Full ques. Paper in attachment
(D) 2 + V2 *
For n > 5 , the expression
1 + 2* + 3x^{2} +4x^{3} + + nx^{n}~^{1}, x*l, is equal to
nx^{n} (l jc) jc^{71} +1
.2
nx^{n} (x 1)  x^{n} +1
h \^{2} (1x)
nx^{11} (x 1) + x^{n} 1
(1 \^{2} {1x)
nx^{n}
^{(D)} /, V (1x)
4. The function /defined on R by
/(x) = 3^{x}+4*5^{x}
has
(A) exactly one zero
(B) exactly two zeros
(C) exactly three zeros
(D) infinitely many zeros
5. The spheres
x^{2} + y^{2} + z^{2} = 1 and x^{2} + (yVi}) +z^{2} =4
intersect at an angle
(A) 0
(B)
(C)
^{(D)} f
6. If Q denotes the region bounded by the Xaxis and the lines y = x and x = 1, then the value of the integral

a 
is 


(C) cos2 
7. Let f be an increasing, differentiable function. If the curve y = f{x) passes through (l, l) and has length
1
then the curve is
(A) y = ln(Vx)l
(B) y = lln(Vx)
(C) y = ln(l + Vx)
(D) y = l + ln(\/x)
8. If the line y = mx, 0 < x < 2 is rotated about the line y = 1, then the area of the generated surface is
(A) 4r(l+ /n)Vl + m
(B) 4"(l + /n^{2})VT+zn
(C) 4zr(l + \/m)Vl + m^{2}
(D) 4(l + 7n)Vl + m^{2}
9. Let D be the region in the first quadrant lying between x^{2} +y^{2} = 1 and x^{2} +y^{2} = 4 . The value of the integral
JJsin (x^{2} +y^{2})dx dy
is
(A) (cos 1  cos 2) 4
(B) (cos 1cos 4) 4
(C) (cos 1cos 2)
2
(D) (cos 1cos 4)
2 f(x, y) = (x + yf  (x + y) +1.
The absolute maximum value and the absolute minimum value of the function on the unit square {(x, 3/) : 0 < x < 1, 0 < y < l}, respectively are
(A) 3 and 
2
(B) and
2 4
(C) 3 and 
4
(D) 2 and 
4
11. For A > 0, the value of the integral
je^{Ax2}dx
0
equals 

