Exam Details
| Subject | ||
| Paper | ||
| Exam / Course | mca | |
| Department | ||
| Organization | central university | |
| Position | ||
| Exam Date | 2013 | |
| City, State | telangana, hyderabad |
Question Paper
Part A
1. It takes 6 technicians a total of 10 hours to install a new equipment from scratch, with each working at the same rate. If six technicians start to install the same equipment at 11:00 am, and one technician per hour is added beginning at 5:00 pm, at what time will the equipment installation be complete?
A. 6:40 pm
B. 7:00 pm
C. 7:20 pm
D. 8:00 pm
2. In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not lie in the same row or column?
A. 56
B. 896
C. 60
D. 768
3. Gopal's Shop sells small cookies in boxes of different sizes. The cookies are priced at Rs.2 per cookie up to 200 cookies. For every additional 20 cookies, the price of the whole lot goes down by 10 paise per cookie. What should be the maximum size of the box (in terms of number of cookies it can hold) that would maximize the revenue?
A. 240
B. 300
C. 400
D . None of these
4. 3 small pumps and a large pump are filling a tank. Each of the three small pumps works at the rate of the large pump. In what fraction of the time that all 4 pumps working together will fill the tank in comparison to the time taken by the large pump
. alone?
A.
B.
C.
D.
5. Four cows are tethered at four corners of a square plot of side 14 meters so that the adjacent cows can just reach one another. There is a small circular pond of area 20 m2 at the centre. The area left ungrazed is:
A. 22 m2
B. 42 m 2
C. 84 m 2
D. 168 m 2
Answer questions 6 and 7 using the
following information: 8 trees, viz.
mango, guava, papaya, pomegranate,
lemon, banana, raspberry and apple
are planted in two rows, of four each
aligned East-West. Lemon is between
mango and apple but just opposite to
guava. Banana is either at the end of
a row and is just immediately to the
right of guava, or Banana is just next
to Guava. Raspberry is at the end of
a row, and mango is at the other end
of the opposite row.
6. Which of the following is always true
1
A. Apple is just next to Lemon
B. Papaya is just next to Apple
C. Raspberry is either to the left or to the right of Pomegranate
D. Pomegranate is diagonally opposite to Banana
7. Which of these is directly opposite Banana?
A. Pomegranate
B. Mango
C. Papaya
D. None of these
8. How many number of times will the digit be written when listing the integers from 1 to 1000?
A.300
B. 271
C. 252
D.304
9. There are 10 positive real numbers nl nz n3·.· nlO· How many triplets of these numbers nz, n3, ... can
be generated such that in each triplet the first number is always less than the second number; and the second number is always less than the third number?
A. 45
B. 90
C. 120
D. 180
10. There were two women amongst other men who took part in a chess tournament. Every participant played two games with every other participant. The number of games which only men played was exactly 104 games more than those played which involved a women. The total number of participants is
A. 13
B. 11
C. 14
D.15
Answer questions 11 and 12 using the following information: In the English alphabet there are 11 symmetric letters that appear the same when looked at in a mirror. Other 15 letters in the alphabet are asymmetric letters.
11. How many four-letter computer passwords can be formed using only the symmetric letters (no repetition allowed)?
A. 7920
B. 330
C. 14640
D. 419430
12. How many three-letter computer passwords (no repetition allowed) can be formed with at least one symmetric letter?
A. 990
B. 2730
C. 12870
D. 15600
13. 'Which letter in the word RUTHLESS has a position from the beginning of the word that is half as much as its position when seen in the alphabet
A. E
B. L
C. H
D. U
14. A wire of some length is bent in circular form and has an area of 308 sq. cm. If the same length of wire is straightened out and bent in the form of a square, the approximate area of the square in sq. cm may be
A. 121
B. 242
C. 308
D. 69.29
15. Consider a square circumscribed by a circle with a radius of 4 units. The area of the square in square units is
A. 16K
B. 16 J2
C.32
D.64
16. A man returns after shooting and catching birds in his bag. He was asked how many birds he had in his bag. He said, "They are all house sparrows but six, they are all pigeons but six, and all doves but six." The number of birds he had in all were
A.36
B. 18
C. 9
.D.27
17. Suppose an ant is placed on one corner of a sugar cube, which has equal sides of 1.5 cm each. If the ant may walk only along the edges of the cube, what is the maximum distance the ant may walk on the cube without retracing its path?
