You are given a $4 \times 4$ chessboard, and asked to fill it with five $3 \times 1$ pieces and one $1 \times 1$ piece. Then, over all such fillings, the number of squares that can be occupied by the $1 \times 1$ piece is
(A) 4
(B) 8
(C) 12
(D) 16 .

A school allowed the students of a class to go to swim during the days March 11th to March 15, 2019. The minimum number of students the class should have had that ensures that at least two of them went to swim on the same set of dates is:
(A) 6
(B) 32
(C) 33
(D) 121 .

Three children and two adults want to cross a river using a rowing boat. The boat can carry no more than a single adult or, in case no adult is in the boat, a maximum of two children. The least number of times the boat needs to cross the river to transport all five people is:
(A) 9
(B) 11
(C) 13
(D) 15 .

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

Let $S_n$ be the set of all $n$-digit numbers whose digits are all 1 or 2 and there are no consecutive 2's. (Example: 112 is in $S_3$ but 221 is not in $S_3$). Then the number of elements in $S_{10}$ is
(A) 512
(B) 256
(C) 144
(D) 89

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

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, $\frac{y}{9x} =$

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student $S _ { 1 }$ gets the previously allotted seat $R _ { 1 }$, and NONE of the remaining students gets the seat previously allotted to him/her is
(A) $\frac { 3 } { 40 }$
(B) $\frac { 1 } { 8 }$
(C) $\frac { 7 } { 40 }$
(D) $\frac { 1 } { 5 }$

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. For $i = 1,2,3,4$, let $T _ { i }$ denote the event that the students $S _ { i }$ and $S _ { i + 1 }$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _ { 1 } \cap T _ { 2 } \cap T _ { 3 } \cap T _ { 4 }$ is
(A) $\frac { 1 } { 15 }$
(B) $\frac { 1 } { 10 }$
(C) $\frac { 7 } { 60 }$
(D) $\frac { 1 } { 5 }$

Five persons $A$, $B$, $C$, $D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is\_\_\_\_

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is $\_\_\_\_$

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$ is $\_\_\_\_$.

Let $X$ be the set of all five digit numbers formed using $1,2,2,2,4,4,0$. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of $38p$ is equal to

Let the set of all relations $R$ on the set $\{ a , b , c , d , e , f \}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal { S }$.
Then the number of elements in $\mathcal { S }$ is $\_\_\_\_$ .

Let $S$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $S$, but 0210222 is NOT in $S$.
Then the number of elements $x$ in $S$ such that at least one of the digits 0 and 1 appears exactly twice in $x$, is equal to $\_\_\_\_$ .

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is
(1) $6 ! 7 !$
(2) $( 6 ! ) ^ { 2 }$
(3) $( 7 ! ) ^ { 2 }$
(4) $7 !$