Then x+y~ is equal to dx ^{J} dy ^{H}
(A) z
(B) 0
(C) 2
(D) 2z
13. Let
f(x) = x^{3}x^{2} +1, 0<ac<l.
Then the absolute minimum value of f(x) is 14
(A)
27
(B) I
9
23
27
(D) 1
14. The slope of the tangent line to the curve
x=a(tsini), y = a(lcosi), te R,
tr * ^{n}
at t = is
2
(A) 1
(B) 0
(C) 1
(D) oo
15. Consider the equations
sin(cosx) = x (1)
and
cos(sinx) = x (2)
for x > 0. Then
(A) Only Equation (1) has a solution
(B) Only Equation (2) has a solution
(C) Both Equations (1) and (2) have solutions
(D) Neither Equation (1) nor Equation (2) has a solution
16. If
 a+h
lim [ e dt = 1, h. j
h*o h
a
then the value of a is
(A) 1
(B) 0
(C) 1
(D) 2
17. Let
fix, y) = x^{3} +;y^{3} +3x^{2} 3y^{2} 5.
Then the local maximum and the local minimum of the function f are at the points
(A) (2,0) and (2, 2), respectively
(B) (2, 0) and (0, 2), respectively
(C) (0, 2) and (2, 0), respectively
(D) (0, 2) and (0, 0), respectively
18. Let 9, 0 < 6 < n be the angle between the planes x  y + z = 3 and 2xz = 4.
The value of 6 is '1'
(A) cos
(B) cos1
(C) cos1
(D) cos1
' 1 ^{N} <Vl5 j
' 3 ' ,Vl5,
19. Let
f(x, y) = xy^{2} +yx^{2}.
Suppose the directional derivative of fin the direction of the unit vector (u_{1}} (l, l) is 1. Then among the following, (u, u_{2}) is
(A) (1,0)
(B) (0,1)
(C) (1,0)
V2V2
20. The point on the sphere x^{2} +y^{2} +z^{2} =1 farthest from the point (l, 2, l) is
(A)
V V V.
1 _2_ J_
VVV.
^{/} 1 2 1 VeVV
21. Consider the dihedral group Z)_{4} = {e, r, r^{2},r^{3}, f, rf, r^{2}f, r^{3}f} with r^{4} = ef^{2} and rf = fr~ The product r^{z}fr~^{1}f~^{l}r^{3}fr corresponds to
(A) 
f 
(B) 
rf 
(C) 
r^{2}f 
(D) 
r^{3}f 
Let p 
and q be distinct 
Suppose S = H n{p,q, 
(A) 
\pQ,P^{q},Q^{P}} 
(B) 
{p + , PQ,P^{9}} 
(C) 
(p,pq, P^{9}) 
(D) 
{p,p + q, pq} 
23. The number of group homomorphisms from the group ( Z_{18}, _{18}) to the group (Z_{30}, _{30} ) is
(A) 3
(B) 4
(C) 5
(D) 6
24. Let <7 = (125) (36) and z = (1456) (23) be two elements of the permutation group on 6 symbols. Then the product a, where crr(i) = cr(r(i)), is
(A) (14) (26) (35)
(B) (13) (26) (45)
(C) (14) (25) (36)
(D) (13) (24) (56)
25. Let G = {ns 1 < /z < 55, gcd (n, 56) = 1} be a multiplicative group modulo 56. Consider the sets
Sj = {1, 9,17, 25, 33, 41} and S_{2} = {l, 15, 29, 43}.
Which one of the following is TRUE?
(A) S_{x} is a subgroup of G but S_{2} is NOT a subgroup of G
(B) is NOT a subgroup of G but S_{2} is a subgroup of G
(C) Both S_{1} and S_{2} are subgroups of G
(D) Neither S_{1} nor S_{2} is a subgroup of G
26. Let G be a group with respect to multiplication. If x = a42 + /?V3 g G then x ^{1} is ot42+/?V3
(A)
(B) 
2a^{2} 
+ 3J3^{2} 
a2 
3 
2 a^{2} 
3 f3^{2} 
a2 
+ 0yl3 
2a^{2} 
3 /3^{2} 
a42 
pS 

^{(D)} 2a^{2} + 3/?^{2} 
27. Let G = {l, 2, , p 1} be the group with respect to multiplication modulo p. If the inverse of 110 in G is 4, then p is of the form
(A) 5n + 1
(B) 5n + 2
(C) 5/z + 3
(D) 5n + 4
28. Consider the alternating group A_{i} = {ctg S_{4} : a is an even permutation}. Which of the following is FALSE?
(A) A_{4} has 12 elements
(B) A_{4} has exactly one subgroup of order 4
(C) A_{4} has a subgroup of order 6
(D) Number of 3 cycles in A_{4} is 8
29. Let P be a 3x3 matrix such that for some c, the linear system Px = c has infinite number of solutions. Which one of the following is TRUE?
(A) The linear system Px = b has infinite number of solutions for all b
(B) Rank (P) = 3
(C) Rank(P)*l
(D) Rank (P)< 2
30. Let
/(jc) = jc^{3} +x^{2} x+15 and g(x) = x^{3} + 2x^{2} x + 15 .
Then, over Q
(A) /"is irreducible and g is reducible
(B) f is reducible and g is irreducible
(C) Both f and g are reducible
(D) Both f and g are irreducible
31. Let

'I 
a 
O' 

r r 


* 
f^{a} 
0 
0^{N} 

rr 

< 
0 
1 
0 

i 
ae R 
and V = 

0 
a 
0 

l 
:ae R 

,0 
0 
1, 

,oJ 




0 
K 



The angle between U and V is
(A) 0
(E) f
(_{C})
4
) f
P =
,1 3 1 Then 8P^{_1} is equal to
v

