## GATE-CS-2002

Question 1 |

4 | |

2 | |

1 | |

0 |

**Linear Algebra**

**GATE-CS-2002**

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Question 1 Explanation:

Rank of the matrix is defined as the maximum number of linearly independent vectors (or) the number of non-zero rows in its row-echelon matrix. A = | 1 1| | 0 0| Since, the matrix A is already in echelon form, Just count the number of non-zero rows to get the rank of the matrix = 1.

Please refer http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29 This solution is contributed by**Anil Saikrishna Devarasetty**.

Question 2 |

The trapezoidal rule for integration give exact result when the integrand is a polynomial of degree:

0 but not 1 | |

1 but not 0 | |

0 or 1 | |

2 |

**Numerical Methods and Calculus**

**GATE-CS-2002**

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Question 2 Explanation:

Question 3 |

The solution to the recurrence equation T(2

^{k}) = 3 T(2^{k-1}) + 1, T (1) = 1, is:2 ^{k} | |

(3 ^{k + 1} - 1)/2 | |

3 ^{log}2k | |

2 ^{log}3k |

**Analysis of Algorithms (Recurrences)**

**GATE-CS-2002**

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Question 3 Explanation:

We have
T (2

^{k}) = 3 T (2^{k-1}) + 1 = 3^{2}T (2^{k-2}) + 1 + 3 = 3^{3}T (2^{k-3}) + 1 + 3 + 9 . . . (k steps of recursion (recursion depth)) = 3^{k}T (2^{k-k}) + (1 + 3 + 9 + 27 + ... + 3^{k-1}) = 3^{k}+ ( ( 3^{k}- 1 ) / 2 ) = ( (2 * 3^{k}) + 3^{k}- 1 )/2 = ( (3 * 3^{k}) - 1 ) / 2 = (3^{k+1}- 1) / 2 Hence, B is the correct choice. Please comment below if you find anything wrong in the above post.Question 4 |

The minimum number of colours required to colour the vertices of a cycle with η nodes in such a way that no two adjacent nodes have the same colour is

2 | |

3 | |

4 | |

n - 2⌊n/2⌋ + 2 |

**Graph Theory**

**GATE-CS-2002**

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Question 4 Explanation:

We need 3 colors to color a odd cycle and 2 colors to color an even cycle.

Question 5 |

In the worst case, the number of comparisons needed to search a singly linked list of length n for a given element is

log _{2} n | |

n/2 | |

log _{2}n - 1 | |

n |

**GATE-CS-2002**

**Discuss it**

Question 5 Explanation:

Singly linked list has uni - directional flow, i.e., it has only one pointer for moving (the next pointer).
In the worst case, for searching an element in the singly linked list, we will have to traverse the whole list (the case when the required element is either the last element or is not present in the list).
So, in the worst case for a list of length n, we will have to go to each node for comparison and thus, we would be needing 'n' comparisons.
Thus, D is the correct choice.
Please comment below if you find anything wrong in the above post.

Question 6 |

Which of the following is true?

The set of all rational negative numbers forms a group under multiplication. | |

The set of all non-singular matrices forms a group under multiplication. | |

The set of all matrices forms a group under multiplication. | |

Both (2) and (3) are true. |

**Set Theory & Algebra**

**GATE-CS-2002**

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Question 6 Explanation:

A group is a set of elements such that any two elements of the group combine to form a third element of the same group. Also, a group must satisfy certain properties:
Closure Property - Any two elements of the set when operated open by an operator form a third element that must also be in the set.
Associative Property - For an expression with three or more operands having the same operator between them, the order of operation does not matter as long as the sequence of operands are not changed. For example, (a + b) + c = a + (b + c).
Identity element Property - Each set must have an identity element, which is an element of the set such that when operated upon with another element of the set, it gives the element itself. For example, a + 0 = a. Here, 0 is the identity element.
Invertibility Property - For each element of the set, inverse should exist.
Now, for the given statements, we have
A is incorrect as it does not satisfies closure property. If we take two negative numbers and multiply them, we get a positive number which is not in the set.
B is correct. The matrices in the set must be non - singular, i.e., their determinant should not be zero, for the inverse to exist (Invertibility Property).
C is incorrect as the inverse of a singular (determinant = 0) matrix does not exist (Invertibility Property violated).
Thus, B is the correct option.
Please comment below if you find anything wrong in the above post.

Question 7 |

The language accepted by a Pushdown Automation in which the stack is limited to 10 items is best described as

Context Free | |

Regular | |

Deterministic Context Free | |

Recursive |

**Context free languages and Push-down automata**

**GATE-CS-2002**

**Discuss it**

Question 7 Explanation:

Pushdown automata is used for context free languages, i.e., languages in which the length of elements is unrestricted and length of one element is related to other. To resolve this problem, we use a stack with no restrictions on length.

But in the given case, length of stack is restricted. Thus, this pushdown automata can only accept languages which can also be accepted by finite state automata and a finite state automata accepts only regular languages.

Thus, B is the correct choice.

Please comment below if you find anything wrong in the above post.

But in the given case, length of stack is restricted. Thus, this pushdown automata can only accept languages which can also be accepted by finite state automata and a finite state automata accepts only regular languages.

Thus, B is the correct choice.

Please comment below if you find anything wrong in the above post.

Question 8 |

"If X, then Y unless Z" is represented by which of the following formulae in propositional logic?
("¬" is negation "^" is conjunction, and "→" is implication)

(X ^ ¬ Z) → Y | |

(X ^ Y) → ¬ Z | |

(X → (Y ^ ¬ Z) | |

(X → Y(^ ¬ Z) |

**Propositional and First Order Logic.**

**GATE-CS-2002**

**Discuss it**

Question 8 Explanation:

The statement "If X then Y unless Z" means, if Z doesn't occur, X implies Y i.e. ¬ Z→ (X→ Y), which is equivalent to Z∨ (X→ Y) (since P→ Q ≡ ¬ P∨ Q), which is then equivalent to Z∨ (¬ X∨ Y). Now we can look into options which one matches with this.

So option∧¬ Z)→ Y = ¬ ((X∧¬ Z))∨ Y = (¬ X∨ Z)∨ Y, which matches our expression. So option A is correct.
Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2002.html

So option

**(a)**is (XQuestion 9 |

A device employing INTR line for device interrupt puts the CALL instruction on the data bus while

A) is active B) HOLD is active C) READY is active D) None of these

A | |

B | |

C | |

D |

**Microprocessor**

**GATE-CS-2002**

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Question 9 Explanation:

**Explanation:**INTR is a signal which if enabled then microprocessor has interrupt enabled .It receives high INTR signal and activates INTA signal, So another request can’t be accepted till CPU is busy in servicing interrupt So (A) is correct option.

Question 10 |

In 8085, which of the following modifies the program counter ?

Only PCHL instruction | |

Only ADD instructions | |

Only JMP and CALL instructions | |

All instructions |

**Microprocessor**

**GATE-CS-2002**

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Question 10 Explanation:

Program counter is the register which has the next location of the program to be executed next. JMP & CALL changes the value of PC. PCHL instruction copies content of registers H & L to PC. ADD instruction after completion increments program counter. So program counter is modified in all
cases. Hence (D) is correct option.

There are 50 questions to complete.