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(Paper) ISAT Mathematics Sample Questions 2010

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ISAT Mathematics Sample Questions 2010

Question 1 : What is the value of x in the triangle shown?
A. 54
B. 59
C. 49
D. 24

Question 2: Last year there were 80 students enrolled in the eighth-grade class. This year the number of students enrolled in the eighth-grade class increased by 10%. How many students are enrolled in the eighth-grade class this year?
A. 88
B. 8
C. 90
D. 81

Question 3 : A company packs its coffee into cylindrical containers. The height of each container is 6 inches, and the radius of the container is 3 inches.
Which is closest to the volume of one of these cylindrical containers?
A. 36 cubic inches
B. 113 cubic inches
C. 54 cubic inches
D. 170 cubic inches

Question 4 : The student council is making snack bags for a class trip. Each snack bag will contain:
• 1 type of drink
• 1 type of cookie
• 1 type of fruit
To make each snack bag, they will choose from 2 types of drinks, 4 types of cookies, and 2 types of fruit.  How many combinations of 1 type of drink, 1 type of cookie, and 1 type of fruit are possible?
A. 48
B. 8
C. 3
D. 16

Question 5 : Which is closest to the circumference of this circle?
A. 14 inches
B. 63 inches
C. 20 inches
D. 28 inches

Question 7 : Which of the following is equivalent to the expression shown?
4x – 5 – 2x – 3
A. 6x + 2
B. 2x + 2
C. 6x – 8
D. 2x – 8

Question 8 : Between which two consecutive integers is ?
A. 6 and 7
B. 100 and 101
C. 75 and 76
D. 17 and 18

Question 10 : Amy has of a yard of string to make bracelets. Each bracelet requires of a yard of string. What is the greatest number of bracelets Amy can make with this length of string?
A. 3
B. 6
C. 4
D. 8

Question 11 : Look at the addition patterns below.
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
How many consecutive odd integers starting with 1 must be added to produce 64?
A. 8
B. 6
C. 7
D. 9

Question 12 : Which point on the number line below represents the value ?
A. Point Q
B. Point S
C. Point P
D. Point R

Question 14 : Paula multiplied a number by 16. Her result is a positive number less than 16. Which of these did Paula multiply by 16?
A. A number greater than one
B. A number less than zero
C. A number between zero and one
D. Zero

Question 15 :

(Download) Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2009)

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Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2009)

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Papers : Chemistry, Mathematics, Biology & Answer Keys of Year 2009

(Download) Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2008)

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Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2008)

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(Download) Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2007)

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Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2007)

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Papers : Physics, Chemistry, Mathematics, Biology & Answer Keys of Year 2007

(Download) Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2006)

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Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2006)

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Papers : Physics, Chemistry, Mathematics, Biology & Answer Keys of Year 2006

(Download) Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2005)

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Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2005)

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Papers : Physics, Chemistry, Mathematics, Biology & Answer Keys of Year 2005

(Download) Karnataka Common Entrance Test (KCET) Previous Papers & Answer Keys (2004)

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(Download) GATE 2010 : Electrical Engineering (EE) Paper

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1.An ammeter has a current range of 0 – 5 A, and its internal resistance is 0.2Ω . In order to change the range to 0 – 25 A, we need to add a resistance of
(A)0.8Ω in series with the meter (B) 1.0Ω in series with the meter
(C)0.04Ω in parallel with the meter (D) 0.05Ω in parallel with the meter

2.As shown in the figure, a negative feedback system has an amplifier of gain 100 with ±10% tolerance in the forward path, and an attenuator of value 9/100 in the feedback path. The overall system gain is approximately:

(A) 10±1%
(B) 10 ±2%
(C) 10 ±5%
(D) 10 ±10%

3.A three-phase, 33kV oil circuit breaker is rated 1200A, 2000MVA, 3s. The symmetrical breaking current is
(A) 1200 A (B) 3600 A (C) 35 kA (D) 104.8 kA

4.Assuming that the diodes in the given circuit are ideal, the voltage V 0 is
(A) 4V
(B) 5V
(C) 7.5V
(D) 12.12V