' 13 
4 
i\ 
(A) 
15 
4 
3 

,io 
0 
2, 

'13 
15 
10'^{N} 
(B) 
4 
4 
0 

ll 
3 
2, 

"13 
10 
^{15}1 
(C) 
4 
0 
4 

ll 
2 
3 , 

M
CO 
4 
^{1}! 
(D) 
10 
0 
2 

115 
4 
3, 
33. Let P be an n xn idempotent matrix, that is, P^{2} = P. Which of the following is FALSE?
(A) P^{T} is idempotent
(B) The possible eigenvalues of P are 0 or 1
(C) The nondiagonal entries of P can be zero
(D) There are infinite number of n x n nonsingular matrices that are idempotent
34. Let V be the vector space of all polynomials with real coefficients. If W is the vector subspace of V generated by
1x, x^{2} x, x^{2} 1 and x^{2} 3x+ 2,
then the dimension of W is
(A) 1
(B) 2
(C) 3
(D) 4
35. Let
P =
Then
(A) P has two linearly independent eigenvectors
(B) P has an eigenvector
(C) P is nonsingular
(D) There exists a nonsingular matrix S such that S^{_1}PS is a diagonal matrix
36. Let u, ve R^{3}, v * 0. Which of the following is FALSE?
is the length of the projection of u along v
(B) If u w = v w for all w e R^{3}, then u = v
(C) u.v = !(u_{+}vfuvf)
(D)  a + v f +1 u  v f = 2 (HI^{2} + M^{2})
37. Let P be a 2x2 matrix such that P^{102} = 0. Then
(A) P^{2} = 0
(B) (JP)^{2}=0
(C) (/ + P)^{2}=0
(D) P = 0
38. Let
0 1
P =
i
The eigenvectors corresponding to the eigenvalues i and  i are respectively
fl') 
X 
and 
J, 

and 
W 
i
i 
and 
(A)
39. The area of the parallelogram with sides x = i + j+k and y =  i + j
is
(A) &
(B) 2V3
(C) 3V2
(D) 6
> > x=i + ./+ &, y = ai + k and z = i + aj .
Then the volume of the parallelopiped with sides x, y and z is
(A) 1 + a+a^{2}
(B) 1 + aa^{2}
(C) 1a + a^{2}
(D) a^{2}+al
41. The solution of the initial value problem xy'y = 0
with y( 1) = 1 is
(A) y{x) = x
1
(B) y{x) = 
x
(C) y(x) = 2x 1
cd)
2x 1
42. Let y(x) = xsinx be one of the solution of an n^{th} order linear differential equation with constant coefficients. Then the minimum value of n is
(A) 1
(B) 2
(C) 3
(D) 4
f_{x}z \
hx + siny dy = 0
3
is
X^{3}v X^{2}V^{2}
(A) + cos y ^{=} c
3 2 J
3 2 2
/r\ ^{x} y ^{x} y
(B)  +h COS y  C
3 2
x^{3} xV (O cosy = c
3 6
X^{3} x^{2}v^{3}
(D) + + cosy=c
3 6 J
44. The general solution of the differential equation
/// rr / a
y +y yy=0
is
(A) +xc_{2} + x^{2}c_{3})e*
(B) ip_{1}+xc_{2}+x^{2}c_{a})e~^{x}
(C) c_{1}e^{x} +(c_{2} + xc_{3})e~^{x}
(D) (cj +xc_{2})c* + c_{3}e^{x}
45. Let
f(x) = 2x^{3} x^{2} +2x5 .
Consider the following statements about the roots of f(x) = 0 P : At least one root is positive.
Q : At least one root is negative.
R : There is a root between x = 1 and x = 2.
Which one of the following is TRUE?
(A) P, Q and R are valid statements
(B) P and Q are valid statements but R is NOT a valid statement
(C) P and R are valid statements but Q is NOT a valid statement
(D) P is a valid statement but Q and R are NOT valid statements
46. The maximum absolute error that occurs in rounding off a number after 6 places of decimal is
(A) 5xl0^{8}
(B) 1(T^{7}
(C) 5xlCT^{7}
(D) 5x10"
47. Which of the following is FALSE?
(A) A unique interpolating polynomial of degree n is obtained from the given values at fixed n +1 points
(B) The Lagrange interpolation formula can be applied to equispaced points
(C) The Newtons forward difference interpolation formula can be applied to nonequispaced points
(D) The trapezoidal rule gives exact value of the integral for linear functions
X 
1 
0 
1 
2 
3 
fix) 
1 
5 
3 
1 
5 
Applying Simpsons one third rule, the value of the integral
3
jY(.x) dx
is
(A) 10
(B) 12
(C) f
(D) 15
49. Consider
f(x) = l + xe~^{x}.
The NewtonRaphson iterative scheme for finding a root of f(x) = 0 is
l + x^{1}e~^{x}"
(A) x_{n+1} =
(x_{n} l)e*
_ ^{x}\ e~^{x}" +x(L + e~*)l
l + xe ^{n}
_{(C}) _{r} _x^{2}ne~^{x}* +x_{n}(le^{x}')+l
vW ^{x}n+l :
l + xe "
50. Consider the following Primal Linear Programming Problem :
Maximize c^{T}x
Subject to JPx = b x>0
The Dual Linear Programming Problem is
(A) Minimize y^{T}b Subject to : P^{T}y = c, y unrestricted
(B) Minimize y^{T}b Subject to : P^{T}y>c, y unrestricted
(C) Minimize y^{T}b Subject to : P^{T}y = c, y>0
(D) Minimize y^{T}b Subject to : P^{T}y>c, y>0
51. If the Primal Linear Programming Problem is unbounded then which of the following is TRUE?
(A) Dual problem is unbounded
(B) Dual problem has a single bounded optimal solution
(C) Dual problem has multiple bounded optimal solutions
(D) Dual problem is infeasible
52. Which of the following pair of linear programming constraints is equivalent to the inequality  x_{x}  x_{2}\ a ?
(A) x_{1}x_{2} <a, x_{2}x_{1} <a
(B) x_{1}x_{2}<a,x_{2}x_{1}<a
(C) x_{1}x_{2}<a, x_{2}x_{1}<a
(D) x_{1}x_{2} < a, x_{2}x_{1}<a
53. Consider the following Linear Programming Problem : Maximize 3xj + 8x_{2}
Subject to 2x_{x} +5x_{2} <10 6! + x_{2} 6 x_{lt} x_{2}>0
The optimal value of the objective function is
(A) 
0 
(B) 
3 