A. 18 cm
B.9cm
C. 10.5 cm
D. 13.5 cm
18. Vhen the big hand of the clock is exactly at the 12 clock position, an ant starts to crawl in a counter clockwise direction from the 6 clock position at a constant speed. On reaching the big hand of the clock, the ant turns around and at the same speed, starts to crawl, in the opposite direction. Exactly 45 minutes after the first meeting with the big hand the ant crosses the big hand for the second time and dies. How long has the ant been crawling?
A. 54 min.
B. 51 min.
C. 1 hr, and 9 min.
D. 1 hr, and 21 min.
19. One third of Shrihari's marks in Math equal half of his English marks. Shrihari noticed that in these two subjects his marks totaled 150. What did Shrihari score in English
A. 15
B. 60
C. 30
B-9
D.90
20. The sum of two digits of a number is
15. If 9 is added to the number then the digits get reversed. Which of the following is FALSE about the number
A. The number is divisible by 3
B. The number has the two digits separated by a difference of one
C. The number is divisible by 6
D. The number is divisible by 9
21. An ant is at a point P in a planar square field. It was observed that P is 13 feet from the corner and 17 feet from corner B (diagonal to and finally 20 feet from a third corner. The area in square feet of the field is
A. 231
B.89
C. 369
D. 169
22. Consider the triangle ABC with sides AB=20 em., AC=l1 em., and BC=13 em. Then the length in em. of the diameter of the semi-circle inscribed within ABC, which lies on AB, and has sides AC and BC as tangents is given by
A.9
B. 11
C. 10
D. 10.5
23. On a straight road XY, 100 meters long, five heavy stones are placed 2 meters apart beginning at the end X. A worker, starting at has to transport all the stones to by carrying
4
only one stone at a time. The minimum distance he has to travel (in meters) is:
-A. 422
B. 480
C. 744
D. 860
24. Given that abca abca, where all c are integers, then which of the following is true
A. c=9
B. a=l
C. b=4
D. None of these
25. Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is:
A.
B.
C.
D.
Part B
A. X n is positive if n is even
B. X n is positive if n is odd
Answer the questions 26-27 based on the algorithm given below: C. X n is negative if n is even
D. None of these
STEP START STEP C FLAG Given that n is an integer, which of STEP INPUT TABLE A[1 .. 10J the following is TRUE about the sum
S 2n 3n
STEP3: FORI=1TO 10 DO IF 1J THEN
A. S is never the square of a rational
N N 1
number
IF (FLAG FALSE) THEN B. S is always the square of a rational number
FLAG TRUE
ENDIF
C. S is the square of a rational num
ENDIF
ber, provided n is even
ENDFOR
D. S is the square of a rational num
STEP OUTPUT C
ber, provided n is odd
STEP END
30. !f then
Given that IS:
10
A.
f
26. What are the values of Nand
B. f(x
N= 4
B. 10 c. (.x
C. C D.
l+xy
D. N=3,C=10
31. In an equilateral triangle, let h. be the
27. What are the values of Nand C if height and r be the radius of the cirthe statement "FLAG TRUE" is
cumcircle. Then the ratio r h is
removed from the algorithm?
A.
A. N C 8
B.2:3
B. N=4,C=10
C. 2n
C. N 8
D. n:V5/2
D. N C 10
32. Let ¢ denote an empty set. The power
28. The nth element of a series is represet of the empty set is sented as Xu X If
A. ¢
X o x and x then which
of the following is always true: B.
c.
D. None of these
33. Let A and B be two non-empty sets with cardinality p and q respectively. Then the total number of relations that can be defined from the set A to set B is
D. p, or or pv
34. Consider the relation f defined
X2 0< X .