5. The fully controlled thyristor converter in the figure is fed from a single-phase source. When the firing angle is 0°, the dc output voltage of the converter is 300 V. What will be the output voltage for a firing angle of 60°, assuming continuous conduction?
(A) 150V
(B) 210V
(C) 300V
(D) 100πV

6.A box contains 4 white balls and 3 red balls. In succession, two balls are randomly selected and removed from the box. Given that the first removed ball is white, the probability that the second removed ball is red is
(A) 1/3 (B) 3/7 (C) 1/2 (D) 4/7

7.The armature resistance of the motor is,
(A)2Ω (B) 3.4Ω (C) 4.4Ω (D) 7.7Ω

8. 25 persons are in a room. 15 of them play hockey, 17 of them play football and 10 of them play both hockey and football. Then the number of persons playing neither hockey nor football is
(A) 2 (B) 17 (C)13 (D) 3  .. [..]

(Download) BITSAT 2009 Exam Paper (Math, Physics, Chemistry English And Logical Reasoning)

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Q1. A boat takes time t to go downstream from point A to point B and time 2t to go upstream from point B to point A . If the speed of the stream is v and the speed of the boat with respect to the stream is u , then which of the following gives the correct relationship between u & v ?
a) u = v
b) u = 3v
c) v = 2u
d) v=3u

Q2. Which of the following is correct about the acceleration of a particle describing a uniform circular motion ?
a) The acceleration is constant both in magnitude and direction
b) The acceleration is constant in direction but not in magnitude
c) The acceleration is constant in magnitude but not in direction
d) The acceleration is neither constant in magnitude nor in direction

Q3. Five identical resistors of resistance 1 each, are connected along the four edges and one of the diagonals of a square. A potential difference of 1 V is applied across the other diagonal. What will be the current in the resistor along the diagonal ?
a) 0.5A
b) 1 A
c) 5A
d) zero ... [..]

(Download) GATE Computer Science (CS) Paper 2010

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1.Which languages necessarily need heap allocation in the runtime environment?
(A) Those that support recursion (B) Those that use dynamic scoping
(C) Those that allow dynamic data structures (D) Those that use global variables

2. One of the header fields in an IP datagram is the Time to Live (TTL) field. Which of the following statements best explains the need for this field?
(A) It can be used to prioritize packets
(B) It can be used to reduce delays
(C) It can be used to optimize throughput
(D) It can be used to prevent packet looping

3. Which one of the following is not a client server application?
(A) Internet chat (B) Web browsing (C) E-mail (D) Ping

4. Let L1 be a recursive language. Let L2 and L3 be languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true?
(A) L2 – L1 is recursively enumerable
(B) L1 – L3 is recursively enumerable
(C) L2 ∩ L1 is recursively enumerable

(Download) Maths Model Practice Paper For AIEEE 2010

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(Paper) GATE | Electronics and Communication Question Paper 2006

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GATE | Electronics and Communication Question Paper 2006

Q.1 — Q.20 Carry One Mark Each.

1. The rank of the matrix
1 1 1
1 -1 0 is:
1 1 1
(A) 0
(B) 1
(C) 2
(D) 3

2. VxVxP, wherePisa vector, is equal to
(A) PxVxP-V2P
(B) V2P+V(V.P)
(C) V2P+VxP
(D) v(v.P)-v2P

3. [[(vxP).ds, where P is a vector, is equal to
(A) 4P.dI
(B) VxVxP.dI
(C) 4VxP.dI
(D) [[[V.Pdv

4. A probability density function is of the form
p(x) = Ke aix1 XE
The value of K is
(A) 0.5
(B) 1
(C) O.5cL
(D) x

5. A solution for the differential equation k(t)+2x(t) = 5(t) with initial condition x(0 —) = 0 is:
(A) e 2u(t)
(B) e2u(t)
(C) e u(t)
(D) &u(t)

6. A low-pass filter having a frequency response H(ja) = A(ai)edoes not produce any phase distortion if
(A) A(a) = Cai2,Ø(ai) = kai3
(B) A(a) = Cai2,Ø(ai) = ko.
(C) A(a)=Ca,Ø(a)=ka2
(D) A(a)=C, )=ka1

