Let $\mathbb { R } ^ { + } = \{ x \in \mathbb { R } : x \geq 0 \}$. For $x \in \mathbb { R } ^ { + }$, denote by $\operatorname{FRAC}( x )$ the fractional part of $x$, i.e., $x - n$ where $n$ is the largest integer that is less than or equal to $x$. Consider the series $\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { FRAC } ( x / n ) } { n }$. Pick the correct statement(s) from below.
(A) The above series converges for all $x \in \mathbb { R } ^ { + } - \mathbb { Z }$.
(B) The above series diverges for some non-negative integer $x$.
(C) The above series defines a continuous function in a neighbourhood of $\frac { 1 } { 2 }$.
(D) The above series defines a continuous function in a neighbourhood of 1.

Let $f _ { n } ( x ) = \frac { 1 } { 1 + x ^ { n } }$. Pick the correct statement(s) from below.
(A) $f _ { n }$ converges uniformly on $[ 0,1 / 2 ]$.
(B) $f _ { n }$ converges uniformly on $[ 0,1 )$.
(C) $f _ { n }$ converges uniformly on $[ 0,2 ]$.
(D) $f _ { n }$ converges pointwise on $[ 0 , \infty )$.

Pick the correct statement(s) from below.
(A) If $f ( z )$ is a function defined on $\mathbb { C }$ that satisfies the Cauchy-Riemann equations at $z = 0$, then $f ( z )$ is complex-differentiable at $z = 0$.
(B) The function $\frac { ( \sin z - z ) \bar { z } ^ { 3 } } { | z | ^ { 6 } }$ is holomorphic on $\{ z \in \mathbb{C} : 0 < | z | < 1 \}$ and has a removable singularity at $z = 0$.
(C) There exists a holomorphic function on $\{ z \in \mathbb { C } : | z | > 3 \}$ whose derivative is $\frac { z } { ( z - 2 ) ^ { 2 } }$.
(D) There exists a holomorphic function on the upper half plane $\{ z \in \mathbb { C } : \mathfrak { I } z > 0 \}$ whose derivative is $\frac { z } { ( z - 2 ) ^ { 2 } \left( z ^ { 2 } + 4 \right) }$.

Pick the correct statement(s) from below.
(A) There exists a maximal ideal $M$ of $\mathbb { Z } [ x ]$ such that $M \cap \mathbb { Z } = ( 0 )$.
(B) If $M$ is a maximal ideal of $\mathbb { Z } [ x ]$, then $\mathbb { Z } [ x ] / M$ is finite.
(C) If $I$ is an ideal of $\mathbb { Z } [ x ]$ such that $\mathbb { Z } [ x ] / I$ is finite, then $I$ is maximal.
(D) The ideal $\left( 7 , x ^ { 2 } - 14 x - 2 \right)$ in $\mathbb { Z } [ x ]$ is maximal.

9. A binary relation $R$ defined on a set $S$ is said to be antisymmetric if for $x , y \in S , x R y$ and $y R x \Longrightarrow x = y$. Let $R _ { 1 } , R _ { 2 }$ be two binary relations defined on a set $S$. The union of $R _ { 1 } , R _ { 2 }$ is the binary relation $U$ defined on $S$ as: For $x , y \in S , x U y \Longleftrightarrow x R _ { 1 } y$ or $x R _ { 2 } y$. The intersection of $R _ { 1 } , R _ { 2 }$ is the binary relation $I$ defined on $S$ as: For $x , y \in S , x I y \Longleftrightarrow x R _ { 1 } y$ and $x R _ { 2 } y$. Which of the following statements is/are true?
(a) A binary relation cannot be both symmetric and antisymmetric.
(b) A binary relation can be both transitive and antisymmetric.
(c) The union of two equivalence relations is always an equivalence relation.
(d) The intersection of two equivalence relations is always an equivalence relation.