3x 3 x 10 and the
relation 9 defined by X2 0 X 2 3x then
A. f is a function but 9 is not
B. both J and 9 are functions
C. f is not a function but 9 is
D. neither of them are functions
35. Let f be subset of Z x where Z denotes the set of integers, and we have J b E Z}. Then f is
A. bijective function
B. injective function
C. surjective
D. none of these
36. For real numbers and nEZ, sin x sin y implies which of the following?
A. J: fiT!
B. x mr
C. X 2n7f ± y
D. x n7f ± y
37.. Show the bits in a 12 bit-register that is holding the number equivalent to decimal 215 in binary coded octal and binary coded decimal
A. 000011000011,001000010101
B. 000011010111, 001000010101
C. 000011000111,001000010111
D. none of the above
38. If 587 the base of the number system is
A. 14
B. 11
C. 9
D. 12
39. At how many points in the xy plane do the graphs of y X 12 and y 2x intersect?
A. One
B. None
C. Two
D. Three
40. Suppose that f is a continuous realvalued function defined OIl the dosed interval Which of the following are true for constants E and x,y E
1. There is C such that
I y.
II. There is D such that
I y that satisfy
Ix-y D.
III.
There is E f'lwh th=tt.
I E Ix Ifx, y
A.
III only
B. I only
C. I and II only
D. I, II and III
41. Suppose that j is a function on the set of real numbers, and is differentiable twice, and that are all negative. Suppose j" has all the three properties in the interval
I. It is increasing,
II. It has a unique zero,
III.
It is unbounded.
Which of the same three properties
does f necessarily have
A.
II only
B.
I only
C.
III only
D.
II and III only
42. Let 2x3 ax2 3x and X3+x2-4x-9. Both give the same reminder when divided by x if a is
A.10
B. 1
C.5
D. -5
43. Consider the unit square formed by points in the plane; and Let P be an arbitrary point chosen in this unit square. Connect P to the points A and B. The probability that the points ABP form an obtuse triangle is
0.5
B. 0.393
C. 0.712
D. 0,25
44. Let j and g be two functions defined on an interval I such that 0 and Vx E and f is strictly increasing in I. Then the product function jg is
A. strictly increasing in I with fg 0
B. strictly increasing in I with fg a
C. strictly decreasing in I with jg
a
D. nothing can be said about it from the given information
45. The derivative offunction where
sinx2
x
a x a
A. exists at x but is not continuous
B. does not exist at x a
C. exists at x and is also continuous
D. none of the above
46. If denotes the largest integer and if (where ever it exists) is given by
A.
B.2x
C.2[x]
D. exists nowhere
47. The sum of the interior angles of a polygon is equal to 86 right angles. Then the number of sides of that polygon is
A.43
B.86
C.30
D.45
48. The area of the polygon whose vertices are and x is equal to
A.2x
B. x2/2
C. 1
D.2
49. Use the notation I x I to denote the absolute value of and denotes the largest integer smaller or equal to
x. For an integer, such that a
let be given by {llx-l 1 x<a
l
a
Which of the following is the set of
points of discontinuity for the function
A. All integers a
B. All integers
C. All a
D. a
50. Consider the function f(x {X2;y2
Which of the following is NOT true for this function
A. f is continuous along the line
o
B. f is continuous along the line
o
C. f i8 constant along the line x
D. f is not continuous at (0.0).
51. Suppose c are all and if D is the determinant below, which of the
following is true about D
abc
b c a cab
1. D II. D III. D 0
A. II only
B. I only
C. both I and II
D. both II and III
52. Suppose C are all and are in a geometric progression as the successive terms .. " .... Then what is the value of the determinant
log a p 1
D below? D 10gb q 1
log c r 1
A.l
B. 0
C. pqr
D. pqr -abc
53. Consider the determinant
1 sin e 1
sin e 1 sin e
-sin g 1
Then 6./2 for all values of lies between
A. 1 and 2
B. 0 and 1
C. 2 and 4
D. >4
54. Which of the following is equal to the nullity of matrix A below
8
10
264
A.l
B. 0
C.2
D.3
55. How many 2-digit or 3-digit numbers ca,n be formed using the digits and are divisible by
A.64
B.56
C.80
D.92
56. Let (256ho x and if 2x then the values of B are respectively
A.