111 
(C) 


7 
(D) 
16 
54. A cow is tied with a pole by a 100 meter long rope. What is the probability that at some point of time the cow is at least 60 meters away from the pole? q
^{(A) 25}
(B)
25
(C) 25
18
25
55. Two letters are chosen one after another without replacement from the English alphabet. What is the probability that the second letter chosen is a vowel?
4_
25
5_
26
_5_
25
1 _1_ 5 26
56. Let X be a binomial random variable with parameters n and p. If the mean and the
3
standard deviation ofX are 3 and , respectively, then what is the value of (n, p)?
2
(A) 4,J
(B) I 6,
2)
9, 
I 3 J
/
(D)
57. Let X,Y,Z be independent Poisson variables, such that E(X) = E(Y) and E(Z) = 2E(X). If P{X = 5, Y = 4) is equal to P(Z = 8), then E (X) is
58. The largest natural number whose base 7 representation has exactly four digits, is
(A) 2400
(B) 6666
(C) 7777
(D) 2401
59. 10s complement of the decimal number 56789 is
(A) 01234
(B) 12345
(C) 43210
(D) 43211
60. Let x = 0.125E + 01, y = (l.0l)_{2} and z = (l.2)_{g} . Which of the following is TRUE?
(A) x, y and z are equal
(B) Only x and y are equal
(C) Only x and z are equal
(D) All x, y and z are different
61. The decimal value of (2l)_{8} x(l0l)_{16} lies in the interval
(A) 3000  3499
(B) 3500  3999
(C) 4000  4499
(D) 4500  4999
62. The binary equivalent of the hexadecimal number A52C is
(A) 1010101101100
(B) 1010010100101100
(C) 1010111000101100
(D) 1010010100101101
63. Let x, y and z be Boolean variables. The number of possible values for the expression
xy + zx
is
(A) 1
(B) 2
(C) 4
(D) 8
64. Let x and y be independent Boolean variables, each taking values 0 or 1 with probabilities
0.5 and 0.5, respectively. The probability that
x + y(x+y) = 1
IS 

