Suppose Roger has 4 identical green tennis balls and 5 identical red tennis balls. In how many ways can Roger arrange these 9 balls in a line so that no two green balls are next to each other and no three red balls are together?
(A) 8
(B) 9
(C) 11
(D) 12

The number of permutations $\sigma$ of $1,2,3,4$ such that $| \sigma ( i ) - i | < 2$ for every $1 \leq i \leq 4$ is
(A) 2
(B) 3
(C) 4
(D) 5.

Let 10 red balls and 10 white balls be arranged in a straight line such that 10 each are on either side of a central mark. The number of such symmetrical arrangements about the central mark is
(A) $\frac { 10 ! } { 5 ! 5 ! }$
(B) $10 !$
(C) $\frac { 10 ! } { 5 ! }$
(D) $2 \cdot 10 !$

A finite sequence of numbers $(a_{1}, \ldots, a_{n})$ is said to be alternating if
$$a_{1} > a_{2}, \quad a_{2} < a_{3}, \quad a_{3} > a_{4}, \quad a_{4} < a_{5}, \ldots$$ $$\text{or} \quad a_{1} < a_{2}, \quad a_{2} > a_{3}, \quad a_{3} < a_{4}, \quad a_{4} > a_{5}, \ldots$$
How many alternating sequences of length 5, with distinct numbers $a_{1}, \ldots, a_{5}$ can be formed such that $a_{i} \in \{1, 2, \ldots, 20\}$ for $i = 1, \ldots, 5$?

If the word PERMUTE is permuted in all possible ways and the different resulting words are written down in alphabetical order (also known as dictionary order), irrespective of whether the word has meaning or not, then the $720 ^ { \text {th} }$ word would be:
(A) EEMPRTU
(B) EUTRPME
(C) UTRPMEE
(D) MEETPUR.

The number of different ways to colour the vertices of a square $PQRS$ using one or more colours from the set \{Red, Blue, Green, Yellow\}, such that no two adjacent vertices have the same colour is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 84 .

Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by 0 or 1.
(a) In how many ways can this be done such that each row sum and each column sum is even?
(b) In how many ways can this be done such that each row sum and each column sum is odd?

How many numbers formed by rearranging the digits of 234578 are divisible by 55?
(A) 0
(B) 12
(C) 36
(D) 72

Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done?
(A) 3
(B) 4
(C) 5
(D) 6

8. How many anagrams, even without meaning, are there of the word ``STUDIARE''? In how many of these anagrams can the word ``ARTE'' be read consecutively, as for example in ``SUARTEDI''?
How many anagrams, even without meaning, are there of the word ``VACANZA''? ``Mathematics knows no races or geographical boundaries; for mathematics, the cultural world is a single nation''
\footnotetext{Maximum duration of the exam: 6 hours. The use of scientific or graphical calculators is permitted provided they are not equipped with symbolic algebraic processing capability and do not have Internet connectivity. The use of a bilingual dictionary (Italian–language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the distribution of the exam. }

32. An $n$-digit number is a positive number with exactly $n$ digits. Nine hundred distinct $n$ digit numbers are to be formed using only the three digits 2,5 and 7 . The smallest value of n for which this is possible is:
(A) 6
(B) 7
(C) 8
(D) 9

15. In a triangle $A B C$, Let $\angle C = n / 2$. If $r$ is the inradius and $R$ is the circum-radius of the triangle, then $2 ( r + R )$ is equal to :
(A) $a + b$
(B) $b + c$
(C) $c + a$
(D) $a + b + c$

16. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even position :
(A) 16
(B) 36

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(C) 60
(D) 180

5. The number of arrangements of the letters of the word BANANA in which the two N' s do not appear adjacently is

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(A) 40
(B) 60
(C) 80
(D) 100

3. Using permutation or otherwise prove that $\frac { n ^ { 2 } ! } { ( n ! ) ^ { n } }$ is an integer, where $n$ is a positive integer.
Sol. Let there be $\mathrm { n } ^ { 2 }$ objects distributed in n groups, each group containing n identical objects. So number of arrangement of these $n ^ { 2 }$ objects are $\frac { n ^ { 2 } ! } { ( n ! ) ^ { n } }$ and number of arrangements has to be an integer. Hence $\frac { \mathrm { n } ^ { 2 } } { ( \mathrm { n } ! ) ^ { \mathrm { n } } }$ is an integer.

3. Using permutation or otherwise prove that $\frac { n ^ { 2 } ! } { ( n ! ) ^ { n } }$ is an integer, where $n$ is a positive integer.
Sol. Let there be $\mathrm { n } ^ { 2 }$ objects distributed in n groups, each group containing n identical objects. So number of arrangement of these $n ^ { 2 }$ objects are $\frac { n ^ { 2 } ! } { ( n ! ) ^ { n } }$ and number of arrangements has to be an integer. Hence $\frac { \mathrm { n } ^ { 2 } } { ( \mathrm { n } ! ) ^ { \mathrm { n } } }$ is an integer.