B.
C.
D.
57. The average temperature of a town in the first four days of a month was 58 degrees. The average for the second, third, fourth. and fifth days was 60 degrees. If the temperatures of the first and fifth days were in the ratio 7:8 the temperature on the fifth day was
A.62
B.64
C.56
D. None of these
58. Two boys A and B speak the truth only 75% and 80% of the time respectively. Lets say both witnessed an incident, what is the percentage of time that the two would contradict each other when narrating the same incident?
A. 25
B. 15
D. 45
59. Solution of the initial value problem y" -6y giveny(O) 10; -10 is
A. 4e-6t lOet
B. e6t e-t
C. e-6t 50 et
7
D. 20e-3t 30e2t
60. When 4 dice are thrown, what is the probability that the same number appears on each of them?
A. 1/36
B. 1/1296
C. 1/216
D. 24/216
61. The meall deviation about the median for the data 12, 10, 18, 19, 21 is
A.5.27
B. 1.01
C.9
D. 10.01
62. Let for x -10 where f is an increasing function then
9
A. is a decreasing function of x
B. is an increasing function of x
C. is increasing for x and
decreFiSing for ·-10 x 0
D. none of these
63. The solution to the indefinite integral
Q
XFLAG:'" 1
du, is given by
NO
YFLAG= 1
A. -sin-1 C
B. 2u2a2J(a2-u2) -sin-1 C
1
C. C
D. 2u2a2J(a2-u2) -sinh-1 C
64. Consider the circles, all of unit radius given by the form
1. The radius of curvature at any
point for these circles may be
also expressed as
A.
B.
C.
D. None of these
65. Solution to the differential equation d4 y 4y 0 is
dx4 .
A. y e-2x (cl cos x c2sinx) A. log y -xy C
e2x (c3 cos x C4 sin .,. 1
B. 1og'" c
B. y e-X(cl cos x c2sinx) y xy
C. log; -xy
eX (C3 cos X C4 sin D. log 1L
xy
c. y e-X(cl cos 2x C2 sin 2x) X
eX (C3 cos2x C4 sin 2x)
Answer the following 3 questions
D. y -eX (Cl cos X C2 cos based on the flowchart given above:
eX (C3 sin x C4 sin
67. What is the output of the flowchart if
66. The solution to the differential equaX -10 and Y 3
tion xy)ydx -xy)xdy
is A. Q and X
10
B. Q 3 and X
C. Q -3andX 1
D. Q= 6and 1
68. Which of the following conditions can be used in place of the condition "IF ((XFLAG 0 AND YFLAG OR (XFLAG 0 AND YFLAG
A. IF (XFLAG -YFLAG
B. IF (XFLAG -YFLAG
C. IF (XFLAG YFLAG
D. IF (XFLAG YFLAG
69. For what values of X and the flowchart is never going to terminate?
A. X> 0andY 0
B. X< 0 and Y 0
C. X=0andY
D. All of the above
70. A person speaks truth only 4 times out of 5. A die is tossed and the person says that the die rolled a six. Find the probability that actually there was a six
A.4/5
B. 2/9
C. 7/9
D.4/9
71. cos A +cos B can also be written as
A. 2cos--sm-
22
B. 2cos cos
C. 2cos sin
. .
D.
22
72. The vectors Ai i+Aj and 2i Ak are coplanar if:
" II: l±2V5
III:
0
Which of the following is correct?
A.
I and III only
B.
I and II only
C.
II and III only
D.
II and III
73. If the vectors ai i+bj and i+j ck (where a,b and c are coplanar, then
A.