(A) 
0 
(B) 
0.25 
(C) 
0.5 
(D) 
0.75 
.s
65. The Boolean expression
(x + y)(y + z)(z+x)
is equal to 
(A) 
xyz 
(B) 
xyz 
(C) 
(x+z)y 
(D) 
{x + z)y 
66. Let X and Y be 4 bit registers with initial contents as 1011 and 1001, respectively. The following sequence of operations are performed on the two registers :
Y<rXY
X<XY
Y<rXY
where denotes XOR operation. The final contents of the two registers are
(A) X =1001, Y = 1011
(B) X =1011, Y= 1001
(C) X = 1011, Y= 1011
(D) X =1001, Y = 1001
The logic circuit diagram shown in Figure 1 is equivalent to the Boolean expression
(A) x + y
(B) x + y
(C) x+y
(D) x+y
The logic circuit diagram given in Figure 2 is equivalent to
(A) AND gate
(B) OR gate
(C) NAND gate
(D) XOR gate
69. BIOS is the acronym for
(A) Binary Input Output Source
(B) Basic Input Output Support
(C) Binary Input Output System
(D) Basic Input Output System
70. The maximum number of characters that can be encoded in a fixed length encoding sehen with n bits is
(A) 2"
(B) n!
(C) n2
(D) n
71. Which of the following is an 8bit processor?
(A) Intel 80286
(B) Intel 8086
(C) Intel 8085
(D) Intel Pentium II
72. For which of the following combinations an SR FlipFlop is set to 1?
(A) S = 0, R = 0
(B) S = 0, R = 1
(C) S = 1, R = 0
(D) S = 1, R = 1
73. For which of the following combinations, a JK FlipFlop will enter into the complement of the present state?
(A) J = 0, K = 0
(B) J = 0, K = 1
(C) J = 1, K = 0
(D) J = 1, K = 1
74. Which of the following is NOT a Software?
(A) Adobe
(B) Browser
(C) Compiler
(D) Device Driver
List 1
1. Operating Systems
P. Pentium Q. Linux R. Router S. Anti Virus
2. Application Software
3. Processor
4. Network
(A) (1, Q), (2, S), (3, P), (4, R)
(B) (1, Q), (2, R), (3, P), (4, S)
(C) (1, P), (2, S), (3, Q), (4, R)
(D) (1, P), (2, R), (3, S), (4, Q)
76. Match the file extensions in List 1 with the corresponding applications in List 2

