CMI Entrance Exam 2022 Question Paper for M.Sc in Data Science

CHENNAI MATHEMATICAL INSTITUTE
M.Sc. Program in Data Science
Entrance Examination 2022

Enter your Admit Card Number : D − −

IMPORTANT INSTRUCTIONS

• Part (A) consists of multiple-choice questions. There may be multiple correct choices. You have to select all
the correct options and no incorrect option to get full marks. There is no partial credit.

• For questions in part (A), you have to provide the answers on the computer. You only have to choose the
appropriate answer(s) from the choices provided. If the answer is parts (a) and (c), choose only (a) and (c).

• For questions in part (B), you have to write your answer with a short explanation in the space provided for
the question.

• Part(A) will be used for screening. Part (B) will be graded only if you score a certain minimum in part
(A). However your scores in both parts will be used while making the final decision.

• For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.: nr , n Pr , n!



etc.

Notation and terminology
• A function f from a set A to a set B is said to be injective (or one-to-one) if f (x) = f (y) implies x = y
for all x, y ∈ A;

• f is said to be surjective (or onto) for every y ∈ B there exists x ∈ A such that f (x) = y;

• f is said to be bijective if it is both injective and surjective;

• f is said to be invertible if there exists a function g from B to A such that f (g(y)) = y for all y ∈ B and
g(f (x)) = x for all x ∈ A.

• For a matrix A, AT denotes the transpose of A. For a square matrix A, |A| denotes the determinant of A
and trace(A) denotes the trace of A — namely the sum of the diagonal elements of A.

• A diagonal matrix is a square matrix D with all off-diagonal entries equal to zero i.e. dij = 0 for all i ̸= j.

• An upper triangular matrix is a square matrix A for which all entries below the diagonal are zero, i.e. aij = 0
for i > j.

• A symmetric matrix is a square matrix S for which sij = sji for all i ̸= j.

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Part A

Multiple-choice questions

1. Let X be a Binomial(n, p) random variable with the probability mass function
 
n k
P(X = k) = p (1 − p)n−k , k = 0, 1, 2, . . . , n.
k

Let T be a random variable defined as:

1 if X = 1,
T =
0 otherwise.

Let E(T ) and V(T ) denote the expectation and the variance of a random variable T . Which of the following
statements is/are true?

(a) E(T 2 ) = np(1 − p)n−1 and V(T ) = np(1 − p)n−1 (1 − (np(1 − p)n−1 ))


(b) E(T ) = np and V(T ) = np(1 − p)
(c) E(T ) = np(1 − p)n−1 and E(T 2 ) = np(1 − p)n−1
(d) E(T ) = p and V(T ) = p(1 − p)

2

2. Consider the following code, in which A is an array indexed from 0 and n is the number of elements in A.
function foo (A , n ) {
L = 0;
R = n - 1;

while ( L <= R ) {
i = ceil (( L + R )/2);

if ( A [ i ] < i ) {
L = i + 1;
} else {
if ( A [ i ] > i ) {
R = i - 1;
} else {
return ( i );
}
}
}

return ( -1);
}

Here, ceil(x) returns the smallest integer bigger than or equal to the number x.
If A = [-5, -4, -3, -2, -1, 4, 6, 8, 10, 12], what will foo(A, 10) return?

(a) -5
(b) -1
(c) 4
(d) 6

3. Which of the following statements is/are true?
n
X 17
(a) For any real number r with |r| > 1, lim 17rn = .
n→∞ 1−r
i=1
(b) Let i be the positive square root of −1 and let x be any real number. Then

(eix − e−ix )
tan x = .
i(eix + e−ix )

1 1 1
(c) + + + · · · < 1.
1·2 2·3 3·4
(d) The function f (x) defined as
(
3 cos 3x, 0 < x < π/6
f (x) =
0 otherwise,

is a probability density function.