B. 1
C I-abc
•
D.O
74. If 2sino:.
l+cosO:+SlnO:
equal to
A. y
B. t
C. 1-y D.l+y
y then l-coso+sino is
l+sina
75. The value of the expression
cos2 X 1 +sinX +sinX cos X cos X 1 -sinX
is equal to
A. sinX
B.O
C. cosX
D.l
1. It takes 6 technicians a total of 10 hours to install a new equipment from scratch, with each working at the same rate. If six technicians start to install the same equipment at 11:00 am, and one technician per hour is added beginning at 5:00 pm, at what time will the equipment installation be complete?
A. 6:40 pm
B. 7:00 pm
C. 7:20 pm
D. 8:00 pm
2. In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not lie in the same row or column?
A. 56
B. 896
C. 60
D. 768
3. Gopal's Shop sells small cookies in boxes of different sizes. The cookies are priced at Rs.2 per cookie up to 200 cookies. For every additional 20 cookies, the price of the whole lot goes down by 10 paise per cookie. What should be the maximum size of the box (in terms of number of cookies it can hold) that would maximize the revenue?
A. 240
B. 300
C. 400
D . None of these
4. 3 small pumps and a large pump are filling a tank. Each of the three small pumps works at the rate of the large pump. In what fraction of the time that all 4 pumps working together will fill the tank in comparison to the time taken by the large pump
. alone?
A.
B.
C.
D.
5. Four cows are tethered at four corners of a square plot of side 14 meters so that the adjacent cows can just reach one another. There is a small circular pond of area 20 m2 at the centre. The area left ungrazed is:
A. 22 m2
B. 42 m 2
C. 84 m 2
D. 168 m 2
Answer questions 6 and 7 using the
following information: 8 trees, viz.
mango, guava, papaya, pomegranate,
lemon, banana, raspberry and apple
are planted in two rows, of four each
aligned East-West. Lemon is between
mango and apple but just opposite to
guava. Banana is either at the end of
a row and is just immediately to the
right of guava, or Banana is just next
to Guava. Raspberry is at the end of
a row, and mango is at the other end
of the opposite row.
6. Which of the following is always true
1
A. Apple is just next to Lemon
B. Papaya is just next to Apple
C. Raspberry is either to the left or to the right of Pomegranate
D. Pomegranate is diagonally opposite to Banana
7. Which of these is directly opposite Banana?
A. Pomegranate
B. Mango
C. Papaya
D. None of these
8. How many number of times will the digit be written when listing the integers from 1 to 1000?
A.300
B. 271
C. 252
D.304
9. There are 10 positive real numbers nl nz n3·.· nlO· How many triplets of these numbers nz, n3, ... can
be generated such that in each triplet the first number is always less than the second number; and the second number is always less than the third number?
A. 45
B. 90
C. 120
D. 180
10. There were two women amongst other men who took part in a chess tournament. Every participant played two games with every other participant. The number of games which only men played was exactly 104 games more than those played which involved a women. The total number of participants is
A. 13
B. 11
C. 14
D.15
Answer questions 11 and 12 using the following information: In the English alphabet there are 11 symmetric letters that appear the same when looked at in a mirror. Other 15 letters in the alphabet are asymmetric letters.
11. How many four-letter computer passwords can be formed using only the symmetric letters (no repetition allowed)?
A. 7920
B. 330
C. 14640
D. 419430
12. How many three-letter computer passwords (no repetition allowed) can be formed with at least one symmetric letter?
A. 990
B. 2730
C. 12870
D. 15600
13. 'Which letter in the word RUTHLESS has a position from the beginning of the word that is half as much as its position when seen in the alphabet
A. E
B. L
C. H
D. U
14. A wire of some length is bent in circular form and has an area of 308 sq. cm. If the same length of wire is straightened out and bent in the form of a square, the approximate area of the square in sq. cm may be
A. 121
B. 242
C. 308
D. 69.29
15. Consider a square circumscribed by a circle with a radius of 4 units. The area of the square in square units is
A. 16K
B. 16 J2
C.32
D.64
16. A man returns after shooting and catching birds in his bag. He was asked how many birds he had in his bag. He said, "They are all house sparrows but six, they are all pigeons but six, and all doves but six." The number of birds he had in all were
A.36
B. 18
C. 9
.D.27
17. Suppose an ant is placed on one corner of a sugar cube, which has equal sides of 1.5 cm each. If the ant may walk only along the edges of the cube, what is the maximum distance the ant may walk on the cube without retracing its path?
