GATE CS 1998

Question 61

a. Find the points of local maxima and minima, if any, of the following function defined in 0≤ x ≤ 6.

  x3-6x+9x-15

  b. Integrate

Question 62

Derive the expression for the number of expressions required to solve a system of linear equations in n unknowns using the Gaussian Elimination Method. Assume that one operation refers to a multiplication followed by an addition.

Question 63

Suppose A={ a, b, c, d } and Π₁ is the following partition of A Π₁= { {a, b, c} {d} } a. List the ordered pairs of the equivalence relations induced by Π₁. b. Draw the graph of the above equivalence relation. c. Let Π₂= {{a},{b},{C},{d}} Π₃= {{a,b,c,d}} and Π₄= {{a,b},{c,d}} Draw a Poset diagram of the poset, ({Π₁,Π₂,Π₃,Π₄}, refines ⟩

Question 64

Let (A, *) be a semi group. Furthermore, for every a and b in A, if a ≠ b, then a*b ≠ b*a. a. Show that for every a in A  a*a = a b. Show that for every a, b in A  a*b*a = a c. Show that for every a, b, c in A  a*b*c = a*c

Question 65

Let M = ({q₀, q₁}, {0, 1}, {z₀, x}, δ, q₀, z₀, ∅) be a pushdown automaton where δ is given by δ(q₀, 1, z₀) = {(q₀, xz₀)} δ(q₀, ε, z₀) = {(q₀, ε)} δ(q₀, 1, X) = {(q₀, XX)} δ(q₁, 1, X) = {(q₁, ε)} δ(q₀, 0, X) = {(q₁, X)} δ(q₀, 0, z₀) = {(q₀, z₀)}

1. What is the language accepted by this PDA by empty stack?
2. Describe informally the working of the PDA

Question 66

Answer the following: a.  Let G1 = (N, T, P, S1) be a CFG where, N= {S1A, B}, T= {a,b} and P is given by

         S1 → aS1b             S1 → aBb
         S1 → aAb              B → Bb
         A → aA                B → b
         A → a

What is L(G₁)? b. Use the grammar in part(a) to give a CFG for L₂ = {aⁱ b^j a^k b^l | i, j, k, l ≥ 1, i= j or k=l} by adding not more than 5 production rule. c.  Is L₂ inherently ambiguous?

Question 67

Answer the following: a. Draw the schematic of an 8085 based system that can be used to measure the width of a pulse. Assume that the pulse is given as a TTL compatible signal by the source which generates it. b.  Write the 8085 Assembly Language program to measure the width of the pulse. State all your assumption clearly.

Question 68

Design a synchronous counter to go through the following states: 1, 4, 2, 3, 1, 4, 2, 3, 1, 4,...........

Question 69

[5 Marks question] Calculate the total time required to read 35 sectors on a 2-sided floppy disk. Assume that each track has 8 sectors and the track-to-track step time is 8 milliseconds. The first sector to be read is sector 3 on track 10. Assume that the diskette is soft stored and the controller has a 1-sector buffer. The diskette spins at 300 RPM and initially, the head is on track 10.

Question 70

[5 Marks question] For a set-associative Cache Organization, the parameters are as follows:

tc - Cache access tine
tm - Main memory access time
l - number of sets
b - block size
k x b - set size

Calculate the hit ratio for a loop executed 100 times where the size of the loop is n X b and n= k X m is a non-zero integer and 1 < m ≤ l. Given the value of the hit ratio for l = 1.