# CS GATE 2014 PAPER 03 (Test 5)

Tag: cs gate 2014 paper 03
Q.1
Consider a hash table with 100 slots. Collisions are resolved using chaining. Assuming simple uniform hashing, what is the probability that the first 3 slots are unfilled after the first 3 insertions?
A. (97 × 97 × 97)/1003
B. (99 × 98 × 97)/1003
C. (97 × 96 × 95)/1003
D.  (97 × 96 × 95)/(3! × 1003)
Explaination / Solution: Workspace
Report
Q.2
With respect to the numerical evaluation of the definite integral, where a and b are given, which of the following statements is/are TRUE?
(I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
(II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.
A. I only
B. II only
C. Both I and II
D. Neither I nor II
Explaination / Solution:  Workspace
Report
Q.3
The value of the integral given below is A. − 2π
B. π
C. −π
D.
Explaination / Solution: Workspace
Report
Q.4
Consider the set of all functions f : {0,1,...,2014} → {0,1...,2014} such that f (f (i)) = i for 0 ≤ i ≤ 2014. Consider the following statements.
P. For each such function it must be the case that for every i, f(i) = i,
Q. For each such function it must be the case that for some i,f(i) = i,
R. Each such function must be onto.
Which one of the following is CORRECT?
A. P, Q and R are true
B. Only Q and R are true
C. Only P and Q are true
D. Only R is true
Explaination / Solution:
No Explaination.

Workspace
Report
Q.5
Let δ denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with δ ≥3, which one of the following is TRUE?
A. In any planar embedding, the number of faces is at least (n/2) + 2
B. In any planar embedding, the number of faces is less than (n/2) + 2
C. There is a planar embedding in which the number of faces is less than (n/2) + 2
D. There is a planar embedding in which the number of faces is at most n/(δ+1)
Explaination / Solution:

We know that v + r = e+2 ⇒ e=n+r-2......(1)
Where V=  n (number of vertices); r = number of faces and
e = number of edges
Given, δ ≥ 3 then 3n ≤ 2e Number of faces is atleast (n/2) + 2

Workspace
Report
Q.6
The CORECT formula for the sentence, “not all rainy days are cold” is
A. ∀d (Rainy (d)∧∼Cold(d))
B. ∀d (~Rainy (d)→Cold(d))
C. ∃d (~Rainy (d)→Cold(d))
D. ∃d (Rainy (d)∧∼Cold(d)) 