A. 18 cm
B.9cm
C. 10.5 cm
D. 13.5 cm
18. Vhen the big hand of the clock is exactly at the 12 clock position, an ant starts to crawl in a counter clockwise direction from the 6 clock position at a constant speed. On reaching the big hand of the clock, the ant turns around and at the same speed, starts to crawl, in the opposite direction. Exactly 45 minutes after the first meeting with the big hand the ant crosses the big hand for the second time and dies. How long has the ant been crawling?
A. 54 min.
B. 51 min.
C. 1 hr, and 9 min.
D. 1 hr, and 21 min.
19. One third of Shrihari's marks in Math equal half of his English marks. Shrihari noticed that in these two subjects his marks totaled 150. What did Shrihari score in English
A. 15
B. 60
C. 30
B-9
D.90
20. The sum of two digits of a number is
15. If 9 is added to the number then the digits get reversed. Which of the following is FALSE about the number
A. The number is divisible by 3
B. The number has the two digits separated by a difference of one
C. The number is divisible by 6
D. The number is divisible by 9
21. An ant is at a point P in a planar square field. It was observed that P is 13 feet from the corner and 17 feet from corner B (diagonal to and finally 20 feet from a third corner. The area in square feet of the field is
A. 231
B.89
C. 369
D. 169
22. Consider the triangle ABC with sides AB=20 em., AC=l1 em., and BC=13 em. Then the length in em. of the diameter of the semi-circle inscribed within ABC, which lies on AB, and has sides AC and BC as tangents is given by
A.9
B. 11
C. 10
D. 10.5
23. On a straight road XY, 100 meters long, five heavy stones are placed 2 meters apart beginning at the end X. A worker, starting at has to transport all the stones to by carrying
4
only one stone at a time. The minimum distance he has to travel (in meters) is:
-A. 422
B. 480
C. 744
D. 860
24. Given that abca abca, where all c are integers, then which of the following is true
A. c=9
B. a=l
C. b=4
D. None of these
25. Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is:
A.
B.
C.
D.
Part B
A. X n is positive if n is even
B. X n is positive if n is odd
Answer the questions 26-27 based on the algorithm given below: C. X n is negative if n is even
D. None of these
STEP START STEP C FLAG Given that n is an integer, which of STEP INPUT TABLE A[1 .. 10J the following is TRUE about the sum
S 2n 3n
STEP3: FORI=1TO 10 DO IF 1J THEN
A. S is never the square of a rational
N N 1
number
IF (FLAG FALSE) THEN B. S is always the square of a rational number
FLAG TRUE
ENDIF
C. S is the square of a rational num
ENDIF
ber, provided n is even
ENDFOR
D. S is the square of a rational num
STEP OUTPUT C
ber, provided n is odd
STEP END
30. !f then
Given that IS:
10
A.
f
26. What are the values of Nand
B. f(x
N= 4
B. 10 c. (.x
C. C D.
l+xy
D. N=3,C=10
31. In an equilateral triangle, let h. be the
27. What are the values of Nand C if height and r be the radius of the cirthe statement "FLAG TRUE" is
cumcircle. Then the ratio r h is
removed from the algorithm?
A.
A. N C 8
B.2:3
B. N=4,C=10
C. 2n
C. N 8
D. n:V5/2
D. N C 10
32. Let ¢ denote an empty set. The power
28. The nth element of a series is represet of the empty set is sented as Xu X If
A. ¢
X o x and x then which
of the following is always true: B.
c.
D. None of these
33. Let A and B be two non-empty sets with cardinality p and q respectively. Then the total number of relations that can be defined from the set A to set B is
D. p, or or pv
34. Consider the relation f defined
X2 0< X .
