**Engineering-Mathematics |Gate-2015|**

**1. **In the LU decomposition of the matrix

| 2 2 |

| 4 9 |

, if the diagonal elements of U are both 1, then the lower diagonal entry l_{22} of L is : **[GATE – 2015]**

a. 5

b. 6

c. 7

d. 8

*Answer : a)*

**2.** If g(x) = 1-x and h(x) = x/x-1, then g(h(x)) / h(g(x)) is: **[GATE – 2015]**

a. -1/x

b. h(x)/g(x)

c. g(x)/h(x)

d. x/(1-x)^{2}

*Answer : b)*

**3.** Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram:

For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L^{3} = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ L^{3} chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then **[GATE – 2015]**

a. pr = 0

b. pr = 1

c. 1/5 < pr < 1

d. 0 < pr ≤ 1/5

*Answer : c)*

**4.** Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b? **[GATE – 2015]**

A = | 1 4 |

| b a |

a. a=6,b=4

b. a=4,b=6

c. a=3,b=5

d. a=5,b=3

*Answer : d)*

**5.** Let G = (V, E) be a simple undirected graph, and s be a particular vertex in it called the source. For x ∈ V , let d(x)denote the shortest distance in G from s to x. A breadth first search (BFS) is performed starting at s. Let T be the resultant BFS tree. If (u, v) is an edge of G that is not in T, then which one of the following CANNOT be the value of d(u) – d(v) ? **[GATE – 2015]**

a. -1

b. 0

c. 1

d. 2

*Answer : d)*

**6. **

a. -4

b. -3

c. -2

d. -1

*Answer : c)*

**7. **Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is _______________. **[GATE – 2015]**

a. 25

b. 24

c. 26

d. 27

*Answer : b)*

**8. **Consider the operations f(X, Y, Z) = X’YZ + XY’ + Y’Z’ and g(X′, Y, Z) = X′YZ + X′YZ′ + XY Which one of the following is correct? **[GATE – 2015]**

a. Only {f} is functionally complete

b. Both {f} and {g} are functionally complete

c. Only {g} is functionally complete

d. Neither {f} nor {g} is functionally complete

*Answer : a)*

**9. **Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true? **[GATE – 2015]**

a. R is symmetric and reflexive but not transitive|

b. R is reflexive but not symmetric and not transitive

c. R is transitive but not reflexive and not symmetric

d. R is symmetric but not reflexive and not transitive

*Answer : d)*

**10.**

**[GATE – 2015]**

a. 0.99

b. 1.00

c. 2.00

d. 3.00

*Answer : a)*

**11.** Consider the following statements:

S1: If a candidate is known to be corrupt, then he will not be elected

S2: If a candidate is kind,he will be elected

Which one of the following statement follows S1 and S2 as per sound inference rule of logic ? **[GATE – 2015]**

a. If a person is known to corrupt, he is kind

b. If a person is not known to be corrupt, he is not kind

c. If a person is kind, he is not known to be corrupt

d. If a person is not kind, he is not known to be corrupt

*Answer : c)*

**12.** The larger of the two eigenvalues of the matrix

is ______. **[GATE – 2015]**

a. 6

b. 7

c. 8

d. 9

*Answer : a)*

**13.** The cardinality of the power set of {0, 1, 2, … 10} is _________ **[GATE – 2015]**

a. 2046

b. 2047

c. 2048

d. 2049

*Answer : c)*

**14.** The number of divisors of 2100 is ______. **[GATE – 2015]**

a. 36

b. 37

c. 38

d. 39

*Answer : a)*

**15.** In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true? **[GATE – 2015]**

a. A tree has no bridges

b. A bridge cannot be part of a simple cycle

c. Every edge of a clique with size 3 is a bridge (A clique is any complete sub graph of a graph)

d. A graph with bridges cannot have a cycle

*Answer : b)*

**16.** Consider six memory partitions of sizes 200 KB, 400 KB, 600 KB, 500 KB, 300 KB and 250 KB, where KB refers to kilobyte. These partitions need to be allotted to four processes of sizes 357 KB, 210KB, 468 KB and 491 KB in that order. If the best fit algorithm is used, which partitions are NOT allotted to any process? **[GATE – 2015]**

a. 200KBand 300 KB

b. 200KBand 250 KB

c. 250KBand 300 KB

d. 300KBand 400 KB

*Answer : a)*

**17. **The number of onto function (surjective function) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ______. **[GATE – 2015]**