Let $M _ { n } ( \mathbb { R } )$ be the space of $n \times n$ real matrices. View $M _ { n } ( \mathbb { R } )$ as a metric space with $$d \left( \left[ a _ { i , j } \right] , \left[ b _ { i , j } \right] \right) : = \max _ { i , j } \left| a _ { i , j } - b _ { i , j } \right|$$ Let $U \subset M _ { n } ( \mathbb { R } )$ be the subset of matrices $M \in M _ { n } ( \mathbb { R } )$ such that $\left( M - I _ { n } \right) ^ { n } = 0$.
(A) $U$ is closed.
(B) $U$ is open.
(C) $U$ is compact.
(D) $U$ is neither closed or open.

Let $G$ be an abelian group and let $H$ be a nontrivial subgroup of $G$, that is, $H$ is a subgroup containing at least two elements. Show that the following two statements are equivalent.
(A) For every nontrivial subgroup $K$ of $G$, the subgroup $K \cap H$ is also nontrivial.
(B) $H$ contains every nontrivial minimal subgroup of $G$ and every element of the quotient group $G / H$ has finite order.

11. Which of the following is/are logically equivalent to $\neg ( P \Longrightarrow Q )$ ?
(a) $\neg P \vee Q$
(b) $\neg P \wedge Q$
(c) $Q \Longrightarrow P$
(d) $P \wedge \neg Q$

Consider the ring $\mathcal { C } ( \mathbb { R } )$ of continuous real-valued functions on $\mathbb { R }$, with pointwise addition and multiplication. For $A \subset \mathbb { R }$, the ideal of $A$ is $I ( A ) = \{ f \in \mathcal { C } ( \mathbb { R } ) \mid f ( a ) = 0$ for all $a \in A \}$. For a subset $I$ of $\mathcal { C } ( \mathbb { R } )$, the zero-set of $I$ is $Z ( I ) = \{ a \in \mathbb { R } \mid f ( a ) = 0$ for all $f \in I \}$. Prove the following:
(A) (3 marks) $Z ( I \cap J ) = Z ( I J )$ for ideals $I$ and $J$ of $\mathcal { C } ( \mathbb { R } )$.
(B) (2 marks) For each $a \in \mathbb { R } , I ( a )$ is a maximal ideal.
(C) (3 marks) The set $\{ f \in \mathcal { C } ( \mathbb { R } ) \mid f$ has compact support $\}$ is a proper ideal, and its zero set is empty.
(D) (2 marks) True/False: For each prime ideal $\mathfrak { p }$ of $\mathcal { C } ( \mathbb { R } ) , Z ( \mathfrak { p } )$ is a singleton set. (Justify your answer.)

Let $f , g , h$ be functions from $\mathbb { R }$ to $\mathbb { R }$ such that $$h ( f ( x ) + g ( y ) ) = x y$$ for all $x , y \in \mathbb { R }$. Show the following:
(A) $(2$ marks$)$ $h$ is surjective.
(B) $(3$ marks$)$ If $f$ is continuous then $f$ is strictly monotone.
(C) $(5$ marks$)$ There do not exist continuous functions $f , g , h$ satisfying $(*)$.

Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ and $g : \mathbb { R } \longrightarrow \mathbb { R }$ be continuous functions. Assume that $g$ is periodic with period 1. Show that $$\lim _ { n \mapsto \infty } \int _ { 0 } ^ { 1 } f ( x ) g ( n x ) d x = \left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right)$$

Prove or disprove each of the statements below.
(A) (4 marks) Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be a continuous function that takes both positive and negative values. Then $f$ has infinitely many zeros.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a continuous function. Then $f$ is not open.

Prove or disprove the following statements:
(A) (5 marks) Suppose that $f ( z )$ is a complex analytic function in the punctured unit disk $0 < | z | < 1$ such that $\lim _ { n \longrightarrow \infty } f \left( \frac { 1 } { n } \right) = 0$ and $\lim _ { n \longrightarrow \infty } f \left( \frac { 2 } { 2 n - 1 } \right) = 1$, then there exists a positive integer $N > 0$ such that $\lim _ { z \rightarrow 0 } \left| z ^ { - N } f ( z ) \right| = \infty$.
(B) (5 marks) There exists a non-zero entire function $f$ such that $f \left( e ^ { 2 \pi i e n ! } \right) = 0$ for all $n \geq 2025$.

