angle between two adjacent strings?
R : I. Did not stole the horse
II. The one who has stolen the horse is arab.
313
335
13
(a)
(b)
(c)
(d) N.O.T
G : I. R never stole the horse
338
338
7
II. The one who has stolen the horse is Jewish
2
INFOMATHS/MCA/MATHS/
INFOMATHS
41. A coin is tossed 10 times and a bag is filled with 49. How many NDD of length 2 can be formed from
white and black ball accordingly by getting heads or
NDD of lengths 20.
tails respectively 10 balls are drawn from the bag
(a) less than 201
with replacement and we get all white balls then,
(b) More than 200
what is probability that all balls in the bag are white?
(c) more than 300 and less than 701
(d) None of these
10
9
2
1
(a)
(b) 9
(c) 9
(d) 9
50.
How many NDD of length 3 can be formed from
9
2
2
2
 2
NDD of length 30?
42. In the equation SHOO + SHOO = BRAIN, if all
(a) less than 201
digits represent distinct numbers, then the sum of
(b) more than 200
digits of minimum possible number of SHOO is
(c) More than 300 & less than 701
(a) less than 18
(d) None
(b) Number lying between 13 – 18
51. How many NDD of length 4 can be formed from
(c) Number lying between 17 – 22
NDD of length 30?
(d) None of these
(a) less than 201
(b) more than 200
Directions : [43 – 45] Group questions
(c) more than 300 & less than 701
NEWTON + ALWAYS = TOTALED
(d) None of these
Here, each letters belongs to a single digit integer. On the 52. If z is represented by abcde and x = 1abcde and y =
basis of the given statement, answer the following
abcde 1 and y = 3x, then z is
questions:
(a) greater than 30001 but less than 40001
43. In the above statement, NEWTON belong to
(b) less than 30000
(a) less than 500001
(c) greater than 40000 but less than 50001
(b) greater than 500000 or less than 600001
(d) None
(c) greater than 600000 or less than 700001
(d) greater than 700000
Directions (Q. Nos. 53-56) These questions are on the
44. In the above statement, ALWAYS belongs to
following information.
(a) less than 500001
From time to time the managing director of a company
(b) greater than 500000 or less than 600001
appoints planning committee, each consisting of exactly
(c) greater than 600000 or less than 700001
three members. Eligible for appointment are three
(d) greater than 700000
executives from finance; B, C and D and three executives
45. In the above statement, D belongs to which integer from operations E, F and M.
or single digit
Any given committee is subject to the following
(a) 0
(b) 1
(c) 3
(d) None of these
restrictions on appointments.
(i) At least one member must be from finance, and at least
Directions : [46-48]: Group Questions
one member must be from operations.
Consider an infinite non-decreasing series in which every (ii) If B is appointed, C cannot be appointed.
natural numbers repeats for n times. The first 12 digit of (iii) Neither D nor F can be appointed unless the other is
the series is [122333444455].
also appointed.
46. The sum of first 200 elements is
(iv) If E is appointed, M must be appointed.
(a) less than 2000
53. Which of the following is an acceptable committee?
(b) greater than 2001 and less than 2301
(a) E, F and M
(b) D, E and F
(c) greater than 2301 and less than 2701
(c) D, F and M
(d) B, D and M
(d) greater than 2701
54. If appointees from operations are in majority in a
47. The smallest value of n, when the sum of numbers
committee, that committee must include
exceed by 1 million
(a) M
(b) F
(c) E
(d) C
(a) less than 40
(b) > 41 and < 50
55. If the restrictions on appointments apply also to a
(c) > 51 and < 60
(d) None of these
four-member committee appointed from the same
48. The sum of all the elements of the sequence starting
group of executives, which of the following will be
from 1001th to 10000th element:
true?
(a) less than 500000
(a) If B is appointed, M must be appointed
(b) greater than 800000 and less than 1300001
(b) If F is appointed, C must be appointed
(c) greater than 500000 and less than 800001
(c) If C is appointed, E must be appointed
(d) none
(d) None of the above will be true
56. If B is appointed to the same committee as M, which
Directions [Q. No. 49 to 51] Group questions
of the following will be true of that committee?
If NDD is described as finite non-decreasing distinct
(a) E is not a committee member
sequence
(b) F is a committee member
Eg  S1 = {1, 2, 4, 7, 31, 100} is a NDD.
(c) Appointees from finance are in majority
And S2 = {1, 2, 2, 5} is not NDD because of 2 is
(d) Appointees from operations are in majority
repeated.
Directions (Q. Nos. 57-60) Read the following
New NDDS can be formed from a given sequence of information to answer these questions.
NDD  example from above sequence S1, NDD {1, Ankit is decorating his room and trying to arrange six
2, 7} and {1, 7, 31, 100} can be formed.
paintings on the East and West walls in his room. The
paintings are each multi colour representations of one of
3
INFOMATHS/MCA/MATHS/
INFOMATHS
the letters of the alphabets E, H, M, O, R, T. Ankit does
not want the three letters on each wall to make any
common English word. Also, the colours of the O and E
do not look good next to each other, nor do the T and O go
well together.
3.
57. If Ankit puts E, H and M on the East wall, which of
the following must he true?
(a) R and M cannot face each other.
(b) O cannot be in the centre of the West wall.
(c) E cannot be in the centre of the East wall.
(d) T and M cannot face each other.
58. If Ankit’s mother is coming to visit and he decides to
celebrate the visit by having his paintings spell
‘MOTHER’, starting with the leftmost painting on
the East wall and going around the room, which of
the following will be false?
(a) O is opposite to E
(b) H is next to E
(c) T is next to O
4.
(d) T is opposite to R
59. Which of the following is not possible?
(a) T and O to be opposite to each other
(b) T, H and R to be on the same wall
(c) H, M and R to the on the same wall
(d) M and O to be opposite to each other
60. If Ankit trades his M painting for another O painting
just like the one he has now, which of the following
must be false ?
(a) O can be on opposite walls in the middle
6.
(b) Either R or H will be next to an O
(c) Either R or H will be next to either T or E
(d) T will be opposite to either O or E
7.
PUNE-2015
1
A
11
C
21
B
31
C
41