3

4. Let E and F be events such that P (E ∩ F c ) = 0.2, P (F ∩ E c ) = 0.3, P ((E ∩ F )c ) = 0.7, where, for an event
E, the notation E c denotes the complement of the event. Then we can conclude that
(a) P (E ∪ F ) = 0.8.
(b) P (E c ∩ F c ) = 0.3.
(c) P (F ) = 0.6.
(d) P (E) = 0.6.
5. Let A, B be n × n invertible matrices of real numbers. Let C = I + AAT , D = I + BAAT B T . We can
conclude that
(a) (AB)−1 = (I + B −1 A−1 )
(b) (AB)−1 = A−1 B −1
(c) C = C T
(d) D = DT
6. A matrix C is said to be symmetric if C T = C. Which of the following is/are true? Let A, B be n × n
matrices.
(a) If A is symmetric and invertible, then A−1 is also symmetric and invertible.
(b) If A and B are symmetric, then C = AB is also symmetric.
(c) If A and B are invertible, then C = AB is also invertible.
(d) If A and B are symmetric, then D = A + B is also symmetric.
7. Which of the following statements is/are true?
(a) Let A be a n × m matrix with n < m. Then there is a nonzero solution y with Ay = 0 only if A has
full row rank.
(b) Let A be a n × m matrix with n > m. There is a nonzero solution y with Ay = 0.
(c) The row rank of an n × m matrix is equal to its column rank only when n = m.
(d) Let A be an n × n matrix. Suppose A = BC, where B has size n × r and C has size r × n. The rank
of A is less than or equal to r.
8. Let n ≥ 5 be a natural number, let X = {x1 , x2 , ..., xn }, and let Y = {y1 , y2 }. Let F be the set of functions
from X to Y and G be the set of bijective functions from X to Y . Then
(a) The number of functions in F equals n2 .
(b) The number of functions in F equals 2n .
(c) The number of functions in G equals n2 − 2.
(d) The number of functions in G equals 0.
9. In order to select a debating team to represent a school, 7 students from class XII and 13 students from class
XI were shortlisted and were undergoing trials. The coach had to select a team of 5 students, out of which
at least two students should be from each class. One out of the 5 students was to be named as team leader,
who was to be from class XII. Two teams with the same members but different leaders are considered to be
two different
 teams.  The number
 of  different teams the coach can select is
7 13 7 13
(a) 2 × × + ×
  2  3  3  2
7 13 7 13
(b) × + ×
2   3  3  2  
7 13 7 13
(c) 2 × × +3× ×
2   3  3   2 
7 13 7 13
(d) 3 × × +2× ×
2 3 3 2

4

10. Which of the following plots correspond to a function whose derivative is continuous?

I. I

I Ciii)
t ( in

(a) (i) and (ii)
(b) (ii) and (iii)
(c) (iii) and (iv)
(d) (ii) and (iv)

5

11. Which of the following plots correspond to a bijective function?

0 . I 0

I 1

.

Liii) ( ir)

(a) (i) and (ii)
(b) (ii) and (iii)
(c) (i) and (iv)
(d) (iii) and (iv)

12. A relation R on the set A = {a, b, c, d} is defined by reading the columns of the following table from top to
bottom. If a column in the table reads (x, y, 1) it means x is related to y in R. If a column in the table reads
(x, y, 0) it means x is not related to y.

a b c d a b c d a b c d a b c d
a a a a b b b b c c c c d d d d
0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0

For instance, from the fifth column we have, (a, b) ∈ R, and from the second we have (b, a) ∈
/ R.
Another relation S on the set A is defined as: for any x, y ∈ A, the pair (x, y) is in S if and only if there
exists z ∈ A such that both (x, z) ∈ R and (z, y) ∈ R hold.
Which of the following pairs are in S?

(a) (a, a)
(b) (b, b)
(c) (c, c)
(d) (d, d)

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13. We say that a subset S of a finite set U is large if |S| > |U \ S|. Here U \ S denotes the elements of U which
are not in S and the notation |T | denotes the number of elements in a set T . Let x be the number of large
subsets of the set X = {1, 2, . . . , 10}, and let y be the number of large subsets of Y = {1, 2, . . . , 9}. Which
of the following is true?

(a) x = 260, y = 256
(b) x = 386, y = 256
(c) x = 386, y = 130
(d) x = 512, y = 256.