7. The values of voltage (V3) across a tunnel-diode corresponding to peak and valley currents are V, and V1 respectively-:- The range of tunnel-diode voltage VD for which the slope of its IVD characteristics is negative would be
(A) V3<O
(B) O<VJ<V?
(C) V,V<V
(D) VdVV

8. The concentration of minority carriers in an extrinsic semiconductor under equilibrium is:
(A) directly proportional to the doping concentration
(B) inversely proportional to the doping concentration
(C) directly proportional to the intrinsic concentration
(D) inversely proportional to the intrinsic concentration

9. Under low level injection assumption, the injected minority carrier current for an extrinsic semiconductor is essentially the
(A) diffusion current
(B) drift current
(C) recombination current
(D) induced current

10. The phenomenon known as “Early Effect” in a bipolar transistor refers to a reduction of the effective base-width caused by
(A) electron-hole recombination at the base
(B) the reverse biasing of the base-collector junction
(C) the forward biasing of emitter-base junction
(D) the early removal of stored base charge during saturation-to-cutoff switching.

11. The input impedance (Z,)and the output impedance (Z0)of an ideal transconductance (voltage controlled current source) amplifier are
(A) Z,=0,Z0=0
(B) Z,=O,Z0=oo
(C) Z,=oo,Z0=O
(D) Z = 00, =

12. An n-channel depletion MOSFET has following two points on its ID VGS curve:
(i) VGS = 0 at ID = l2mA and
(ii) VGS = —6 Volts at ID =
Which of the following Q-points will give the highest trans-conductance gain for small signals?
(A) VGS = —6 Volts
(B) VGS = —3 Volts
(C) VGS = 0 Volts
(D) VGS = 3 Volts

13. The number of product terms in the minimized sum-of-product expression obtained through the following K-map is (where “d” denotes don’t care states)
1001
0d00
00d 1
1001
(A) 2
(B) 3
(C) 4
(D) 5

14. The Dirac delta function 8(t) is defined as
(A) 8(t)=1 t=O.
L° otherwise
100 t=O
(B) 8(t)=.
LO otherwise
(C) 8(t) = = . and [8(t)dt = 1
L otherwise
(D) 8(t)=4°° t=O and [8(t)dt=1
L otherwise

15. The open-loop transfer function of a unity-gain feedback control system is given by The gain margin of the system in dB is given by
(A) 0
(B) 1
(C) 20
(D)

16. The electric field of an electromagnetic wave propagating in the positive zdirection is given by
E=asin(at—flz)+asinot—flz+. The wave is
(A) linearly polarized in the z-direction
(B) elliptically polarized
(C) left-hand circularly polarized
(D) right-hand circularly polarized

17. A transmission line is feeding 1 Watt of power to a horn antenna having a gain of 10 dB. The antenna is matched to the transmission line. The total power radiated by the horn antenna into the free-space is:
(A) 10 Watts
(B) 1 Watt
(C) 0.1 Watt
(D) 0.01 Watt

18. The eigenvalues and the corresponding eigenvectors of a 2 x 2 matrix are given by Eigenvalue Eigenvector
A1=8
A2=4 v2=[hil
The matrix is:
(A) 6 2 L2 6
(B) 6 L6 4
(C) r L 2
(D) 8 L8 4

19. For the function of a complex variable W = lnZ (where, W=u+jv and Z=x+jy),the u=constant lines get mapped in Z-plane as
(A) set of radial straight lines
(B) set of concentric circles
(C) set of confocal hyperbolas
(D) set of confocal ellipses

20. Three companies, X, Y and Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below. Given that a computer is defective, the probability that it was supplied by Y is:
(A) 0.1
(B) 0.2
(C) 0.3
(D) 0.4

21. For the matrix [ jthe eigenvalue corresponding to the eigenvector is:
(A) 2
(B) 4
(C) 6
(D) 8

22. For the differential equation + k2y = 0 the boundary conditions are Company °h of computers supplied Probability of being defective
X         60°h                 0.01
Y         30°h                 0.02
Z         10°h                 0.03

