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)/100^{3}

**B. ** (99 × 98 × 97)/100^{3}

**C. ** (97 × 96 × 95)/100^{3}

**D. ** (97 × 96 × 95)/(3! × 100^{3})

**Answer : ****Option A**

**Explaination / Solution: **

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With respect to the numerical evaluation of the definite integral, where a and b
are given, which of the following statements is/are TRUE?

**A. ** I only

**B. ** II only

**C. ** Both I and II

**D. ** Neither I nor II

**Answer : ****Option C**

**Explaination / Solution: **

(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.

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The value of the integral given below is

**A. ** − 2π

**B. ** π

**C. ** −π

**D. ** 2π

**Answer : ****Option A**

**Explaination / Solution: **

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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.

**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

**Answer : ****Option B**

**Explaination / Solution: **

No Explaination.

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?

No Explaination.

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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)

**Answer : ****Option A**

**Explaination / Solution: **

We know that v + r = e+2 ⇒ e=n+r-2......(1)

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

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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))

**Answer : ****Option D**

**Explaination / Solution: **

Given statement is

Given statement is

(Since p→q ≡~ p∨q and let r(d) be rainy day, c(d) be cold day)

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