28. A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m . The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N . The maximum possible value of angular velocity of ball (in radian/s) is [Figure]
(A) 9
(B) 18
(C) 27
(D) 36

ANSWER: D

1. A meter bridge is set-up as shown, to determine an unknown resistance ' $X$ ' using a standard 10 ohm resistor. The galvanometer shows null point when tapping-key is at 52 cm mark. The end-corrections are 1 cm and 2 cm respectively for the ends A and B . The determined value of ' $X$ ' is [Figure]
   (A) 10.2 ohm
   (B) 10.6 ohm
   (C) 10.8 ohm
   (D) 11.1 ohm

ANSWER:B

1. A $2 \mu \mathrm {~F}$ capacitor is charged as shown in figure. The percentage of its stored energy dissipated after the switch $S$ is turned to position 2 is [Figure]
   (A) $0 \%$
   (B) $20 \%$
   (C) $75 \%$
   (D) $80 \%$

ANSWER: D

SECTION - II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE may be correct.

35. The phase space diagram for a ball thrown vertically up from ground is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]

ANSWER: D

1. The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $\mathrm { E } _ { 1 }$ and $\mathrm { E } _ { 2 }$ are the total mechanical energies respectively. Then
   (A) $\quad E _ { 1 } = \sqrt { 2 } E _ { 2 }$
   (B) $E _ { 1 } = 2 E _ { 2 }$
   (C) $E _ { 1 } = 4 E _ { 2 }$
   (D) $E _ { 1 } = 16 E _ { 2 }$ [Figure]

ANSWER: C

1. Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]

ANSWER: B

Paragraph for Question Nos. 38 and 39

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let ' $N$ ' be the number density of free electrons, each of mass ' $m$ '. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ' $\omega _ { \mathrm { p } }$ ', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega _ { \mathrm { p } }$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

38. Taking the electronic charge as ' $e$ ' and the permittivity as ' $\varepsilon _ { 0 }$ ', use dimensional analysis to determine the correct expression for $\omega _ { p }$.
(A) $\sqrt { \frac { N e } { m \varepsilon _ { 0 } } }$
(B) $\sqrt { \frac { m \varepsilon _ { 0 } } { N e } }$
(C) $\sqrt { \frac { N e ^ { 2 } } { m \varepsilon _ { 0 } } }$
(D) $\sqrt { \frac { m \varepsilon _ { 0 } } { N e ^ { 2 } } }$

ANSWER: C

1. Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $\mathrm { N } \approx 4 \times 10 ^ { 27 } \mathrm {~m} ^ { - 3 }$. Take $\varepsilon _ { 0 } \approx 10 ^ { - 11 }$ and $\mathrm { m } \approx 10 ^ { - 30 }$, where these quantities are in proper SI units.
   (A) 800 nm
   (B) 600 nm
   (C) 300 nm
   (D) 200 nm

ANSWER: B

SECTION - IV (Total Marks : 28)

(Integer Answer Type)

This section contains $\mathbf { 7 }$ questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9 . The bubble corresponding to the correct answer is to be darkened in the ORS.

40. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of $0.3 \mathrm {~m} / \mathrm { s } ^ { 2 }$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is $( P / 10 )$. The value of $P$ is
[Figure]
ANSWER: 4

1. A block is moving on an inclined plane making an angle $45 ^ { \circ }$ with the horizontal and the coefficient of friction is $\mu$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define $\mathrm { N } = 10 \mu$, then N is

ANSWER: 5

47. Let $\left( x _ { 0 } , y _ { 0 } \right)$ be the solution of the following equations
$$\begin{aligned} ( 2 x ) ^ { \ln 2 } & = ( 3 y ) ^ { \ln 3 } \\ 3 ^ { \ln x } & = 2 ^ { \ln y } . \end{aligned}$$
Then $x _ { 0 }$ is
(A) $\frac { 1 } { 6 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 2 }$
(D) 6

ANSWER: C

1. The value of $\int _ { \sqrt { \ln 2 } } ^ { \sqrt { \ln 3 } } \frac { x \sin x ^ { 2 } } { \sin x ^ { 2 } + \sin \left( \ln 6 - x ^ { 2 } \right) } d x$ is
   (A) $\frac { 1 } { 4 } \ln \frac { 3 } { 2 }$
   (B) $\frac { 1 } { 2 } \ln \frac { 3 } { 2 }$
   (C) $\ln \frac { 3 } { 2 }$
   (D) $\frac { 1 } { 6 } \ln \frac { 3 } { 2 }$