(i) y=Oforx=Oand
(ii) y=Oforx=a
The form of non-zero solutions of y (where mvaries over all integers) are m,rx
(A) y=AmsIn
m a
m,rx
(B) y= AmCO5
m a
(C) Y=4m
mrx
(D) y=Amea

23. Consider the function f (t) having Laplace transform
2 2 Re[sl>O
S +a)0
The final value of f(t)would be:
(A) 0
(B) 1
(C) —1 f(oo) 1
(D)

24. As x is increased from — to 00, the function ex f(x)= 1 + ex
(A) monotonically increases
(B) monotonically decreases
(C) increases to a maximum value and then decreases
(D) decreases to a minimum value and then increases

25. The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by
(A) RL network only
(B) RC network only
(C) LC network only
(D) RC as well as RL networks

26. In the figure shown below, assume that all the capacitors are initially uncharged. If v,(t)=lOu(t) Volts,v0(t) is given by
(A) Se °°° Volts
(B) 8(1 — e °°°) Volts
(C) 8u(t) Volts
(D) 8 Volts

27. Consider two transfer functions G1(S)=s2+as+b and G2(S)=s2+as+b• Vo(t) The 3-dB bandwidths of their frequency responses are, respectively
(A) %Ja2 _4b,%Ja2 +4b
(B) %Ja2 + 4b, %Ja2 — 4b

28. A negative resistance Rneg is connected to a passive network N having driving point impedance Z1 (s)as shown below. For Z2 (s)to be positive real,
Z2(s)

Image 1

 (A) R I<ReZ1(ja),Va neg —
(B) R I —
(C) R 1< ImZ1(ja),va neg —
(D) R <zz (jai),Vai neg— 1

29. In the circuit shown below, the switch was connected to position 1 at t < 0 and at t = 0, it is changed to position 2. Assume that the diode has zero voltage drop and a storage time t5. For 0 < t < t5,v is given by (all in Volts)
(A) v=—5
(B) v=–5
 
Image 2

 (D) —5 <v <0

30. The majority carriers in an n-type semiconductor have an average drift velocity v in a direction perpendicular to a uniform magnetic field B. the electric field E induced due to Hall effect acts in the direction
(A) v x B
(B) Bxv
(C) along v
(D) opposite to v

31. A heavily doped n—type semiconductor has the following data: Hole-electron mobility ratio : 0.4
Doping concentration : 4.2x 108 atoms/m3
Intrinsic concentration : 1.5x 10 atoms/m3
The ratio of conductance of the n —type semiconductor to that of the intrinsic semiconductor of same material and at the same temperature is given by
(A) 0.00005
(B) 2,000
(C) 10,000
(D) 20,000

32. For the circuit shown below; assume that the zener diode is ideal with a breakdown voltage of 6 Volts. The waveform observed across R is:
(A)6V
(B)l2sinot R VR
(C)12V

Image 3

 33. A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-5 number is represented by its corresponding 3-bit binary code.

For example, the base-5 number 24 will be represented by its BCP code 010100. In this numbering system, the BCP code 100010011001 corresponds to the
following number in base-5 system
(A) 423
(B) 1324
(C) 2201
(D) 4231

(Paper) GATE: Computer Science (CS) Question Paper Year 2006 (4)

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68. Consider the relation enrolled (student, course) in which (student, course) is the primary key, and the relation paid (student, amount) where student is the primary key. Assume no null values and no foreign keys or integrity constraints. Given the following four queries:

Queryl: select student from enrolled where student in (select student from paid)
Query2: select student from paid where student in (select student from enrolled)
Query3: select E.student from enrolled E, paid P where E.student = P.student
Query4: select student from paid where exists  (select * from enrolled where enrolled.student = paid.student)

Which one of the following statements is correct?
(A) All queries return identical row sets for any database
(B) Query2 and Query4 return identical row sets for all databases but there exist databases for which Queryl and Query2 return different row sets.
(C) There exist databases for which Query3 returns strictly fewer rows than Query2
(D) There exist databases for which Query4 will encounter an integrity violation at runtime. 