a. 36

b. 37

c. 38

d. 39

*Answer : a)*

**18. **Perform the following operations on the matrix

(i) add the third row to the second row (ii) Subtract the third column from the first column The determinant of the resultant matrix is ____________ **[GATE – 2015]**

a. 0

b. 1

c. 2

d. 3

*Answer : a)*

**19.** Which one of the following well formed formulae is a tautology? **[GATE – 2015]**

a. ∀x ∃y R(x,y)↔ ∃y ∀x R(x,y)

b. (∀x [∃y R(x,y)→S(x,y)])→ ∀x∃y S(x,y)

c. [∀x ∃y (P(x,y)→R(x,y)]↔[∀x ∃y ( ¬ P(x,y)∨R(x,y)]

d. ∀x ∀y P(x,y)→ ∀x ∀y P(y,x)

*Answer: c)*

**20.** A graph is self-complementary if it is isomorphic to its complement for all self-complementary graphs on n vertices, n is : **[GATE – 2015]**

a. A multiple of 4

b. Even

c. Odd

d. Congruent to 0 mod 4, or, 1 mod 4

*Answer : d)*

**21.** Let f(x) = x ^{-1(1/3) } and A denote the area of the region bounded bu f(x) and rhe X-axis, when x varies from -1 to1. Which of the followin statements is/are TRUE? I) f is continuous in [-1,1] II)f is not bounded in [-1,1] III) A is nonzero and finite : **[GATE – 2015]**

a. II only

b. III only

c. II and III only

d. I, II and III

*Answer : c)*

Engineering-Mathematics |Gate-2015|

**22.** Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X to YY. Let f be randomly chosen from F. The probability of f being one-to-one is ______. **[GATE – 2015]**

a. 0.95

b. 0.96

c. 0.97

d. 0.98

*Answer : a)*

Engineering-Mathematics |Gate-2015|

**23.** In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking, the person replies the following:

“The result of the toss is head if and only if I am telling the truth.”

Which of the following options is correct? **[GATE – 2015]**

a. The result is tail

b. The result is head

c. If the person is of Type 2, then the result is tail

d. If the person is of Type 1, then the result is tail

*Answer : b)*

**24.** Suppose U is the power set of the set S = {1, 2, 3, 4, 5, 6}. For any T ∈ U, let |T| denote the number of element in T and T’ denote the complement of T. For any T, R ∈ U, let T \ R be the set of all elements in T which are not in R. Which one of the following is true? **[GATE – 2015]**

a. ∀X ∈ U (|X| = |X’|)

b. ∃X ∈ U ∃Y ∈ U (|X| = 5, |Y| = 5 and X ∩ Y = ∅)

c. ∀X ∈ U ∀Y ∈ U (|X| = 2, |Y| = 3 and X \ Y = ∅)

d. ∀X ∈ U ∀Y ∈ U (X \ Y = Y’ \ X’)

*Answer : d)*

**25. **The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3} is _____ . **[GATE – 2015]**

a. 15

b. 16

c. 17

d. 18

*Answer : a)*

Engineering-Mathematics |Gate-2015|

**26.** Consider a machine with a byte addressable main memory of 2^{20} bytes, block size of 16 bytes and a direct mapped cache having 2^{12} cache lines. Let the addresses of two consecutive bytes in main memory be (E201F)_{16} and (E2020)_{16}. What are the tag and cache line address (in hex) for main memory address (E201F)_{16}? **[GATE – 2015]**

a. E, 201

b. E, 201

c. E, E20

d. 2, 01F

*Answer : a)*

**27. **The velocity v (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time t(in minutes) as follows:

t 2 4 6 8 10 12 14 16 18 20

v 10 18 25 29 32 20 11 5 2 0

The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is _________. **[GATE – 2015]**

a. 309.34

b. 309.33

c. 309.35

d. 309.36

*Answer : b)*

Engineering-Mathematics |Gate-2015|

**28.** If the following system has non-trivial solution,

px + qy + rz = 0

qx + ry + pz = 0

rx + py + qz = 0

then which one of the following options is True? **[GATE – 2015]**

a. p-q+r = 0 or p = q = -r

b. p+q-r = 0 or p = -q = r

c. p+q+r = 0 or p = q = r

d. p-q+r = 0 or p = -q = -r

*Answer : c)*

Engineering-Mathematics |Gate-2015|

**29.** Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) ∈ R if and only if p – s = q – r. Which one of the following is true about R? **[GATE – 2015]**

a. Both reflexive and symmetric

b. Reflexive but not symmetric

c. Neither reflexive nor symmetric

d. Not reflexive but symmetric

*Answer : d)*

Engineering-Mathematics |Gate-2015|