14. A ternary tree starts with a single root node at the top of the tree. Each node in the tree can have up to
three nodes as its children. No node in the tree is the child of two different nodes. A node which has no
children is called a leaf node.
The children of a node are drawn below it, connected by edges. The level of a node v in the ternary tree is
the number of edges in the (unique) path from the root node to v. Thus, for instance, the root node is at
level 0, and each child of the root node is at level 1.
A complete ternary tree is a ternary tree in which (i) each non-leaf node has exactly three children, and (ii)
all leaf nodes are at the same level. This latter level is called the height of the complete ternary tree. The
completeNternary trees of heights 0, 1, and 2, respectively are shown in the figure below, where we use the
symbol to denote a node:

N N N

N N N

N N N

N N N N N N N N N

What is the total number of nodes in a complete ternary tree of height 9?

(a) 211 − 1
10
(b) 2 3+2
10
(c) 3 2−1
10
(d) 3 2+1

15. Which of the following expressions take integral values for all integers n > 100 ?
n
(a) 3 2−1
n
(b) 4 2−1
n
(c) 4 3−1
n
(d) 5 2−1

7

16. At a conference attended by 1235 people, some attendees shake hands with other attendees. As a part of
the local COVID-19 tracing protocol the organizers ask each attendee to note down the number of other
attendees with whom they shook hands. Let N be the sum of all the numbers noted down by the attendees.
That is, N is the sum, taken over all attendees, of the number of other attendees with whom they shook
hands. Which of the following is guaranteed to be true about N ?

(a) N is always a multiple of 1235
(b) N is always a multiple of 2
(c) N is always a multiple of 3
(d) N is always a multiple of 5

17. Let n be a positive integer and let k be a positive integer which is greater than 1. Then n can be represented
as
n = nt k t + nt−1 k t−1 + · · · + n1 k + n0
for a unique set of positive integers n0 , . . . nt . This expression is called the representation of n in base k
and is denoted by (nt nt−1 . . . n0 )k . For example the integer 264 expressed in base 3 is represented as
(1 0 0 2 1 0)3 because 264 = 1 × 35 + 2 × 32 + 1 × 3.
Which of the following expressions represent the integer (1 1 0 1 1 1 1 0)2 ?

(a) (1 1 0 0)8
(b) (1 3 4 2)5
(c) (4 3 2)7
(d) (2 2 0 2 0)3

18. There are two villages X and Y in a faraway land. It is known that each person from village X always tells
the truth, and that each person from village Y always lies.
You meet three people A, B, C who are from these villages. You are told that A and B are from the same
village. A says, “If B is from village X, then I am from village Y ”. Now C says, “If I am from village X,
then 2 + 2 = 4”.
What can you infer about the villages to which A, B, and C belong?

(a) A, B are from village X and C is from village Y .
(b) All three of them are from village Y .
(c) A, B are from village Y and C is from village X.
(d) The given information is insufficient to infer the villages to which A, B, C belong.

8

19. A stick is fixed on the ground at an angle of 60 degrees as shown in Figure 1. The part of the stick which
is above the ground—which is the part that is shown in Figure 1—has a length of 10 metres. At a certain
time T1 during the day, the sun is directly overhead at this location and its rays strike the ground at an
angle of 90 degrees. This is shown in Figure 2. At a certain later time T2 the sun’s rays strike the ground
at an angle of 60 degrees as shown in Figure 3. While Figures 2 and 3 show only the sun’s rays for the sake
of clarity, the stick is present at this location at times T1 and T2 .

Sunrays Sunrays

Stick

60° 60°
90°

Ground Ground Ground

Figure 1 Figure 2 Figure 3

What are the lengths of the shadows of the stick at times T1 and T2 , respectively?
√ √
(a) 5 3 metres and 10 3 metres
√
(b) 5 metres and 10 3 metres
(c) 5 metres and 10 metres
√
(d) 5 3 metres and 10 metres

20. A park in Chennai has walkways as depicted in the figure. The straight path from A to B is 100 metres
long. The curved part is a semicircle whose centre coincides with the midpoint of line segment AB.

A B

A woman and her dog take a walk in this park, both starting from point A at the same time. The woman
takes the straight path from A to B, while the dog goes around the semicircular path. The woman and the
dog both reach point B at the same time. If the woman takes 10 minutes to reach from point A to point B,
which of the following statements about the woman’s and dog’s walks is/are true?