69. Consider the relation enrolled (student, course) in which (student, course) is the primary key, and the relation paid (student, amount) where student is the primary key. Assume no null values and no foreign keys or integrity constraints. Assume that amounts 6000, 7000, 8000, 9000 and 10000 were each paid by 20% of the students. Consider these query plans (Plan 1 on left, Plan 2 on right) to "list all courses taken by students who have paid more than x" enrolled paid enrolled paid 1' 1' Probe index Sequential on student scan, select amount > x A disk seek takes 4ms, disk data transfer bandwidth is 300 MB/s and checking a tuple to see if amount is greater than x takes lOps. Which of the following statements is correct?

(A) Plan 1 and Plan 2 will not output identical row sets for all databases
(B) A course may be listed more than once in the output of Plan 1 for some data bases
(C) For x = 5000, Plan 1 executes faster than Plan 2 for all databases
(D) For x = 9000, Plan I executes slower than Plan 2 for all databases.

Common Data Questions:

Common Data for Questions 71, 72, 73:
The 2 vertices of a graph G corresponds to all subsets of a set of size n, for n < 6. Two vertices of G are adjacent if and only if the corresponding

(Paper) GATE: Computer Science (CS) Question Paper Year 2006 (3)

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51. Consider the following recurrence:
T(n)=2T(r*i1)+1,T(1) = 1

Which one of the following is true?
(A) T(n) = e(loglogn)
(B) T(n) = e(logn)
(C) T(n)=8(sJ)
(D) T(n)rz8(n) 

52. The median of n elements can be found in O(n)time. Which one of the following is correct about the complexity of quick sort, in which median is selected as pivot?
(A) 8(n)
(B) e(nlogn)
(C) 8(n2)
(D) 8(n3) 

53. Consider the following C-function in which a[nl and b[mlare two sorted integer arrays and c[n + mibe another integer array.

void xyz(int a[], mt b [1, mt c []){
mt i,j,k;
i=j=k=O;
while ((i<n) && (j<m))
if (a[i] < b[j]) c[k++] = a[i++];
else c[k++] =

Which of the following condition(s) hold(s) after the termination of the while loop?
(i) j<m,k=n+j—1, and a[n—i1<b[jl ifi=n
(ii) i<n,k=m+i—1, and b[m—i1a[i1 ifj=m 

(A) only (i)
(B) only (ii)
(C) either (i) or (ii) but not both
(D) neither (i) nor (ii) 

54. Given two arrays of numbers a1,...,a and b1,...,b where each number is 0 or 1, the fastest algorithm to find the largest span (i,j)such that a, + a,1 + ... + a = b, + b,1 + ... + b, or report that there is not such span,
(A) Takes Q(3n) and c(2)time if hashing is permitted
(B) Takes 0(n3) and c(n25)time in the key comparison model
(C) Takes e(n)time and space
(D) Takes o(J) time only if the sum of the 2n elements is an even number

56. Consider the following code written in a pass-by-reference language like FORTRAN and these statements about the code. subroutine swap(ix,iy)
it = ix
Li: ix=iy
L2: iy=it
end
ia = 3
ib = 8
call swap (ia, ib+5)
print , ia, ib
end

Si: The compiler will generate code to allocate a temporary nameless cell, initialize it to i3, and pass the address of the cell swap
S2: On execution the code will generate a runtime error on line Li
S3: On execution the code will generate a runtime error on line L2
S4: The program will print i3 and 8
S5: The program will print i3 and -2

Exactly the following set of statement(s) is correct:
(A) Si and S2
(B) Si and S4
(C) S3
(D) Si and S5 

57. Consider this C code to swap two integers and these five statements: the

(Paper) GATE: Computer Science (CS) Question Paper Year 2006 (2)

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26. Which one of the first order predicate calculus statements given below correctly expresses the following English statement? Tigers and lions attack if they are hungry or threatened.
(A) vx[(tiger(x) A lion(x)) — {(hungry(x) v threatened(x)) — attacks(x))1
(B) Vx [(tiger (x) v lion (x)) — {(hungry (x) v threatened (x)) A attacks (x)j
(C) Vx[(tiger(x) v lion(x)) — {attacks(x) — (hungry(x) v threatened(x)))1
(D) vx[(tiger(x) v lion(x)) — {(hungry(x) v threatened(x)) — attacks(x))1

27. Consider the following propositional statements:
    Pl:((AAB)C))((AC)A(BC))
    P2:((AvB)C))((A—C)v(B_C))
Which one of the following is true?