Listi 

List 2 
1. 
mp3 
P. 
image 
2. 
xls 
Q. 
music 
3. 
jpeg 
R. 
database 
4. 
mdb 
S. 
spread sheet 
(A) (1, Q), (2, S), (3, R), (4, P)
(B) (1, Q), (2, S), (3, P), (4, R)
(C) (1, Q), (2, P), (3, S), (4, R)
(D) (1, Q), (2, R), (3, P), (4, S)
(A) Magnetic Tape
(B) Hard Disk
(C) Floppy Disk
(D) CD
78. Which of the following is a valid C directive?
(A) # include <stdio.h>;
(B) # include <stdio.h>
(C) include <stdio.h>;
(D) include <stdio.h>
79. Consider the following declaration in C struct student {
char name [12] ; float gradepoint ;
J;
struct student MCA [5] ;
The number of bytes needed to store the array MCA is
(A) 16
(B) 25
(C) 70
(D) 80
80. Consider the following C statements
P : for (i = 0; i < 8; i+ = 3) {printf ("*");}
Q : for (i = 4; i > 0; i = 2) {printf ("*");}
R : for (i = 0; i <= 9; i+ = 3) {printf ("*");}
S : for (i = 0; i < 7; i + +) {if (i %3 = = 0) printf ("*");}
Which one of the following is a TRUE statement?
(A) P, Q, R and S give the same output
(B) P and S give the same output
(C) Q and R give the same output
(D) P, Q and S give the same output
81. Consider the following program segment {intx, i, j ;
x = 0;
for (t = 0; i < 19; i + +) for (j = i + l; j < 20; j + +)
* ++;}
The value of x after executing the segment is
(A) 171
(B) 190
(C) 342
(D) 380
82. Let f: N> N be defined as
w \ _ f 1> if n = 1 or n = 2
\f(n 1) + f (n  2), otherwise.
What is the value of /"(IO) ?
(A) 34
(B) 45
(C) 55
(D) 89
83. Consider the following program segment
{int n = 1; float x, term, float sum = 1; term = 1; while (n < 51)
{
term * = x*x/(n*(n +1));
sum + = term ; n + = 2;
}
}
For a given x the value of sum approximates the function
(A) sin*
(B) cosac
(C) e1
(D) e*2
84. Consider the following program void swap (int a, int b)
{int temp ; temp = a ; a = b ; b =a ;
}
void main ()
{int x, y; x = 2; y = 3;
swap (x,y);
printf ("re = %d y = %d \n", x, y);
}
The output of the program is
(A) x = 2 y2
(B) x = 2 y = 3
(C) x = 3 y = 2
(D) x = 3 y = 3
85. What is the output of the following C program?
void fun (int * p)
{int i, sum = 0 ; for (i = 2; <4; ++ i) sum + = * (p + i); printf ("%d", sum);
}
void main ()
{int a[ 5 ] = {10, 20, 30, 40, 50}; fun (a +1);
}
(A) 
90 
(B) 
120 
(C) 
130 
(D) 
140 
86. Consider the following C program segment
int gradepoint ; char ch; switch (ch) { case A : {gradepoint = 10 ;} case B : {gradepoint = 8 ; break ; } case C : {gradepoint = 6 ; } default : {gradepoint = 0 ;}}
Executing the program segment for ch = A, B\ C gradepoints are respectively
(A) 10, 8, 6
(B) 10, 8, 0
(C) 8, 8, 6
(D) 8, 8, 0
87. Consider the following C program
void main ()
{
int i, s ;
for (i = 0;; i + +)
{s = s + i / (i2); if (i > 5) break;
}
}
Which one of the following is a TRUE statement?
(A) There is a syntax error
(B) There is a type mismatch error
(C) There is a runtime error
(D) There is no runtime error
88. The unit place of the number 27^{82} is
(A) 1
(B) 3
(C) 7
(D) 9
89. The number of all functions is
(A) m(ml)(mn+l)
(B) n (rel) (/im+l)
(C) mn
(D) nm
90. The number of ways in which 4 boys and 5 girls can sit in a row so that there is a girl between any two boys is
(A) 4! 5!
(B) 3 (4! 5!)
(C) 5 (4! 5!)
(D) 15 (4! 5!) '
91. The next term in the series 191, 211, 232, 254, is
(A) 267
(B) 276
(C) 277
(D) 287
92. If sinx+cosx= then sin(2x) is
(A) 1 a2
(B) a2 1
(C) 1+a2
(D) a2
93. A student computes the sum of squares of the first 40 natural numbers and gives j incorrect answer 22019. By mistake, the student forgot to add the square of one of t numbers. The missed number is
(A) 5
(B) 7
(C) 9
(D) 11
94. For a,be Z, define a relation aRb if ab > 0. Then the relation R is
(A) symmetric, reflexive and transitive
(B) symmetric and reflexive but NOT transitive
(C) symmetric and transitive but NOT reflexive
(D) reflexive and transitive but NOT symmetric
95. What is the sum of the interior angles of an n vertex simple polygon?
(A) {n2)n
(B)
(C) l^{n+1}
96. Who among the following is NOT a Nobel Laureate?
(A) Amartya Sen
(B) J.C. Bose
(C) Muhammad Yunus
(D) S. Chandrasekhar
97. Who is the father of Bhishma in the Mahabharata?
(A) Bharat
(B) Devavrata
(C) Parashar
(D) Shantanu
98. Which country won the 2006 FIFA World Cup?
(A) Argentina
(B) France
(C) Germany
(D) Italy
99. Which of the following diseases is NOT caused by mosquito bite?
(A) Dengue
(B) Encephalitis
(C) Malaria
(D) Typhoid
is
(A) 2V2
(B) 3V2
(C) 2W2
(D) 3+V2
2 +
2 +
2+.
2 +
^{}1
l + xe ^{n}
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PAPER Indian Institute of Technology Guwahati (IITG) 2007 JAM Computer Applications  Question Paper
