**RRB Group-D 2018 Aptitude Test, Sample Paper of RRB Ahmedabad**

The RRB officials have already announced that the RRB Group-d Results 2018-2019 is expected to be released on 24^{th} April 2019 on the Official website www.indianrailways.gov.in. Once the RRB Group-D 2^{nd} Stage exam or PET Results are declared, qualified candidates will need to appear for the next stage Aptitude Test in the Computer Examination. It would be also important for the aspirants to note that the sample paper will be available on rrbntpcresults.com. It is a must for the aspirants to note that the Psychological tests have been designed for different categories of job in critical safety categories, viz., Assistant Station Master, Assistant Drivers, and Motorman. Aspirants, Clerks, Office Assistant, ITI Posts will be given the test battery prescribed for the job for which they have applied for The Aptitude test for Computer Efficiency Test or CBT, will be now in Computer-based mode.

Also, it is important to note that the Computer Aptitude Test is conducted as per the selection criteria approved by the Ministry of Railways. The Aptitude tests for RRB Group-D Examination are administered on the candidates appearing for selection as safety category staff on Indian Railways through different RRBs at the entry level of Assistant Station Master, Assistant Loco Pilot, Group-D Posts like Office Assistant, Peon, Safai Karmi, ITI Vacancy and Motorman for assessment of cognitive ability of the candidates with a purpose to ensure that the candidates possessing the desired level of attributes, essential for safe train operation, are selected. For candidates to qualify the aptitude test, the T-score should be 52 in each test. T-score is the normalized score statistically formulated for application all over. It is important to note that the copies of question paper, OMR sheet, and answer key cannot be provided. The re-evaluation of answer sheet is an action and not information to be provided to the applicant under RTI Act 2005.

Candidates must also note that the scores obtained in RRB examination and thereafter in the aptitude test as a part thereof are appointed in the ratio of 70:30 for the preparation of the final merit list. As weightage to aptitude test is 30, the composite score of a dummy subject can be calculated as follows: Composite T-score of a candidate having 5 tests in a battery is = 300. The max T-score a candidate can obtain having 5 tests in a battery is (80×5) = 400.

Short Questions

1. Consider a 12 hour digital clock (one that takes values from 00:00 to 11:59). Look at it

at a random point during a 12 hour period.

What is the probability that you see at least one digit taking the value ‘1’?

What is the probability that you see exactly one digit taking the value ‘1’?

2. Let a, b be positive integers and let p be a prime factor of a

b − 1.

Show that either gcd(p, a − 1) or gcd(b, p − 1) must be greater that 1.

3. Alice and Bob are given a set of five biased coins. They both estimate the probability

that each coin will show a head when flipped, and each coin is then flipped once. These

are the estimates and values observed:

Coin 1 2 3 4 5

Alice’s estimates 0.4 0.7 0.2 0.9 0.4

Bob’s estimates 0.2 0.8 0.3 0.6 0.3

Observed Heads Heads Tails Tails Heads

Whom would you say is better at estimating the bias of the coins, and why?

4. Find an example of a function f from [0, 1] to [0, 1] with the following properties:

i. f is continuous;

ii. f(0) = 0;

iii. f(1) = 1;

iv. f(x) is locally constant almost everywhere.

5. You have a large rectangular cake, and someone cuts out a smaller rectangular piece from

the middle of the cake at a random size, angle and position in the cake (see the picture

below). Without knowing the dimensions of either rectangle, using one straight (vertical)

cut, how can you cut the cake into two pieces of equal area?

6. A random number generator produces independent random variates x0, x1, x2, . . . drawn

uniformly from [0, 1], stopping at the first xT that is strictly less than x0. Prove that

the expected value of T is infinite. Suggest, with a brief explanation, a plausible value

of P r(T = ∞) for a real-world (pseudo-)random number generator implemented on a

computer.

7. Prove that there does not exist a four-digit square number n such that n ≡ 1 mod 101.

8. The Fourier transform of a function f(t) is defined as:

F(ω) = Z ∞

−∞

f(t) e

−2πiωt dt.

Express the Fourier transform of f(αt) in terms of F, α and ω.

9. A multiset is a collection of objects, some of which may be repeated. So, for example,

{1, 3, 5, 3, 2, 7, 2} is a multiset. The multiplicity of an element in a multiset is the number

of times that element occurs in the multiset. Intersections of multisets can be produced

in the obvious way: given multisets A and B, the multiplicity of an element in A ∩ B is

the minimum of its multiplicities in each of A and B.

Given 2 multisets A amd B, with unordered elements, outline an algorithm for generating

the intersection A ∩ B.