ANSWER: A

49. Let $\vec { a } = \hat { i } + \hat { j } + \hat { k } , \vec { b } = \hat { i } - \hat { j } + \hat { k }$ and $\vec { c } = \hat { i } - \hat { j } - \hat { k }$ be three vectors. A vector $\vec { v }$ in the plane of $\vec { a }$ and $\vec { b }$, whose projection on $\vec { c }$ is $\frac { 1 } { \sqrt { 3 } }$, is given by
(A) $\hat { i } - 3 \hat { j } + 3 \hat { k }$
(B) $- 3 \hat { i } - 3 \hat { j } - \hat { k }$
(C) $3 \hat { i } - \hat { j } + 3 \hat { k }$
(D) $\quad \hat { i } + 3 \hat { j } - 3 \hat { k }$

ANSWER: C

1. Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt { 2 } \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt { 2 } \sin \theta \}$ be two sets. Then
   (A) $P \subset Q$ and $Q - P \neq \varnothing$
   (B) $Q \not \subset P$
   (C) $P \not \subset Q$
   (D) $P = Q$

ANSWER: D

1. Let the straight line $x = b$ divide the area enclosed by $y = ( 1 - x ) ^ { 2 } , y = 0$, and $x = 0$ into two parts $R _ { 1 } ( 0 \leq x \leq b )$ and $R _ { 2 } ( b \leq x \leq 1 )$ such that $R _ { 1 } - R _ { 2 } = \frac { 1 } { 4 }$. Then $b$ equals
   (A) $\frac { 3 } { 4 }$
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { 1 } { 3 }$
   (D) $\frac { 1 } { 4 }$

ANSWER:B

1. Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 6 x - 2 = 0$, with $\alpha > \beta$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n }$ for $n \geq 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } }$ is
   (A) 1
   (B) 2
   (C) 3
   (D) 4

ANSWER: C 53. A straight line $L$ through the point $( 3 , - 2 )$ is inclined at an angle $60 ^ { \circ }$ to the line $\sqrt { 3 } x + y = 1$. If $L$ also intersects the $x$-axis, then the equation of $L$ is
(A) $y + \sqrt { 3 } x + 2 - 3 \sqrt { 3 } = 0$
(B) $y - \sqrt { 3 } x + 2 + 3 \sqrt { 3 } = 0$
(C) $\sqrt { 3 } y - x + 3 + 2 \sqrt { 3 } = 0$
(D) $\sqrt { 3 } y + x - 3 + 2 \sqrt { 3 } = 0$
ANSWER:B

SECTION - II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE may be correct. 54. The vector(s) which is/are coplanar with vectors $\hat { i } + \hat { j } + 2 \hat { k }$ and $\hat { i } + 2 \hat { j } + \hat { k }$, and perpendicular to the vector $\hat { i } + \hat { j } + \hat { k }$ is/are
(A) $\hat { j } - \hat { k }$
(B) $- \hat { i } + \hat { j }$
(C) $\hat { i } - \hat { j }$
(D) $- \hat { j } + \hat { k }$

ANSWER: AD

1. Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$. If $P ^ { T }$ denotes the transpose of $P$, then $M ^ { 2 } N ^ { 2 } \left( M ^ { T } N \right) ^ { - 1 } \left( M N ^ { - 1 } \right) ^ { T }$ is equal to
   (A) $M ^ { 2 }$
   (B) $- N ^ { 2 }$
   (C) $- M ^ { 2 }$
   (D) $M N$

ANSWER : MARKS TO ALL

1. Let the eccentricity of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be reciprocal to that of the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$. If the hyperbola passes through a focus of the ellipse, then
   (A) the equation of the hyperbola is $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 2 } = 1$
   (B) a focus of the hyperbola is $( 2,0 )$
   (C) the eccentricity of the hyperbola is $\sqrt { \frac { 5 } { 3 } }$
   (D) the equation of the hyperbola is $x ^ { 2 } - 3 y ^ { 2 } = 3$

ANSWER: BD 57. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that
$$f ( x + y ) = f ( x ) + f ( y ) , \quad \forall x , y \in \mathbb { R } .$$
If $f ( x )$ is differentiable at $x = 0$, then
(A) $f ( x )$ is differentiable only in a finite interval containing zero
(B) $f ( x )$ is continuous $\forall x \in \mathbb { R }$
(C) $f ^ { \prime } ( x )$ is constant $\forall x \in \mathbb { R }$
(D) $f ( x )$ is differentiable except at finitely many points
ANSWER: BC, BCD

SECTION - III (Total Marks : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph $\mathbf { 2 }$ multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 58 to 60

Let $a , b$ and $c$ be three real numbers satisfying
$$\left[ \begin{array} { l l l } a & b & c \end{array} \right] \left[ \begin{array} { l l l } 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{array} \right] = \left[ \begin{array} { l l l } 0 & 0 & 0 \end{array} \right]$$