Consider all possible permutations of the letters of the word ENDEANOEL.
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) The number of permutations containing the word ENDEA is
(B) The number of permutations in which the letter E occurs in the first and the last positions is
(C) The number of permutations in which none of the letters D, L, N occurs in the last five positions is
(D) The number of permutations in which the letters A, E, O occur only in odd positions is
Column II
(p) $5 !$
(q) $2 \times 5 !$
(r) $7 \times 5 !$
(s) $21 \times 5 !$

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
(A) 55
(B) 66
(C) 77
(D) 88

41. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is
(A) 75
(B) 150
(C) 210
(D) 243

ANSWER : B

1. Let $f ( x ) = \left\{ \begin{array} { r l } x ^ { 2 } \left| \cos \frac { \pi } { x } \right| , & x \neq 0 \\ 0 , & x = 0 \end{array} , x \in \mathbb { R } \right.$, then $f$ is
   (A) differentiable both at $x = 0$ and at $x = 2$
   (B) differentiable at $x = 0$ but not differentiable at $x = 2$
   (C) not differentiable at $x = 0$ but differentiable at $x = 2$
   (D) differentiable neither at $x = 0$ nor at $x = 2$

ANSWER : B

1. The function $f : [ 0,3 ] \rightarrow [ 1,29 ]$, defined by $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 1$, is
   (A) one-one and onto.
   (B) onto but not one-one.
   (C) one-one but not onto.
   (D) neither one-one nor onto.

ANSWER: B

1. If $\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } + x + 1 } { x + 1 } - a x - b \right) = 4$, then
   (A) $a = 1 , b = 4$
   (B) $a = 1 , b = - 4$
   (C) $a = 2 , b = - 3$
   (D) $a = 2 , b = 3$
2. Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a = z ^ { 2 } + z + 1$ is real. Then $a$ cannot take the value
   (A) - 1
   (B) $\frac { 1 } { 3 }$
   (C) $\frac { 1 } { 2 }$
   (D) $\frac { 3 } { 4 }$

ANSWER : D

1. The ellipse $E _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E _ { 2 }$ passing through the point $( 0,4 )$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E _ { 2 }$ is
   (A) $\frac { \sqrt { 2 } } { 2 }$
   (B) $\frac { \sqrt { 3 } } { 2 }$
   (C) $\frac { 1 } { 2 }$
   (D) $\frac { 3 } { 4 }$

ANSWER : C

1. Let $P = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix and let $Q = \left[ b _ { i j } \right]$, where $b _ { i j } = 2 ^ { i + j } a _ { i j }$ for $1 \leq i , j \leq 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is
   (A) $2 ^ { 10 }$
   (B) $2 ^ { 11 }$
   (C) $2 ^ { 12 }$
   (D) $2 ^ { 13 }$

ANSWER : D

1. The integral $\int \frac { \sec ^ { 2 } x } { ( \sec x + \tan x ) ^ { 9 / 2 } } d x$ equals (for some arbitrary constant $K$ )
   (A) $- \frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } - \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
   (B) $\frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } - \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
   (C) $- \frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } + \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$
   (D) $\frac { 1 } { ( \sec x + \tan x ) ^ { 11 / 2 } } \left\{ \frac { 1 } { 11 } + \frac { 1 } { 7 } ( \sec x + \tan x ) ^ { 2 } \right\} + K$

ANSWER : C

1. The point $P$ is the intersection of the straight line joining the points $Q ( 2,3,5 )$ and $R ( 1 , - 1,4 )$ with the plane $5 x - 4 y - z = 1$. If $S$ is the foot of the perpendicular drawn from the point $T ( 2,1,4 )$ to $Q R$, then the length of the line segment $P S$ is
   (A) $\frac { 1 } { \sqrt { 2 } }$
   (B) $\sqrt { 2 }$
   (C) 2
   (D) $2 \sqrt { 2 }$

ANSWER : A

1. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x ^ { 2 } + y ^ { 2 } = 9$ is
   (A) $20 \left( x ^ { 2 } + y ^ { 2 } \right) - 36 x + 45 y = 0$
   (B) $20 \left( x ^ { 2 } + y ^ { 2 } \right) + 36 x - 45 y = 0$
   (C) $36 \left( x ^ { 2 } + y ^ { 2 } \right) - 20 x + 45 y = 0$
   (D) $36 \left( x ^ { 2 } + y ^ { 2 } \right) + 20 x - 45 y = 0$

ANSWER : A

SECTION II: Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 51. Let $\theta , \varphi \in [ 0,2 \pi ]$ be such that $2 \cos \theta ( 1 - \sin \varphi ) = \sin ^ { 2 } \theta \left( \tan \frac { \theta } { 2 } + \cot \frac { \theta } { 2 } \right) \cos \varphi - 1$, $\tan ( 2 \pi - \theta ) > 0$ and $- 1 < \sin \theta < - \frac { \sqrt { 3 } } { 2 }$. Then $\varphi$ cannot satisfy
(A) $0 < \varphi < \frac { \pi } { 2 }$
(B) $\frac { \pi } { 2 } < \varphi < \frac { 4 \pi } { 3 }$
(C) $\frac { 4 \pi } { 3 } < \varphi < \frac { 3 \pi } { 2 }$
(D) $\frac { 3 \pi } { 2 } < \varphi < 2 \pi$