51
D
2
A
12
A
22
B
32
B
42
C
52
C
3
C
13
B
23
C
33
A
43
B
53
C
4
D
14
A
24
A
34
B
44
C
54
C
5
A
15
B
25
A
35
A
45
A
55
C
6
D
16
D
26
B
36
A
46
C
56
D
7
D
17
A
27
C
37
B
47
B
57
B
8
C
18
D
28
A
38
D
48
B
58
D
9
A
19
C
29
A
39
C
49
A
59
C
10
D
20
C
30
C
40
A
50
D
60
A
na
0
8.
a
0
[ f(x) is a periodic with a]

0
2.
f  x  dx  nk
[
dy
 2y  0
dx
dy
 2 y
dx
dy

 2dx
y 
 log y = - 2x + c
 y = Ae-2x
 lim y  lim Ae2 x  0

x 
x 
Hence, all solution of equation approaches to zero
when x  
ke
We have, y 
x 1
k
x 1
k
k .e x 1

2 x 1 2 x 1
k
k
dy

dx
x  1.e
k
x 1



 k x  1 1
x 1
2
x 1
dy ke

3/ 2
dx
2  x  1
k
Ans. (d) Set containing 3 elements
2
 Total number of relation is 23  29  512
Ans. (d) Equation of curve is y = 4e-x/4
Put, x = 0 in y = 4e-x/4, we get
y = 4e0  y = 4
[ e0 = 1]
dy
1
 4e  x / 4 
 e  x / 4
dx
4
 dy 
 
 e 0
 dx  0,4
=-1
Hence, equation of tangent at (0, 4)
y – 4 = - x(x – 0)
x+y=4
f  x  dx  n  f  x  dx
na
Ans. (c) We have,
392
7