(a) The average speed of the woman’s walk is between 0.15ms−1 and 0.17ms−1
(b) The average speed of the woman’s walk is between 0.17ms−1 and 0.20ms−1
(c) The average speed of the dog’s walk is between 0.25ms−1 and 0.27ms−1
(d) The average speed of the dog’s walk is between 0.27ms−1 and 0.30ms−1

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Part (B) - Short-answer questions

For questions in part (B), you have to write your answer with a short explanation in the space provided for the
question in your answer sheet. If you need more space, you may continue on the pages provided for rough work.
Any such overflows must be clearly labeled.

1. There are three different approaches that climbers can take to reach the summit of Mount Chernoff, namely,
Approach-I, Approach-II, and Approach-III. Each attempt at the summit uses exactly one of these ap-
proaches. From climbers’ logs it is known that Approach-I is used in 40% of the attempts, and Approach-II
and Approach-III are each used in 30% of the attempts. From the same logs it is also known that in 10%
of the attempts which used Approach-I the climber lost their way and had to be rescued. Similarly, 15%
of the attempts that used Approach-II and 17% of the attempts which used Approach-III resulted in the
climber losing their way.
It is reported one day that climber Chebyshev has lost her way. What is the probability that she took
Approach-II?

10

Description for the next four questions:

Dasholytics Inc runs an online analytics dashboard. The company employs three machines—Server 1,
Server 2, and Server 3—to serve the data for the dashboard. At any point in time one of these three
machines has the job of serving the data, and the other two are kept in standby mode. Dasholytics uses a
server scheduler (software) that decides which of the three machines should serve the data at any given point
of time. The scheduler switches between data servers without disrupting the data feed to the dashboard.
The machine which is serving the data sometimes fails to do so; in this case an alert is sent out to the
Dasholytics team and they investigate and fix the problem. The time for which a machine fails to serve
data is accounted as service outage caused by that machine. For technical reasons the server scheduler is
deactivated when there is such an outage; the outage gets over only when the issue with the server is fixed
and it is put back online.
The figures below describe the server usage and outage statistics as compiled over the last one year (365
days). Please use this information to answer the questions that follow.

Fig. (a) Fig. (b)

Figure 1: Figure (a) shows the total service outage caused by each server, as a percentage of the total time that
it served data. Figure (b) shows the total time that each server served data, as a percentage of the total duration
over which this information was collected.

2. Express the total service outage caused by all the three servers, as a percentage of the total duration over
which the information was collected.

3. What is the expected number of hours of service outage in an year (365 days)?

4. As part of their due diligence, a potential customer of Dasholytics picks a random point in time during the
year and checks the status of the dashboard. What is the probability that the customer finds that there was
a service outage at this point in time, and that Server 1 was the server responsible for serving data at that
time?

5. An alert that a service outage has been caused by one of the three servers, was sent out to the Dasholyltics
team. What is the probability that this outage was caused by Server 1?

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6. In a question paper, there are 5 questions in part A, 4 in part B, and 3 in part C. In how many ways can a
candidate make up her choice, if she has to select 3 from part A, 2 from part B, and 2 from part C?

Description for the next two questions
The probability density function of a normal distribution with mean µ and variance σ 2 is of the form
1 x−µ 2
f (x) = √ e−0.5( σ ) , − ∞ < x < ∞.
σ 2π
Let X be a random variable with mean E(X) = µ P and variance Var(X) = σ 2 . Let X1 , X2 , . . . , Xn be n
1
independently sampled values of X, and let X̄ = n i Xi be the sample mean. The central limit theorem
states that if the sample size n is large enough then X̄ approximately follows the normal distribution with
mean µ and variance σ 2 /n. That is, X̄ ≈ N (µ, σ 2 /n). This in turn implies that
√
n(X̄ − µ)
Y = ≈ N (0, 1).
σ

Let Z be a random variable that follows the normal distribution with mean 0 and variance 1. For any real
number a let Φ(a) = P(Z < a) be the probability that Z takes values smaller than a. Then
2
e−u /2
Z a
Φ(a) = P(Z < a) = √ du.
−∞ 2π

For solving the next two problems you may assume the following approximations: Φ(−2) = 0.02, Φ(−1) =
0.16, Φ(0) = 0.5, Φ(1) = 0.84, Φ(2) = 0.98.
The weekly number of sales S at a certain car dealership is known to follow a probability distribution with
mean E(X) = 10 and variance Var(X) = 144. A performance audit picks a random sample of 36 weekly
sales figures S1 , S2 , . . . , S36 from the last two years. They find that the sample mean is 10 and the sample
variance is 144. Use this information to answer the next two questions. You may assume that n = 36 is a
large enough sample size.