16. In the following code the operator \% denotes the remainder after integer division. That is: for positive integers $\mathrm { a } , \mathrm { b }$ the value $\mathrm { a } \% \mathrm {~b}$ is the remainder obtained when a is divided by b .
\begin{verbatim} function fizzbuzz(n) { count = 0; for i from 0 to (n-1) { if ((i % 3) == 0) and ((i % 5) != 0) { count = count + 1; } } return(count); } \end{verbatim}
What does fizzbuzz(100) return?
(a) 27
(b) 33
(c) 45
(d) 60

Let $X \subseteq \mathbb { R } ^ { n }$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma ^ { \prime } ( 0 )$, where $\gamma : ( - \epsilon , \epsilon ) \longrightarrow X$ is a differentiable function with $\gamma ( 0 ) = p$. ($\epsilon \in \mathbb { R } , \epsilon > 0$.) The tangent space of $X$ at $p$ is the $\mathbb { R }$-vector space of all the tangent vectors at $p$. Think of $\mathrm { GL } _ { n } ( \mathbb { C } )$ as a subspace of $\mathbb { R } ^ { 2 n ^ { 2 } }$, with the euclidean topology.
Let $G : = \left\{ A \in \mathrm { GL } _ { 2 } ( \mathbb { C } ) \mid A ^ { * } A = A A ^ { * } = I _ { 2 } , \operatorname { det } A = 1 \right\}$.
(A) (2 marks) Show that every tangent vector of $\mathrm { GL } _ { n } ( \mathbb { C } )$ at $I _ { n }$ is of the form $\gamma _ { A } ^ { \prime } ( 0 )$ where $A$ is a $n \times n$ complex matrix and $\gamma _ { A } : \mathbb { R } \longrightarrow \mathrm { GL } _ { n } ( \mathbb { C } )$ is the function $t \mapsto e ^ { t A }$.
(B) (3 marks) Show that the tangent space of $G$ at $I _ { 2 }$ is $V : = \left\{ \left. \left[ \begin{array} { c c } i a & z \\ - \bar { z } & - i a \end{array} \right] \right\rvert \, a \in \mathbb { R } , z \in \mathbb { C } \right\}$.
(C) (5 marks) Consider the homeomorphism $\Phi : G \longrightarrow \mathbb { S } ^ { 3 }$ (where $\mathbb { S } ^ { 3 }$ denotes the unit sphere in $\mathbb { R } ^ { 4 }$) given by $$\left[ \begin{array} { c c } \alpha & \beta \\ \bar { \beta } & \bar { \alpha } \end{array} \right] \mapsto ( \Re ( \alpha ) , \Im ( \alpha ) , \Re ( \beta ) , \Im ( \beta ) )$$ Define a 'multiplication' on $V$ by $[ A , B ] = \frac { A B - B A } { 2 }$. Determine the multiplication on the tangent space at $\Phi \left( I _ { 2 } \right)$ induced by the derivative $D \Phi$. (Hint: The map $( A , B ) \longrightarrow [ A , B ]$ is $\mathbb { R }$-bilinear.)

17. Alpha and Beta are inhabitants of an island of knights and knaves, where knights always tell the truth and knaves always lie. Alpha and Beta are alone at a beach when Alpha says: "At least one of us is a knave." And Beta says: "We are both knaves." Which of the following is/are true?
(a) Alpha and Beta are both knights
(b) Alpha and Beta are both knaves
(c) Alpha is a knight and Beta is a knave
(d) Alpha is a knave and Beta is a knight

Questions 18 to 20 are based on the following code.

The two arguments to the function Mystery(A, n) in the code below are: (i) an integer array A indexed from 0 , and (ii) the number n of elements in A . Each element of A is an integer from the set $\{ 1,2 , \ldots , n \}$. The expression [0] $* ( \mathrm { n } + 1 )$ creates an array, indexed from 0 , that contains $n + 1$ zeroes.
\begin{verbatim} function Mystery(A, n) { found = False; value = None; B = [0] * (n+1); for i from 1 to n { B[A[i]] = B[A[i]] + 1; } for i from 1 to n { if (found == False) { if (B[A[i]] == 1) { found = True; value = A[i]; } } } if (found == True) { return(value); } else { return(None); } } \end{verbatim}
Answer the next three questions about this function. [0pt]