41664 744
Now,
ANSWER WITH EXPLANATIONS
1. Ans. (a) We have, f(x) = f(x + a)
i.e. f(x) is a periodic function with period a

Required probability 
 f  x  dx  k ]
a
0
Ans. (a) Total number of square on chess board = 64
Total number of ways of chosen three square from
64! 64  63  62

 41664
64 square is 64 C3 
61!3!
3  2 1
The square are diagonal line
 Total number of favourable out comes is
4(3C3 + 4C3 + 5C3 + 6C3 + 7C3) + 2  8C3
= 4(1 + 3 + 10 + 20 + 35) + 2  56
= 280 + 112 = 392
4
Ans. (c) We have, y = x3 – 6x2 + 9x + 1
dy
 3 x 2  12 x  9
dx
dy
 0,
dx
Then, 3x2 – 12x + 9 = 0
(x – 3) (x – 1) = 0
f(x) is increasing x  (- , 1) and (3, )
f(x) is decreasing x  (1, 3)
Graph of f(x) is
INFOMATHS/MCA/MATHS/
INFOMATHS

f ' 9  9
f 9

3 3
3 [
3
f(9) = 9, f ' (9) = 3]
13. Ans. (b) We have, (13)27
 (134)6  133
(28561)6  2197
Digit of unit place of (28561)6 is 1.
And digit of unit place of (28561)6  2197 = 7
Hence, digit of unit place of (13)27 = 7.
Symmetric rotation about the point of f(x) is
 1 3 5 1 
,

   2,3
2 
 2
9.
14. Ans. (a) We have, f(x) = e-|x|
e  x , x  0
 f  x   x
e , x0
f(x) is continuous at x = 0
If lim f  x   lim f  x   f  0
Ans. (a) We have, y = aebx
dy

 abebx
dx
dy

 by
dx
d 2 y bdy
 2 
dx
dx
2
d 2 y  dy 
 y 2  
dx
 dx 
x 0
x 0
x 0
f(x) is continuous at x = 0
e x x  0
Now, f '  x    x
x0
 e
f(x) is discontinuous at x = 0
If LHD of f(0) = RHD = 0 of f(0)
LHD of f ' (0) = - 1 and RHD of f '(0) = 1
f(x) is not differentiable at x = 0.
2
y
x 0
 lim e x  lim ex  f  0  1  1  1
d 2 y  dy 
   0
dx 2  dx 
1
15. Ans. (b) We have, lim 1  ax  x
x 0
10. Ans. (d) We have, f(x + y) = f(x) . f(y)
Differentiating w.r.t.x, y as constant, we get
 f ' (x + y) = f ' (x) . f(y)
…(i)
Put x = 0 and y = 6 in Eq. (i), we get
 f ' (0 + 6) = f '(0) . f(6)
 f '(6) = f '(0) . f(6)
 f '(6) = 10  3
= 30
[ f '(0) = 10, f(6) = 3]
1
 lim
1  ax  1
e x 0
x
g x
 lim

a
[
lim[
f  x ]    lim
 f  x   1 g  x ]
x 0
xa
e
 e-a
x a
e
16. Ans. (d) (3n + 690)2 = 492 k04
Taking square in both sides, we get
3n  690  492k 04
3n + 690  702.70
 3n + 690  702.70
702.70  690
n
3
n4
3n + 690  701.4
 3n  114
n  3.8
 Hence, n = 4
11. Ans. (c) Let f(x) = f(x + 1)1/3 – (x – 1)1/3

1
1
1
f ' x  


2/3
2/3
3   x  1
 x  1 
For maxima and minima,
f ' (x) = 0

1
1
1
 

0
2/3
2/3
3   x  1
 x  1 
x=0
Now, f '(x) is increasing if, x  (-, 0) and
decreasing in x  (0, )
17. Ans. (a) We have, y = x2 – 5x + 6
 f(x) is maximum at x = 0
dy