7. (a) What is the probability that the average number of sales in a week will be more than 8?
(b) What is the total number of sales over the 36 weeks which were sampled?

8. Check if the following statement(s) are correct. Briefly explain your reasons.

(a) The probability that the average number of sales in a week will be more than 8 but less than 14 is
Z 14
1 u−10 2
√ e−0.5( 12 ) du
8 12 2π

(b) The probability that a salesperson would be able to sell, on an average, ten or more products in a week
is 50%.

9. Two particles move in opposite directions around a circular track. The first moves at a constant speed of 10
m/s. The speed of the second increases at a constant rate of 2 m/s every second. The particles are at the
same position A at time 0, with the second particle being momentarily at rest at t = 0. We are told that
the second meeting of these particles after time 0 takes place at the point A. We assume that the particles
magically cross each other the first time they meet with no change in their instantaneous velocities. Given
this information:

(a) What is the circumference of the track?
(b) At what time did the particles meet for the first time after time 0?

12

10. Words are formed using the characters 0 and 1. The length of a word is the number of characters in it. We
say there is a path from word x to word y if starting from the word x you can get the word y by applying
the following sequence of transformation rules finitely many times in any order.
• Replacing an occurrence of the string 101 by a 0.
• Replacing an occurrence of the string 010 by a 1.
• Replacing a 0 by a 101.
• Replacing a 1 by by a 010.
If there is a path from word x to word y we say x and y are equivalent or that they are in the same equivalence
class. For example the four letter word 1011 is equivalent to the two letter word 01, since the prefix 101 in
1011 can be replaced by a 0 to get 01.
We say x has a shorter description if there is a word of shorter length equivalent to it.
(a) State true or false: for any word x there is a unique shortest word in its equivalence class.
(b) How many three letter words are there in this language which have no shorter descriptions and which
are all in different equivalance classes?
11. You are given an 8 × 8 chessboard and a large supply of T -shaped tiles and square-shaped tiles of the kind
shown below.

Note that the squares in these two shaped tiles are of the same size as the black/white squares on the
chessboard.
We wish to cover the squares of the chess board using these tiles. Each square of the chessboard must be
occupied by exactly one square from a tile; we are not allowed to break the tiles.
(a) Is it possible to cover the chessboard with 16 T shaped tiles? Why?
(b) Is it possible to cover the chessboard with 15 T shaped tiles and one square tile? Why?
12. To guard against login attempts by bots, Alphabet Bank of India uses a captcha challenge on their net
banking login page. The captcha text is a sequence of five letters from the English alphabet.
(a) An early version of the captcha text consisted of a sequence of five distinct letters picked uniformly
at random from the lower-case English alphabet, {a, b, c, . . . , z}. Two examples of such captcha text
are: qfhdk and smgft. An example of such captcha text which has its first two letters in increasing
alphabetical order is: vxztr.
What is the probability that the captcha text generated in this way has the first two letters in
increasing alphabetical order, and why?
(b) After a couple of years the bank noticed that some bots had become very good at cracking the captcha.
So they updated their software so that it now generates the captcha text by picking five distinct letters
uniformly at random from the English alphabet, where the case of each letter can be either upper or
lower. That is, the letters are now picked from the set {a, b, c, . . . , z, A, B, C, . . . , Z}. Note that the
same letter does not appear in both lower and upper case in the same captcha text.
Two examples of the captcha text generated by this version are: QFhDk and sMgfT. Examples of such
captcha text with the first two letters in increasing alphabetical order are: VxPTr and vXptR. Note that
the case of the letter does not matter when deciding the alphabetical order.
What is the probability that the captcha text generated in this way has the first two letters in increasing
alphabetical order, and why?

13

13. Among 40 bicycle gears, 28 are broken or rusted but not both, 6 are non-defective (i.e. neither rusted nor
broken), and the number of broken gears equals the number of rusted ones. How many gears are rusted?