(A) P1 is a tautology, but not P2
(B) P2 is a tautology, but not P1
(C) P1 and P2 are both tautologies
(D) Both P1 and P2 are not tautologies 

28. A logical binary relation a, is defined as follows:
Let be the unary negation (NOT) operator, with higher precedence then o. Which one of the following is equivalent to A A B?
(A) ('-'.'AOB)
(B) -'.'(AO'-..'B)
(C) ".'(".'Ao".'B)
(D) AOB) 

29. If s is a string over (0 + 1)* then let n0 (s) denote the number of 0's in s and n1 (s)the number of l's in s. Which one of the following languages is not regular?
(A) L = {s (0 + 1)*n0 (s) is a 3-digit prime
(B) L = {s E (0 + 1)* for every prefix s' of s, fl0 (s') — n1 (s') 2}
(C) L={sE(0+1)*n0(s)_n1(s)4}
(D) L = {s E (0 + 1) j n0 (s) mod 7 = n1 (s) mod 5 = 0)

30. For SE (0+1)*let d(s)denote the decimal value of s(e.g.d(101)= 5). Let L = {s E (0 + 1) j d (s) mod 5 = 2 and d (s) mod 7 = 4) Which one of the following statements is true?
(A) L is recursively enumerable, but not recursive
(B) L is recursive, but not context-free
(C) L is context-free, but not regular
(D) L is regular

31. Let SHAM3 be the problem of finding a Hamiltonian cycle in a graph G =(V,E)with V divisible by 3 and DHAM3 be the problem of determining if a Hamiltonian cycle exists in such graphs. Which one of the following is true?

A B AoB

True True True
True False True
False True False
False False True 

(A) Both DHAM3 and SHAM3 are NP-hard
(B) SHAM3 is NP-hard, but DHAM3 is not

(Paper) GATE: Computer Science (CS) Question Paper Year 2006

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Q.1 — Q.20 Carry One Mark Each
1. Consider the polynomial p(x) = a0 +a1x+a2×2 +a3×2, where a1 (- 0,Vi. The minimum number of multiplications needed to evaluate pon an input x is:
(A) 3
(B) 4
(C) 6
(D) 9 

2. Let X,Y,Z be sets of sizes x, y and z respectively. Let W = XxY and E be the set of all subsets of W. The number of functions from Z to E is:
(A) Z2XY
(B) Zx2
(C) Z2XY
(D) 2XYZ 

3. The set {1,2,3,5,7,8,91 under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is false?
(A) It is not closed
(B) 2 does not have an inverse
(C) 3 does not have an inverse
(D) 8 does not have an inverse 

4. A relation R is defined on ordered pairs of integers as follows: (x,y)R(u,v)if x < u andy> v. Then R is:
(A) Neither a Partial Order nor an Equivalence Relation
(B) A Partial Order but not a Total Order
(C) A Total Order
(D) An Equivalence Relation

5. For which one of the following reasons does Internet Protocol (IP) use the timeto- live (TTL) field in the IP datagram header?
(A) Ensure packets reach destination within that time
(B) Discard packets that reach later than that time
(C) Prevent packets from looping indefinitely
(D) Limit the time for which a packet gets queued in intermediate routers. 

6. Consider three CPU-intensive processes, which require 10, 20 and 30 time units and arrive at times 0, 2 and 6, respectively. How many context switches are needed if the operating system implements a shortest remaining time first scheduling algorithm? Do not count the context switches at time zero and at the end.
(A) 1
(B) 2
(C) 3
(D) 4 

7. Consider the following grammar. Consider the following LR(0) items corresponding to the grammar above. Given the items above, which two of them will appear in the same set in the canonical sets-ofitems for the grammar?

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