Let $\mathbb { F } _ { q }$ be the finite field with $q$ elements and $P \in \mathbb { F } _ { q } [ x ]$ be a monic irreducible polynomial of even degree $2 d$. Then show that $P$, when considered as a polynomial in $\mathbb { F } _ { q ^ { 2 } } [ x ]$, decomposes into a product $P = Q _ { 1 } Q _ { 2 }$ of irreducible polynomials $Q _ { i }$ in $\mathbb { F } _ { q ^ { 2 } } [ x ]$ with $\operatorname { deg } \left( Q _ { i } \right) = d$.

18. What does the function call Mystery ([1, 2, 3, 3, 2, 1], 6) return?
(a) None
(b) 1
(c) 2
(d) 3 [0pt]

Show that the power series $\sum _ { n = 1 } ^ { \infty } z ^ { n ! }$ represents an analytic function $f ( z )$ in the open unit disk $\Delta$ centred at 0. Show that $f ( z )$ cannot be extended to a continuous function on any connected open set $U$ such that $U$ is strictly larger than $\Delta$.

19. What does the function call Mystery ([1, 2, 4, 3, 2, 1], 6) return?
(a) None
(b) 2
(c) 3
(d) 4 [0pt]

It is known that there exist surjective continuous maps $I \longrightarrow I ^ { 2 }$ where $I = [ 0,1 ]$ is the unit interval.
(A) (4 marks) Using the above result or otherwise, show that there exists a surjective continuous map $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$.
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a surjective continuous map. Let $\Gamma = \{ ( x , f ( x ) ) \mid x \in \mathbb { R } \} \subset \mathbb { R } ^ { 3 }$. Show that $\mathbb { R } ^ { 3 } \setminus \Gamma$ is path connected.

20. What does the function call Mystery ([6, 5, 4, 3, 2, 5], 6) return?
(a) 3
(b) 4
(c) 5
(d) 6

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. A girl writes five consecutive positive integers on a blackboard. She then erases one of them. The sum of the remaining four numbers is 2025 . What number did she erase?
2. A toy company currently sells 1,000 toys each month at a price of ₹500 per toy. To increase their sales, the company is considering to lower the price. Market research shows that for every ₹10 decrease in the price, the number of toys sold increases by 100 . However, the price cut applies to all the toys sold.
   (a) By how much should the company reduce the price, so as to maximize its monthly revenue?
   (b) What would be the maximum revenue per month that the company can achieve?

Assume that the decrease in price is an integer multiple of ₹1.
3. 15 balls are placed independently and uniformly at random into 15 bins numbered from 1 to 15 . The probability that a ball ends up in a particular bin is $\frac { 1 } { 15 }$.
(a) What is the probability that the first 5 balls go into different bins, conditioned on the event that first four balls are in different bins?
(b) What are the possible values of the expected number of balls in bin 1 conditioned on the event that the first 13 balls are in different bins?
4. There are 18 chocolates in a bag, of which 7 are green, 6 are blue, and 5 are red. We pick chocolates one at a time from the bag without replacement.
(a) What is the probability that the first and the third chocolate are green?
(b) What is the probability that after picking twelve chocolates, only chocolates of one colour remain in the bag?
5. Let $M$ be an $n \times n$ matrix. We define an elementary row operation on $M$ to be one of the following: i Interchanging some two rows of $M$. ii Multiplying a row in $M$ by a non-zero scalar. iii Adding a scalar multiple of one row of $M$ to another row of $M$. An $n \times n$ matrix is said to be elementary if it is the result of a single elementary row operation performed on the $n \times n$ identity matrix. Which of the following statements is/are true? Justify your answer with a short proof, if the statement is true, otherwise, provide a counterexample. a Every elementary operation on a $n \times n$ matrix $A$ can be performed by multiplying $A$ by an elementary $n \times n$ matrix on the right. b An elementary row operation on a $n \times n$ matrix $A$ results in a matrix with the same determinant as that of $A$. 6. Consider the following polynomial of positive degree $n$
$$P _ { n } ( x ) = 1 + 2 x + 3 x ^ { 2 } + \ldots + ( n + 1 ) x ^ { n }$$
Show that there is no real number $r$ such that $P _ { n } ( r ) = 0$ when $n$ is even. 7. How many positive integers less than 1000 are neither divisible by 3 nor divisible by 5 ? Explain how you arrived at your answer. 8. Three hostel friends Amar, Prem and Raj are suspected of breaking a window. They made the following statements when questioned by the warden:

• Amar: I did not break it. Prem is lying.
• Prem: Amar is telling the truth. Raj broke the window.
• Raj: I did not break it. Either Amar is telling the truth or Prem is telling the truth.