14. Consider the following code, in which A and B are arrays indexed from 0 and lenA and lenB are the numbers
of elements in A and B, respectively.
function foo (A , B , lenA , lenB ) {
acc = 0;

for i = 0 to ( lenA - 1) {
acc = A [ i ]^ acc ;
}
print ( acc );

for i = 0 to ( lenB - 1) {
acc = B [ i ]^ acc ;
}
print ( acc );
}

Here, a^b represents the bitwise Exclusive OR function over variables a and b.
Given integers a and b, we write them in binary, padded by zeros to the left to make them of equal length.
We then apply Exclusive OR to these binary representations bitwise. The operation a^b denotes the in-
teger value obtained by performing bitwise Exclusive OR on the binary values of a and b. For example,
3^4 = 011^100 = 111 = 7, and 9^5 = 1001^0101 = 1100 = 12. The truth table for the Exclusive OR
function is provided.

a b a^b
0 0 0
0 1 1
1 0 1
1 1 0

Let A = [1, 3, 3, 5, 5] and B = [9, 5, 5, 3, 3, 1]. What are the two values printed by foo(A,B,5,6),
in order?

14

15. Consider the following code, in which A and B are arrays indexed from 0 and lenA and lenB are the numbers
of elements in A and B, respectively.
function foo (A , B , lenA , lenB ) {
sum = lenA + lenB ;
i = 0;
j = 0;

for t = 0 to ( sum - 1) {
if ( A [ i ] < B [ j ]) {
i = i + 1;
} else {
if ( A [ i ] > B [ j ]) {
j = j + 1;
} else {
return A [ i ];
}
}
}

return ( -1);
}

Let A = [2, 4, 6, 7, 8, 9, 10] and B = [1, 3, 5, 7, 9, 11, 13]. What does foo(A,B,7,7) return?

16. Consider the following code, in which A is an integer array of length n indexed from 0, and x is an integer.
function foo (A ,x , n ) {
found = False ;

while ( found != True ) {
i = randInt (0 , n );

if ( A [ i ] == x ) {
found = True ;
}
}
return ( i );
}

Here, randInt(0,n) returns an integer picked uniformly at random from the range {0, 1, . . . , (n − 1)}.
All integers present in array A are distinct, and integer x is present in array A. Suppose we make the call
foo(A,x,n). What is the expected number of times that the call randInd(0, n) is made from within this
call to foo(A,x,n)?

17. A graph consists of a finite set of vertices, and edges between some (unordered) pairs of these vertices. The
graphs in this question have no loops (an edge between a vertex and itself) or multiple edges (more than
one edge between the same pair of vertices). A vertex v in a graph is said to be a global vertex if there is an
edge between v and every other vertex in the graph.
A graph G is constructed as follows: First, its vertex set is assigned to be the set {v1 , v2 , . . . , v10 }. Next,
between each pair of distinct vertices vi , vj , an edge is added with probability 21 .

(a) What is the probability that vertex v1 is a global vertex in graph G?
(b) What is the expected number of global vertices in graph G?

15

18. The sum of two positive integers a and b is 48 and their least common multiple is 189. Find a and b.

19. Recall that for an n × n matrix A, det(A) = det(AT ). Here AT is the transpose of matrix A. An n × n
matrix A is said to be skew-symmetric if ai,j = −aj,i , where aij denotes the element in the ith row and j th
column,for all 1 ≤ i, j ≤ n and all elements are real.

(a) Is the following statement true or false? Why?
If n is odd, then the determinant of an n × n-skew symmetric matrix is 0.
(b) Give an example of a 2 × 2 skew-symmetric matrix whose determinant is non-zero.

20. An object X is viewed from a camera placed at point C1 . The camera is then moved to the point C2 and
X is again viewed from this new position. Suppose H is the plane containing the points C1 and C2 . Let H ′
be a plane parallel to H lying between the object X and the camera positions. Let f denote the distance
between H and H ′ . Let X1 , X2 be the points of intersection of H ′ with the line segments joining X with C1
and C2 respectively.
Let x1 be the distance between the reference axis L1 and X1 . Similarly let x2 be the distance between the
reference axis L2 and X2 . The axes L1 and L2 are both perpendicular to the camera plane H. The difference
|x1 − x2 | is called the disparity in views. Find a formula for the depth z i.e. distance of the object from the
camera plane in terms of f , the distance b between the cameras, and the disparity.

• ✗

n

H
:¥"

H
C
l

I

,
← b →
1

ti
Cz
:
↓ .
Z

×

16