3x 3 x 10 and the
relation 9 defined by X2 0 X 2 3x then
A. f is a function but 9 is not
B. both J and 9 are functions
C. f is not a function but 9 is
D. neither of them are functions
35. Let f be subset of Z x where Z denotes the set of integers, and we have J b E Z}. Then f is
A. bijective function
B. injective function
C. surjective
D. none of these
36. For real numbers and nEZ, sin x sin y implies which of the following?
A. J: fiT!
B. x mr
C. X 2n7f ± y
D. x n7f ± y
37.. Show the bits in a 12 bit-register that is holding the number equivalent to decimal 215 in binary coded octal and binary coded decimal
A. 000011000011,001000010101
B. 000011010111, 001000010101
C. 000011000111,001000010111
D. none of the above
38. If 587 the base of the number system is
A. 14
B. 11
C. 9
D. 12
39. At how many points in the xy plane do the graphs of y X 12 and y 2x intersect?
A. One
B. None
C. Two
D. Three
40. Suppose that f is a continuous realvalued function defined OIl the dosed interval Which of the following are true for constants E and x,y E
1. There is C such that
I y.
II. There is D such that
I y that satisfy
Ix-y D.
III.
There is E f'lwh th=tt.
I E Ix Ifx, y
A.
III only
B. I only
C. I and II only
D. I, II and III
41. Suppose that j is a function on the set of real numbers, and is differentiable twice, and that are all negative. Suppose j" has all the three properties in the interval
I. It is increasing,
II. It has a unique zero,
III.
It is unbounded.
Which of the same three properties
does f necessarily have
A.
II only
B.
I only
C.
III only
D.
II and III only
42. Let 2x3 ax2 3x and X3+x2-4x-9. Both give the same reminder when divided by x if a is
A.10
B. 1
C.5
D. -5
43. Consider the unit square formed by points in the plane; and Let P be an arbitrary point chosen in this unit square. Connect P to the points A and B. The probability that the points ABP form an obtuse triangle is
0.5
B. 0.393
C. 0.712
D. 0,25
44. Let j and g be two functions defined on an interval I such that 0 and Vx E and f is strictly increasing in I. Then the product function jg is
A. strictly increasing in I with fg 0
B. strictly increasing in I with fg a
C. strictly decreasing in I with jg
a
D. nothing can be said about it from the given information
45. The derivative offunction where
sinx2
x
a x a
A. exists at x but is not continuous
B. does not exist at x a
C. exists at x and is also continuous
D. none of the above
46. If denotes the largest integer and if (where ever it exists) is given by
A.
B.2x
C.2[x]
D. exists nowhere
47. The sum of the interior angles of a polygon is equal to 86 right angles. Then the number of sides of that polygon is
A.43
B.86
C.30
D.45
48. The area of the polygon whose vertices are and x is equal to
A.2x
B. x2/2
C. 1
D.2
49. Use the notation I x I to denote the absolute value of and denotes the largest integer smaller or equal to
x. For an integer, such that a
let be given by {llx-l 1 x<a
l
a
Which of the following is the set of
points of discontinuity for the function
A. All integers a
B. All integers
C. All a
D. a
50. Consider the function f(x {X2;y2
Which of the following is NOT true for this function
A. f is continuous along the line
o
B. f is continuous along the line
o
C. f i8 constant along the line x
D. f is not continuous at (0.0).
51. Suppose c are all and if D is the determinant below, which of the
following is true about D
abc
b c a cab
1. D II. D III. D 0
A. II only
B. I only
C. both I and II
D. both II and III
52. Suppose C are all and are in a geometric progression as the successive terms .. " .... Then what is the value of the determinant
log a p 1
D below? D 10gb q 1
log c r 1
A.l
B. 0
C. pqr
D. pqr -abc
53. Consider the determinant
1 sin e 1
sin e 1 sin e
-sin g 1
Then 6./2 for all values of lies between
A. 1 and 2
B. 0 and 1
C. 2 and 4
D. >4
54. Which of the following is equal to the nullity of matrix A below
8
10
264
A.l
B. 0
C.2
D.3
55. How many 2-digit or 3-digit numbers ca,n be formed using the digits and are divisible by
A.64
B.56
C.80
D.92
56. Let (256ho x and if 2x then the values of B are respectively
A.