You know that exactly one of them lied and the other two told the truth. Then, who broke the window? Justify. 9. A cloth bag labeled $X$ contains two apples, bag $Y$ contains two oranges and bag $Z$ one apple and one orange. You pick a bag at random and then remove one fruit from that bag at random. Suppose you removed an apple. What is the probability that the fruit remaining in the bag is also an apple? Justify.

Questions 10 and 11 are based on the following description.

The following question appeared in a quiz: "Write the pseudocode for a function Closest $( A , n , x )$ that takes an array $A$, a positive integer $n$, and an integer $x$ as arguments. The elements of $A$ are all integers less than $2 ^ { 64 }$, and $n$ is the number of elements in $A$. The call Closest $( A , n , x )$ should return an integer $y$ such that: (i) $y \neq x$, (ii) $y$ is present in $A$, and (iii) there is no $z \neq x$ in $A$ where $| z - x | < | y - x |$ holds. If $A$ has no such element $y$, then the function should return the special value None."
A student submitted the code below as the answer to this question. In the code the array A is indexed from 0 , and MAXINT $= 2 ^ { 64 } - 1$. The call abs( $z$ ) returns the absolute value $| z |$ of integer $z$.
\begin{verbatim} function Closest(A, n, x) { minVal = MAXINT; for i from 0 to (n-1) { absDiff = abs(x - A[i]); if (absDiff < minVal) { minVal = absDiff; y = A[i]; } } if (minVal != MAXINT) { return(y); } else { return(None); } } \end{verbatim}
This answer turned out to be wrong; this function gives the correct answer for some valid inputs, and wrong answers for other valid inputs. Answer the next two questions about this function. 10. What do the following function calls return?
(a) Closest $( [ - 10,2,10 ] , 3,8 )$ [0pt] (b) Closest ([0,-5,4], 3, 7) 11. Give one example of (i) an input array A with exactly 3 elements and (ii) an integer x for which the call Closest(A, 3, x) returns a wrong answer. What is this wrong answer? What is the correct answer? 12. A sequence of five natural numbers $s _ { 1 } \leq s _ { 2 } \leq s _ { 3 } \leq s _ { 4 } \leq s _ { 5 }$ satisfy the following conditions:

• $\sum _ { i = 1 } ^ { 5 } s _ { i } = 35$
• $\sum _ { i = 1 } ^ { 3 } s _ { i } = 15$
• $\sum _ { i = 3 } ^ { 5 } s _ { i } = 27$
• $s _ { 2 }$ is even
• $s _ { 4 } - s _ { 2 } = 2$

Find all such sequences that satisfy the above conditions. 13. Five executives of a company namely CEO (chief executive officer), CFO (chief financial officer), COO (chief operating officer), CTO (chief technology officer), CMO (chief marketing officer) are to be seated around a circular table.

• The CEO must sit next to the CFO.
• The COO must not sit next to the CTO.
• The CTO must not sit next to the CEO.

In how many distinct ways can they be seated? (Rotations of the same arrangement are considered the same). 14. Evaluate the following limit:
$$\lim _ { x \rightarrow 0 } \frac { e ^ { 2 x } - 2 x - \cos x } { x \sin x }$$

1. Find the determinant of the following matrix:

$$A = \left[ \begin{array} { c c c c } 1 & 1 & 4 & 3 \\ 5 & 8 & 26 & 21 \\ 2 & 3 & 10 & 8 \\ 3 & 5 & 16 & 13 \end{array} \right]$$

1. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function which is twice differentiable everywhere and suppose that $f ^ { \prime } ( q ) = 0$ for every rational number $q$. Find the value of $f \left( \pi ^ { 2 } \right) - f ( \pi )$.
2. Consider the following $2 \times 2$ matrix

$$B = \left[ \begin{array} { c c } 0 & r \\ - 1 & 0 \end{array} \right] ,$$
where $r$ is a nonzero real number. Find $B ^ { 2025 }$, i.e., the matrix obtained by multiplying $B$ with itself 2025 times.