ANSWER : ACD

1. Let $S$ be the area of the region enclosed by $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$, and $x = 1$. Then
   (A) $S \geq \frac { 1 } { e }$
   (B) $S \geq 1 - \frac { 1 } { e }$
   (C) $S \leq \frac { 1 } { 4 } \left( 1 + \frac { 1 } { \sqrt { e } } \right)$
   (D) $S \leq \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { e } } \left( 1 - \frac { 1 } { \sqrt { 2 } } \right)$

ANSWER : ABD

1. A ship is fitted with three engines $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$. The engines function independently of each other with respective probabilities $\frac { 1 } { 2 } , \frac { 1 } { 4 }$ and $\frac { 1 } { 4 }$. For the ship to be operational at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$ denote respectively the events that the engines $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$ are functioning. Which of the following is (are) true ?
   (A) $P \left[ X _ { 1 } { } ^ { c } \mid X \right] = \frac { 3 } { 16 }$
   (B) $P$ [Exactly two engines of the ship are functioning $\mid X ] = \frac { 7 } { 8 }$
   (C) $P \left[ X \mid X _ { 2 } \right] = \frac { 5 } { 16 }$
   (D) $P \left[ X \mid X _ { 1 } \right] = \frac { 7 } { 16 }$

ANSWER : BD

1. Tangents are drawn to the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$, parallel to the straight line $2 x - y = 1$. The points of contact of the tangents on the hyperbola are
   (A) $\left( \frac { 9 } { 2 \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
   (B) $\left( - \frac { 9 } { 2 \sqrt { 2 } } , - \frac { 1 } { \sqrt { 2 } } \right)$
   (C) $( 3 \sqrt { 3 } , - 2 \sqrt { 2 } )$
   (D) $( - 3 \sqrt { 3 } , 2 \sqrt { 2 } )$

ANSWER : AB

1. If $y ( x )$ satisfies the differential equation $y ^ { \prime } - y \tan x = 2 x \sec x$ and $y ( 0 ) = 0$, then
   (A) $y \left( \frac { \pi } { 4 } \right) = \frac { \pi ^ { 2 } } { 8 \sqrt { 2 } }$
   (B) $y ^ { \prime } \left( \frac { \pi } { 4 } \right) = \frac { \pi ^ { 2 } } { 18 }$
   (C) $y \left( \frac { \pi } { 3 } \right) = \frac { \pi ^ { 2 } } { 9 }$
   (D) $y ^ { \prime } \left( \frac { \pi } { 3 } \right) = \frac { 4 \pi } { 3 } + \frac { 2 \pi ^ { 2 } } { 3 \sqrt { 3 } }$

ANSWER : AD

SECTION III : Integer Answer Type

This section contains $\mathbf { 5 }$ questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive). 56. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as $f ( x ) = | x | + \left| x ^ { 2 } - 1 \right|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

ANSWER : 5

1. The value of $6 + \log _ { \frac { 3 } { 2 } } \left( \frac { 1 } { 3 \sqrt { 2 } } \sqrt { 4 - \frac { 1 } { 3 \sqrt { 2 } } \sqrt { 4 - \frac { 1 } { 3 \sqrt { 2 } } \sqrt { 4 - \frac { 1 } { 3 \sqrt { 2 } } \ldots } } } \right)$ is

ANSWER : 4

1. Let $p ( x )$ be a real polynomial of least degree which has a local maximum at $x = 1$ and a local minimum at $x = 3$. If $p ( 1 ) = 6$ and $p ( 3 ) = 2$, then $p ^ { \prime } ( 0 )$ is

ANSWER : 9

1. If $\vec { a } , \vec { b }$ and $\vec { c }$ are unit vectors satisfying $| \vec { a } - \vec { b } | ^ { 2 } + | \vec { b } - \vec { c } | ^ { 2 } + | \vec { c } - \vec { a } | ^ { 2 } = 9$, then $| 2 \vec { a } + 5 \vec { b } + 5 \vec { c } |$ is

ANSWER : 3

1. Let $S$ be the focus of the parabola $y ^ { 2 } = 8 x$ and let $P Q$ be the common chord of the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y = 0$ and the given parabola. The area of the triangle $P Q S$ is

ANSWER : 4

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k - 20 =$

Six cards and six envelopes are numbered $1,2,3,4,5,6$ and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
(A) 264
(B) 265
(C) 53
(D) 67

Let $n$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac { m } { n }$ is

Words of length 10 are formed using the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}, \mathrm{H}, \mathrm{I}, \mathrm{J}$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9x} =$

The number of 5 digit numbers which are divisible by 4, with digits from the set $\{ 1,2,3,4,5 \}$ and the repetition of digits is allowed, is $\_\_\_\_$.

Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac { 1 } { 5 ! } ( \beta - \alpha )$ is $\_\_\_\_$ .