B.
C.
D.
57. The average temperature of a town in the first four days of a month was 58 degrees. The average for the second, third, fourth. and fifth days was 60 degrees. If the temperatures of the first and fifth days were in the ratio 7:8 the temperature on the fifth day was
A.62
B.64
C.56
D. None of these
58. Two boys A and B speak the truth only 75% and 80% of the time respectively. Lets say both witnessed an incident, what is the percentage of time that the two would contradict each other when narrating the same incident?
A. 25
B. 15
D. 45
59. Solution of the initial value problem y" -6y giveny(O) 10; -10 is
A. 4e-6t lOet
B. e6t e-t
C. e-6t 50 et
7
D. 20e-3t 30e2t
60. When 4 dice are thrown, what is the probability that the same number appears on each of them?
A. 1/36
B. 1/1296
C. 1/216
D. 24/216
61. The meall deviation about the median for the data 12, 10, 18, 19, 21 is
A.5.27
B. 1.01
C.9
D. 10.01
62. Let for x -10 where f is an increasing function then
9
A. is a decreasing function of x
B. is an increasing function of x
C. is increasing for x and
decreFiSing for ·-10 x 0
D. none of these
63. The solution to the indefinite integral
Q
XFLAG:'" 1
du, is given by
NO
YFLAG= 1
A. -sin-1 C
B. 2u2a2J(a2-u2) -sin-1 C
1
C. C
D. 2u2a2J(a2-u2) -sinh-1 C
64. Consider the circles, all of unit radius given by the form
1. The radius of curvature at any
point for these circles may be
also expressed as
A.
B.
C.
D. None of these
65. Solution to the differential equation d4 y 4y 0 is
dx4 .
A. y e-2x (cl cos x c2sinx) A. log y -xy C
e2x (c3 cos x C4 sin .,. 1
B. 1og'" c
B. y e-X(cl cos x c2sinx) y xy
C. log; -xy
eX (C3 cos X C4 sin D. log 1L
xy
c. y e-X(cl cos 2x C2 sin 2x) X
eX (C3 cos2x C4 sin 2x)
Answer the following 3 questions
D. y -eX (Cl cos X C2 cos based on the flowchart given above:
eX (C3 sin x C4 sin
67. What is the output of the flowchart if
66. The solution to the differential equaX -10 and Y 3
tion xy)ydx -xy)xdy
is A. Q and X
10
B. Q 3 and X
C. Q -3andX 1
D. Q= 6and 1
68. Which of the following conditions can be used in place of the condition "IF ((XFLAG 0 AND YFLAG OR (XFLAG 0 AND YFLAG
A. IF (XFLAG -YFLAG
B. IF (XFLAG -YFLAG
C. IF (XFLAG YFLAG
D. IF (XFLAG YFLAG
69. For what values of X and the flowchart is never going to terminate?
A. X> 0andY 0
B. X< 0 and Y 0
C. X=0andY
D. All of the above
70. A person speaks truth only 4 times out of 5. A die is tossed and the person says that the die rolled a six. Find the probability that actually there was a six
A.4/5
B. 2/9
C. 7/9
D.4/9
71. cos A +cos B can also be written as
A. 2cos--sm-
22
B. 2cos cos
C. 2cos sin
. .
D.
22
72. The vectors Ai i+Aj and 2i Ak are coplanar if:
" II: l±2V5
III:
0
Which of the following is correct?
A.
I and III only
B.
I and II only
C.
II and III only
D.
II and III
73. If the vectors ai i+bj and i+j ck (where a,b and c are coplanar, then
A.
B. 1
C I-abc
•
D.O
74. If 2sino:.
l+cosO:+SlnO:
equal to
A. y
B. t
C. 1-y D.l+y
y then l-coso+sino is
l+sina
75. The value of the expression
cos2 X 1 +sinX +sinX cos X cos X 1 -sinX
is equal to
A. sinX
B.O
C. cosX
D.l