 2x  5
Hence, f(0) = (1)1/3 – (-1)1/3 = 2
dx
 Maximum value of f(x) is 2.
 dy 
 65 1
 
 dx 3,0
f  x  3
12. Ans. (a) We have, lim
x 9
 dy 
x 3
And  
 4  5  1
 dx  2,0
f ' x
0
  dy 

 dy 
2 f  x
 
 lim
  dx 

[ Apply L’Hospital’s rule]
dx




x 9
1
 3,0 
 2,0  
tan   
0
  dy 
 dy  
2 x
1       
dx
  3,0  dx  2,0 
5
[if k = 9]
[if k = 0]
INFOMATHS/MCA/MATHS/
INFOMATHS
tan  

11
11
23. Ans. (c) Three numbers can be selected out of (2n +
n  4n2  1
1) integer is 2n  1c3 
3
Total number of AP with common difference with 1,
2, 3, …. n are (2n – 1), (2n – 3) …., respectively.
Favourable outcomes = 2n – 1 + 2n – 3 + …. + 1
= n2
3n 2
3n
 2
Required probability 
2
4
n
1
n  4n  1
1

0
Hence,  

2
18. Ans. (d) Let x be a element of s.
xs
Now, P(S) be the power set of s.
 {x}  P(S)
Hence,  s  P(S)
dr
100  0.5%
r
Area of circle = r2
 A = r2
dA

 2 r
dr
 dA = 2 rdr
dA
2 rdr

100 
100
A
A
dA
2 rdr

 100 
 100
A
 r2
dr
 2 100
r
= 2  0.5 = 1%
Hence, change in area of circle 1%.
[ s = {x}]
24. Ans. () We have,
19. Ans. (c) We have, xy = ex-y
Taking log both the sides, we get
log xy = log ex-y
 y log x = (x – y)
 Differentiating w.r.t. x, we get
y
dy
dy
dy
x y
  log x
 1
  log x  1 
x
dx
dx
dx
x
dy
x y
dy
1

1
 


  
0
dx x  log x  1
 dx 1,1 1 log1  1
20. Ans. (c) We have,
x2 = 8y and x – 2y + 8 = 0
Solving these equation, we get
x = - 4, or 8.
25. Ans. (a) The total number of ways in which each of
the five persons can leave the cabin at any of the 7
floors = 75
Five persons can leave the cabin at 5 different floor
in 7p5 ways.
7
p
 Required probability  5 5
7
 Area of shaded region
26. Ans. (b) It is given that, probability of passing
atleast once in 75%
i.e. P(A  B  C) = P(A) + P(B) + P(C)
- P(A  B) – P(B  C) – P(A  C) + P (A  B  C)
75

 P  A  P  B   P  C   P  A  B 
100
- P(B  C) – P(A  C) + P(A  B  C) …(i)
Probability of passing at least two is 50%.
i.e. 50% = P(A  B) + P(B  C) + P(A  C)
- 2P (A  B  C)
…(ii)
And probability of passing exactly two is 40%.
40% = P(A  B) + P(B  C) + P(A  C)
- 3P (A  B C)
…(iii)
Solving Eqs. (ii) and (iii), we get
P(A  B  C) = 10%
1
i.e. pmc 
10
Solving eqs. (i) and (ii), we get
P(A) + P(B) + P(C) = 135%
135 27

