# Gate IT 2007

 Question 1
Suppose there are two coins. The first coin gives heads with probability 5/8 when tossed, while the second coin gives heads with probability 1/4. One of the two coins is picked up at random with equal probability and tossed. What is the probability of obtaining heads ?
 A 7/8 B 1/2 C 7/16 D 5/32
Gate IT 2007
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Question 1 Explanation:
Either 1st coin is selected or second coin is selected
Probability of 1st coin to be selected = ½
And probability of getting head on it = 5/8
Or
Probability of 2nd coin to be selected = 1/2
And pobability of getting head on 2nd coin = ¼
Probability of getting head (overall) = (Probability of selecting 1st coin * Probability of head on it) +(Probability of selecting 2nd coin * Probability of getting head on it) = (1/2)*(5/8) + (1/2)*(1/4) = 7/16
 Question 2
Let A be the . What is the maximum value of xTAx where the maximum is taken over all x that are the unit eigenvectors of A?
 A 5 B (5 + √5)/2 C 3 D (5 - √5)/2
Gate IT 2007
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Question 2 Explanation:
|M-λ.I| = 0, where λ is the eigen values and I is the identity matrix
|A-(λ*I)| = 0
(3-λ)(2-λ)-1 = 0
6-3λ -2λ + λ2+1=0
λ2-5λ+5=0
λ = (5+√5)/2 and (5-√5)/2,
λ = (5+√5)/2 is max value another root with negative sign which will not be max value.
For, λ=5+5√2, xTAx=[18.131 21.231 21.231 34.331]
For, λ=5−5√2, xTAx=[Tex]\begin{pmatrix} 14.300&−0.700\\ 6.200&1.200 \end{pmatrix}[/Tex]
Hence, for λ=5+5√2 the value of xTAx is maximum.
 Question 3
Consider a weighted undirected graph with positive edge weights and let uv be an edge in the graph. It is known that the shortest path from the source vertex s to u has weight 53 and the shortest path from s to v has weight 65. Which one of the following statements is always true?
 A weight (u, v) < 12 B weight (u, v) ≤ 12 C weight (u, v) > 12 D weight (u, v) ≥ 12
Graph Shortest Paths    Graph Theory    Gate IT 2007
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Question 3 Explanation:
The minimum weight happens when (S,U) + (U,V) = (S,V) Else (S,U) + (U,V) >= (S,V) Given (S,U) = 53, (S,V) = 65 53 + (U,V) >= 63 (U,V) >= 12. This solution is contributed by Anil Saikrishna Devarasetty
 Question 4
In the Spiral model of software development, the primary determinant in selecting activities in each iteration is
 A Iteration size B Cost C Adopted process such as Rational Unified Process or Extreme Programming D Risk
Software Engineering    Gate IT 2007
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Question 4 Explanation:
Spiral model is used to discover all risks associated as early as possible.
 Question 5
Which of the following systems is a most likely candidate example of a pipe and filter architecture ?
 A Expert system B DB repository C Aircraft flight controller D Signal processing
Computer Organization and Architecture    Gate IT 2007
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Question 5 Explanation:

Reference: Software Architecture: A Case Based Approach By Vasudeva Varma, Varma Vasudeva
 Question 6
A processor takes 12 cycles to complete an instruction I. The corresponding pipelined processor uses 6 stages with the execution times of 3, 2, 5, 4, 6 and 2 cycles respectively. What is the asymptotic speedup assuming that a very large number of instructions are to be executed?
 A 1.83 B 2 C 3 D 6
Computer Organization and Architecture    Gate IT 2007
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Question 6 Explanation:
For non pipeline processor,
It takes, 12 cycles to complete 1 instruction
So, for n instructions it will take 12n cycle
For pipelined processor,
Each stage time = max{each stage cycles} = max{3, 2, 5, 4, 6 and 2} = 6 cycles
So, for n instructions it will take = 6*6+ (n-1)*6 {6*6 for 1st instruction and for rest of n-1 it will take 6}
For a large number of instructions:
Limn->∞ 12n/36 + (n-1)*6 = 12/6 =2
 Question 7
Which of the following input sequences for a cross-coupled R-S flip-flop realized with two NAND gates may lead to an oscillation ?
 A 11, 00 B 01, 10 C 10, 01 D 00, 11
Digital Logic & Number representation    Gate IT 2007
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Question 7 Explanation:
RS flip flop using NAND gates. So, 00 input causes indeterminate state which MAY lead to oscillation.
 Question 8
The following circuit implements a two-input AND gate using two 2-1 multiplexers.

What are the values of X1, X2, X3?
 A X1=b, X2=0, X3=a B X1=b, X2=1, X3=b C X1=a, X2=b, X3=1 D X1=a, X2=0, X3=b
Digital Logic & Number representation    Gate IT 2007
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Question 8 Explanation:

F = (bX1' + aX1)X3 + X2X3'
By putting following values: X1 = b, X2 = 0, X3 = a
We will get F = ab.
 Question 9
Consider an ambiguous grammar G and its disambiguated version D. Let the language recognized by the two grammars be denoted by L(G) and L(D) respectively. Which one of the following is true ?
 A L (D) ⊂ L (G) B L (D) ⊃ L (G) C L (D) = L (G) D L (D) is empty
Regular languages and finite automata    Gate IT 2007
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Question 9 Explanation:
By changing grammar, language will not change here. {as converting NFA to DFA language will not be changed}
 Question 10
Processes P1 and P2 use critical_flag in the following routine to achieve mutual exclusion. Assume that critical_flag is initialized to FALSE in the main program.
 get_exclusive_access ( ) { if (critical _flag == FALSE) { critical_flag = TRUE ; critical_region () ; critical_flag = FALSE; } } Consider the following statements.
i. It is possible for both P1 and P2 to access critical_region concurrently.