1. If the point $P ( a , b , c )$, with reference to (E), lies on the plane $2 x + y + z = 1$, then the value of $7 a + b + c$ is
   (A) 0
   (B) 12
   (C) 7
   (D) 6

ANSWER: D

1. Let $\omega$ be a solution of $x ^ { 3 } - 1 = 0$ with $\operatorname { Im } ( \omega ) > 0$. If $a = 2$ with $b$ and $c$ satisfying (E), then the value of

$$\frac { 3 } { \omega ^ { a } } + \frac { 1 } { \omega ^ { b } } + \frac { 3 } { \omega ^ { c } }$$
is equal to
(A) - 2
(B) 2
(C) 3
(D) - 3

ANSWER: A

1. Let $b = 6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x ^ { 2 } + b x + c = 0$, then

$$\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { \alpha } + \frac { 1 } { \beta } \right) ^ { n }$$
is
(A) 6
(B) 7
(C) $\frac { 6 } { 7 }$
(D) $\infty$

ANSWER: B

Paragraph for Question Nos. 61 and 62

Let $U _ { 1 }$ and $U _ { 2 }$ be two urns such that $U _ { 1 }$ contains 3 white and 2 red balls, and $U _ { 2 }$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U _ { 1 }$ and put into $U _ { 2 }$. However, if tail appears then 2 balls are drawn at random from $U _ { 1 }$ and put into $U _ { 2 }$. Now 1 ball is drawn at random from $U _ { 2 }$. 61. The probability of the drawn ball from $U _ { 2 }$ being white is
(A) $\frac { 13 } { 30 }$
(B) $\frac { 23 } { 30 }$
(C) $\frac { 19 } { 30 }$
(D) $\frac { 11 } { 30 }$

ANSWER: B

1. Given that the drawn ball from $U _ { 2 }$ is white, the probability that head appeared on the coin is
   (A) $\frac { 17 } { 23 }$
   (B) $\frac { 11 } { 23 }$
   (C) $\frac { 15 } { 23 }$
   (D) $\frac { 12 } { 23 }$

ANSWER: D

SECTION - IV (Total Marks : 28)

(Integer Answer Type)

This section contains $\mathbf { 7 }$ questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9 . The bubble corresponding to the correct answer is to be darkened in the ORS. 63. Consider the parabola $y ^ { 2 } = 8 x$. Let $\Delta _ { 1 }$ be the area of the triangle formed by the end points of its latus rectum and the point $P \left( \frac { 1 } { 2 } , 2 \right)$ on the parabola, and $\Delta _ { 2 }$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\frac { \Delta _ { 1 } } { \Delta _ { 2 } }$ is

ANSWER:2

1. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { 100 }$ be an arithmetic progression with $a _ { 1 } = 3$ and $S _ { p } = \sum _ { i = 1 } ^ { p } a _ { i } , 1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m = 5 n$. If $\frac { S _ { m } } { S _ { n } }$ does not depend on $n$, then $a _ { 2 }$ is ANSWER : 3, 9, 3 \& 9 BOTH
2. The positive integer value of $n > 3$ satisfying the equation

$$\frac { 1 } { \sin \left( \frac { \pi } { n } \right) } = \frac { 1 } { \sin \left( \frac { 2 \pi } { n } \right) } + \frac { 1 } { \sin \left( \frac { 3 \pi } { n } \right) }$$
is

ANSWER: 7

1. Let $f : [ 1 , \infty ) \rightarrow [ 2 , \infty )$ be a differentiable function such that $f ( 1 ) = 2$. If

$$6 \int _ { 1 } ^ { x } f ( t ) d t = 3 x f ( x ) - x ^ { 3 }$$
for all $x \geq 1$, then the value of $f ( 2 )$ is

ANSWER : MARKS TO ALL

1. If $z$ is any complex number satisfying $| z - 3 - 2 i | \leq 2$, then the minimum value of $| 2 z - 6 + 5 i |$ is

ANSWER: 5

1. The minimum value of the sum of real numbers $a ^ { - 5 } , a ^ { - 4 } , 3 a ^ { - 3 } , 1 , a ^ { 8 }$ and $a ^ { 10 }$ with $a > 0$ is

ANSWER: 8

1. Let $f ( \theta ) = \sin \left( \tan ^ { - 1 } \left( \frac { \sin \theta } { \sqrt { \cos 2 \theta } } \right) \right)$, where $- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }$. Then the value of

$$\frac { d } { d ( \tan \theta ) } ( f ( \theta ) )$$
is

ANSWER: 1