Instructions for Questions 18,19, and 20:

Read the following description carefully and answer the questions that follow. Use all the information provided. Clearly show all your calculations.

Description:

NorthCool Beverages Pvt. Ltd. is a leading beverage company that operates across six northern states of India. During the past summer the company launched an aggressive marketing campaign to promote its range of flavoured drinks. The charts below summarise the sales data collected during this campaign. 18. What is the revenue, in Lakhs, from the sales of the mango flavoured drink? 19. NorthCool Beverages plans to launch a new variant, "Lemon Max", in the two states with the highest sales of the lemon flavour. The company expects Lemon Max to generate additional revenue equal to $20 \%$ of the current lemon flavour sales in those two states. What would be the percentage revenue share of lemon flavour drinks if their expectations are met?
[Figure]
Figure 1: Revenue share by flavour, and sales of lemon-flavoured drink by state

1. On average, for every ₹ 1 lakh in total revenue from the lemon-flavoured drink, 10,000 litres of lemonflavoured drink are sold across the six states. Estimate the total volume (in litres) of lemon-flavoured drink sold by NorthCool Beverages across all six states.

The following table shows the manufacturing cost per unit, selling price, and sales volume for two products A and B produced by a company last year.

Category	Product A	Product B
Manufacturing Cost	$a _ { 11 }$	$a _ { 12 }$
Selling Price	$a _ { 21 }$	$a _ { 22 }$

Sales Volume	First Half	Second Half
A	$b _ { 11 }$	$b _ { 12 }$
B	$b _ { 21 }$	$b _ { 22 }$

Represent the above tables as matrices $A = \left( \begin{array} { l l } a _ { 11 } & a _ { 12 } \\ a _ { 21 } & a _ { 22 } \end{array} \right)$ and $B = \left( \begin{array} { l l } b _ { 11 } & b _ { 12 } \\ b _ { 21 } & b _ { 22 } \end{array} \right)$ respectively, and let the product of these two matrices be $A B = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$. When the profit per unit is defined as the selling price minus the manufacturing cost, select all correct statements from . [3 points]
ㄱ. $a + b$ is the total manufacturing cost of products sold in the first half of last year. ㄴ. $c + d$ is the total selling amount of products sold throughout last year. ㄷ. $d - b$ is the total profit from products sold in the second half of last year.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

As shown in the figure below, for a natural number $n$, $n$ terms $$\left[ \frac { n } { 1 } \right] \left[ \frac { n } { 2 } \right] \left[ \frac { n } { 3 } \right] , \cdots , \left[ \frac { n } { n } \right]$$ are arranged in the $n$-th row from column 1 to column $n$ in order. (Here, $[ x ]$ is the greatest integer not exceeding $x$.)
Select all correct statements from . [4 points]
ㄱ. In row $n$, the number of terms with value 1 is $\left[ \frac { n + 1 } { 2 } \right]$. ㄴ. In row 100, the number of terms with value 3 is 8. ㄷ. In column 3, the number of terms with value 5 is 5.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ

As shown in the figure below, for a natural number $n$, the $n$ terms $$\left[ \frac { n } { 1 } \right] , \left[ \frac { n } { 2 } \right] , \left[ \frac { n } { 3 } \right] , \cdots , \left[ \frac { n } { n } \right]$$ are arranged in the $n$-th row from column 1 to column $n$ in order. (Here, $[ x ]$ is the greatest integer not exceeding $x$.)
Which of the following in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. In the $n$-th row, the number of terms with value 1 is $\left[ \frac { n + 1 } { 2 } \right]$. ㄴ. In the 100th row, the number of terms with value 3 is 8. ㄷ. In the 3rd column, the number of terms with value 5 is 5.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