10. Sort the following 16 numbers into 4 sets of 4 and give an explanation for each grouping:

{1, 2, 3, 4, 4, 5, 5, 8, 9, 11, 13, 17, 25, 32, 49, 128}.

11. Given

log2

(11) = 3 +

1

2 +

1

5 +

1

1 +

1

1 +

1

1 +

1

25 +

1

1 + · · ·

which is larger, 2128 or 1137?

12. Write n! = 2a

b where b, n ∈ N and a ∈ N ∪ {0}. Prove that a < n.

Longer Questions

1. Alice and Bob play Rock, Paper, Scissors until one or the other is 5 wins ahead. They

generate their wins at random, so, in each round, the outcomes are equiprobably win,

lose or draw.

After 10 rounds, Alice is 1 ahead. After 13 rounds, one or the other is 1 ahead. At round

20, one of them attains the 5 win lead and the game ends. What is the probability that

Alice is the ultimate winner?

2. You are monitoring a data stream which is delivering very many 32-bit quantities at a

rate of 10 Megabytes per second. You know that either:

A: All values occur equally often, or

B: Half of the values occur 210 times more often than others (but you dont know

anything about which would be the more common values).

You are allowed to read the values off the data stream, but you only have 220 bytes of

memory.

Describe a method for determining which of the two situations, A or B, occurs. Roughly

how many data values do you need to read to be confident of your result with a probability

of 0.999? [This is about the 3 sigma level – 3 standard deviations of a normal distribution.]

3. A 2 × N rectangle is to be tiled with 1 × 1 and 2 × 1 tiles. Prove that the number of

possible tilings tends to kxN as N gets large. Find x, to 2 decimal places.

4. The Prime Power Divisors (PPD) of a positive integer are the largest prime powers that

divide it. For example, the PPD of 450 are 2, 9, 25. Which numbers are equal to the sum

of their PPD?

5. 353, 46, 618, 30, 877, 235, 98, 24, 107, 188, 673 are successive large powers of an integer x

modulo a 3-digit prime p. Find x and p.

6. (X1, Y1),(X2, Y2), . . .(Xn, Yn) are n ordered points in the Cartesian plane that are successive

vertices of a non-intersecting closed polygon.

Describe how to find efficiently a diagonal (that is, a line joining 2 vertices) that lies

entirely in the interior of the polygon.

Repeated application of this process will completely triangulate the interior of the polygon.

Estimate the worst case number of arithmetic operations needed to complete the

triangulation.

7. A semigroup is a set S with an associative operation, which we will write as multiplication.

That is, x(yz) = (xy)z for all x, y, z ∈ S.

a. Suppose that a finite semigroup S has the property that for all x there is an integer

n > 1 such that x

n = x. Is it true that S is, in fact, a group?

b. Suppose that a finite semigroup S has the property that for all x there is an integer

n such that x

n+1 = x

n

. Show that the only subsets of S which form a group are of

size 1.

8. A finite state machine M consists of a finite set of states, each having two arrows leading

out of it, labelled 0 and 1. Each arrow from state x may lead to any state (both may lead

to the same state, which might be x).

One state is labelled the “initial” state; one or more states are labelled “accept”. The

machine reads a finite binary string by starting at “initial” and considering each symbol

in the string in turn, moving along the arrow with that label to the next state until the

end of the string is reached. If the final state is labelled “accept”, the machine accepts

the string.

The language, L(M) of the maching is the set of all the finite binary strings that the

machine accepts.

a. Design a machine whose language is all palindromic strings of length 6 (i.e., strings

for which the last 3 symbols are a reflection of the first 3).

b. Suppose that L(M) is not finite. Show that the number of strings in L(M) of length

n or less grows linearly with n.

c. Show that there is no machine such that L(M) consists of all twice-repeated strings.

9. Suppose P is a permutation on {0, 1, . . . , n − 1}. We want to know the length of the

longest monotonically increasing subsequence. That is, the largest m such that there

exist monotonically increasing j1, j2, . . . , jm for which P(j1), P(j2), . . . , P(jm) are also

monotonically increasing.

Describe an algorithm that will determine m using O(n2) time and O(n) memory.

10. Given 541 points in the interior of a circle of unit radius, show that there must be a subset

of 10 points whose diameter (the maximum distance between any pair of points) is less

than √

2/4.

11. Show how the edges of a cube (8 vertices; 12 edges) can be directed so that some vertices

have all edges pointing out (i.e., directed away from the vertex), and the remainder have

one edge out and two in.

Suppose the edges of a 5-dimensional hypercube (32 vertices, 80 edges) are directed so

that all vertices have either 5 edges pointing out, or 4 edges in and 1 out. Show that

every 4-dimensional subcube must contain precisely 6 of the 5-out vertices.