 i.e. p + m + c 
100 20
8
 x  8  x 2 
 x 2 8 x x3 

dx



   
4  2  8 
 4 2 24  4
 64 64 64   16 32 8 
          = 36
2 3   4 2 3
 4
8
21. Ans. (b) The total number of ways in which two
machine can be selected out of four machine is 4C2 =
6. If only two tests are required to identify faulty
machine then, in first two tests faulty machines are
identified.
So, favourable outcomes = 1
1
Required probability 
6
22. Ans. (b) We have,
 v ' u dx
 u  v " dx   u ' v " dx  dx  c
 uv '  u ' v ' dx  c
 uv ' u  v ' dx   [u "  v ' dx].dx  c
27. Ans. (c) Since, set A contains n elements, So it has
2n subsets.
 Set P can be chosen in 2n ways.
Similarly, set Q can be chosen in 2n ways.
 P and Q can be chosen in 2 n – 2n = 4n ways.
 uv ' u ' v   u " v.dx  c
Hence, k   u " v
6
INFOMATHS/MCA/MATHS/
INFOMATHS
3
3
Suppose, P contains r elements, where r varies from
97
 4   96 4   1   97 



O to n, then, P can be chosen in nCr ways for 0 to be
 
4
  25   25  
100
25
25








25
 
disjoint from A, it should be chosen from the set of
all subsets of set consisting of remaining (n – r)
elements.
39. Ans. (c) We have, a = bk, where k is positive integers
This can be done in 2n-r ways.
a>b
 P and Q can be chosen in nCr . 2n-r ways.
169 = 132
But, r can vary from 0 to n.
Hence, option (c) is correct.
 Total number of disjoint sets P and Q
n
53. Ans. (c) D and F should be appointed together, so
n
  n cr 2n  r  1  2   3n
option (a) and (d) are ruled out.
r 0
E and M must be appointed together, so option (b) is
n
3n  3 
ruled out. Option (c) is correct because it satisfies all
So, required probability  n   
the conditions
4
4
54. Ans. (c) This committee has two executives from
operation and one executive from finance.
Executives from operations are E, F and M. E and F
cannot be selected because of (iv). So, the committee
must include M.
30. Ans. (c) We have, f(x)3 = 4x2
 f ' (x3) . 3x2 = 16x3
Put x = 2, we get
f ' (23) . 3.22 = 16.23
16  2 32
 f ' 8 

3
3
35. Ans. (a) We have,
In OPB,
OBP = 90
and
POB = 30
OB
 cos 30 
OP
55. Ans. (c) A four member committee can be formed
with B, D, and M. So, option (a) is correct.
56. Ans. (d) If B and M are appointed then, cannot be
appointee D and F have to be appointed together. So,
the third person appointed must be E. So, appointees
from operations are in majority.
57. Ans. (b) O cannot be in the centre of the West wall.
Because the only ways in which rest three letters O,
R, T to the displayed in East wall, when O is in the
centre are ROT and TOR. And in both the cases O
and T are next to each other which is not
permissible.
58. Ans. (d) Letter of the word MOTHER can be
arrange in the following ways
 OP 
2
3
[ OB = r = 1]
 Locus of p is circle of radius
2
.
All option are correct except (d).
3
59. Ans. (c) If H, M and R on the wall, then three letters
on the other wall will be T O E. and in all the
possibilities either O and E will be next to each other
or O and T will be next to each other and it will be
violation of the condition.
36. Ans. (a) We have, P(A) = 0.5
P(A  B)  0.3
P(A) + P(B) – P(A  B)  1
P(B)  1 + P(A  B) – P(A)
P(B)  1 + 0.3 – 0.5
P(B)  0.8
60. Ans. (a) If painting M is exchanged from O and if, O
is in the middle of each wall, then the positions of all
38. Ans. (d) Let P be the probability getting product of
the six painting will be as under
two-digit number is 18.
i.e. such number are 29, 36, 63, 92
4
P 
100
4
96
Now, any position of paintings E and T will violate
q  1

100 100
the conditions.
Required probability is P(X  3) = P(X=3)+P(x=4)
= 4c3 p3q + 4c4 p4
3
4
 4   96   4 
 4

 
 

 100   100   100 
7
INFOMATHS/MCA/MATHS/